Welcome

My name is Parker Lamb - I'm an undergraduate Physics and Astronomy double-major at the University of Washington.

Here you'll find notes on courses I care to take them for and some projects I'm working on while here. If you have any questions for me email me at aethio@uw.edu.

Chapter 5 - Magnetostatics

Reference "Introduction to Electrodynamics" (5e) by David Griffiths.

Say instead of static electric charges (electrostatics), we start dealing with moving electric charges (electrodynamics).

Moving charges will generate a magnetic field around them. If the rate of moving charges (current $I$ ) is constant, we're working with magnetostatics. The $B$ -field for some current $I$ can be modeled by the right-hand rule:

Magnetic fields interact with one another. Using the Lorentz force law, $F_{mag} = Q (v \times B)$ Or, in the presence of an electric field as well, $F_{mag} = Q (E + v \times B)$

Todo: cyclotron motion?

Note: magnetic fields do no work. $d W_{mag} = F_{mag} \cdot d l = Q (v \times B) \cdot v d t = 0$ The objects that generate the $B$ -fields do work; but it's an apparently subtle distinction that the textbook does not expand on yet.

Current

Current is some charge per unit time through some cross-sectional area, defined such that $1 ampere = 1 Coulomb /1 second$ .

If we had some line charge of $λ$ Coulombs traveling at velocity $v$ , $I = λ v$ . In cases where we're not just along a straight line, $v$ is a vector and as such current is too: $I = λ v$ The force on this line of charge is $F_{mag} = I \int d l \times B$ for a constant current $I$ .

Surface and Volume Currents

If we have some charges flowing across some $d l$ on a surface, or across some area $d A$ on a volume, we represent them with surface current density $K$ and volume current density $J$ .

Similar to a line charge, if our surface has density $σ$ (or $ρ$ for a volume charge) and charges move at velocity $v$ : $K = σ v J = ρ v$ For some volume, the charge conservation equation (or continuity equation) says any charge flowing out of a volume means $\nabla \cdot J = - \frac{\partial ρ}{\partial t} = 0 (in magnetostatics)$

Note: $\partial ρ / \partial t$ means change in charge density per change in time, meaning $δ J$ . This is zero in magnetostatics.

Biot-Savart Law

The magnetic field of some steady-state line current is

$B (r) = \frac{μ _{0}}{4 π} I \int \frac{d l ^{'} \times R ^}{R ^{2}}$

Note: $B$ has units of newtons per ampere-meter (or Tesla): $1 T = 1 N/A \cdot m$ .

where $r$ is the "test point" where you'd like to know the magnetic field, $r^{'}$ points to some (temporary) infinitesimal charge line element $d l^{'}$ , and $R$ points from the charge element to the test point.

Note: the integration is along the current path.

$μ_{0}$ is the "permeability of free space": $μ_{0} = 4 π \times 1 0^{- 7} N/A^{2}$

The B.S. law also works for surface and volume charges: $B_{surf} (r) = \frac{μ _{0}}{4 π} K \int \frac{d A ^{'} \times R ^}{R ^{2}} B_{vol} (r) = \frac{μ _{0}}{4 π} J \int \frac{d τ ^{'} \times R ^}{R ^{2}}$ where $d A^{'} = d x^{'} d y^{'}$ and $d τ^{'} = d x^{'} d y^{'} d z^{'}$ , converting to polar or spherical as necessary.

Divergence and Curl of $B$

The curl of any magnetic field is proportional to the current density (or contained current): $\nabla \times B = μ_{0} J$ where $J$ is a volume current density. It's also related to total current $I$ by taking a surface integral bounding the volume: $I_{enc} = \int J \cdot d A$

This means stronger currents have higher-magnitude $B$ -fields, and stronger $B$ -fields enclose a higher current / current density within.

The divergence of a magnetic field is always zero. $\nabla \cdot B = 0$

A magnetic field will circle around a wire but will not expand outward.

Ampère's Law

For some circular magnetic field (circumference $2 π r$ ) around a wire, the path integral of it is independent of radius: $\oint B \cdot d l = \frac{μ _{0} I}{2 π} \int_{0}^{2 π} d ϕ = \frac{μ _{0} I}{2 π} 2 π = μ_{0} I$

The radius of the path integral (circle, circ. $2 π r$ ) increases at the same rate as the magnitude of the $B$ -field decreases - i.e. it's independent of radius.

This is known as Ampère's Law: $\oint B \cdot d l = μ_{0} I_{enc}$ The current enclosed by some path integral along the $B$ -field is proportional to the enclosed current $I_{enc}$ . It's like the magnetostatics equivalent to Gauss's law as it relates to Coulomb's law.

For a surface current along an infinite plane, the Ampèrean loop might be a rectangle perpendicular to the current.

Above, the $\overset{z}{^}$ components of the Amp. loop aren't aligned with $B$ , so we only care about the $\overset{y}{^}$ components - i.e. $\oint B \cdot d l = B \oint d l B (2 l) B = μ_{0} I_{enc} = μ_{0} K l = \frac{μ _{0} K}{2} \overset{y}{^} (for z > 0, negative if z < 0)$

Note that Ampère's law only works for

Infinite straight lines / cylinders
Infinite planes
Infinite solenoids
Toroids (see Griffiths Ex. 5.10)

Magnetic Vector Potential $A$

Magnetic vector potential $A$ is the magnetostatics equivalent to electric potential $V$ .

$\nabla \cdot B = 0$ . If we instead write this as $\nabla \cdot (\nabla \times A) = 0$ , $B = \nabla \times A$ Griffiths has a further derivation of this in 5.4.1. Ultimately, we get to the magnetic Poisson's equation equivalent: $\nabla^{2} A = - μ_{0} J$ $A (r) = \frac{μ _{0}}{4 π} \int \frac{J ( r ^{'} )}{R} d τ^{'}$ We can pull out $J$ if constant. Use $K (r^{'}) d a^{'}$ or $I d l^{'}$ for surface and volume currents respectively.

The direction of $A$ will almost always be the same as that of current.

Dipole Moment of $A$

The explicit multipole expansion of $A$ is in Griffiths 5.4.3. I've only included the dipole moment for brevity, and because the monopole moment has no proof of existence & higher-order terms are rarely helpful.

The dipole moment tends to dominate magnetic vector potential multipole expansions (no monopole given $\nabla \cdot B = 0$ ), and is represented as $A_{dip} (r) = \frac{μ _{0}}{4 π} \frac{m \times r ^}{r ^{2}}$ where $m$ is the magnetic dipole moment $m = I \int d a = I a (a is vector area of Amp \overset{e}{ˋ} ran loop)$ this is independent of origin, since $a$ is centered at $r = 0$ .

Chapter 6 - Magnetic Fields in Matter

Reference "Introduction to Electrodynamics" (5e) by David Griffiths.

If a material is placed in a $B$ -field, it will acquire a magnetic polarization (or magnetization). Different materials acquire different polarizations depending on their atomic structures.

Paramagnets: magnetization parallel to applied $B$ , materials with odd # of electrons.
Diamagnets: magnetization antiparallel to applied $B$ , materials with an even # of electrons.
Ferromagnets: magnetization persists on the material even after the applied $B$ -field is removed, and is determined by the whole "magnetic history" of the object.

All electrons act as magnetic dipoles.

Imagine a magnetic dipole as pointing from south to north (the Gilbert model). It's an inaccurate model at small scales according to Griffiths, but he recommends it for intuition.

Torques and Forces on Magnetic Dipoles

Magnetic dipoles will experience some torque $N$ in an applied field, where $N = m \times B$ and $m$ is the magnetic dipole moment. For paramagnetic materials (odd electrons), $m$ will be roughly in the same direction as the applied $B$ -field.

For a current loop, $m = I ab$ where $a, b$ are side lengths.

In a uniform field the net force on the dipole is zero, though this is not the case for nonuniform fields.

For an infinitesimal loop with dipole moment $m$ in field $B$ , the force on the loop is $F = \nabla (m \cdot B)$

Diamagnets

Diamagnetism affects all materials, but is much weaker than paramagnetism, so is most easily observed in materials with an even number of electrons.

When an external $B$ -field is applied to a material, individual electrons will speed up according to $Δ v = \frac{e RB}{2 m _{e}}$ where $R$ is the "radius" of the electron from the nucleus of an atom. This increase in orbital speed will change the dipole moment $Δ m = - \frac{1}{2} e (Δ v) (R) \overset{z}{^} = - \frac{e ^{2} R ^{2}}{4 m _{e}} B$ this change in the dipole moment is antiparallel to $B$ as shown above.

Magnetization

Magnetization $M$ is $M = magnetic dipole moment m per unit volume$ For a paramagnet, perhaps suspended above a solenoid, the magnetization would be positive/upward, and force downward. For a diamagnet, the magnetization would be instead downward, and force upward.

In general in a nonuniform field, paramagnets are attracted into the field, and diamagnets are repelled away.

Note: $M$ is an average over a wildly complex set of infinitesimal dipoles and "smooths out" the dipole into a macroscopic view.

Note: both diamagnetism and paramagnetism are quite weak compared to, for instance, ferromagnetism, and so are often neglected in experimental calculations.

Field of a Magnetized Object

For some single dipole, the magnetic vector potential is $A (r) = \frac{μ _{0}}{4 π} \frac{m \times R ^}{R ^{2}}$ For a magnetized object with magnetization $M$ , $A (r) = \frac{μ _{0}}{4 π} \int \frac{M ( r ^{'} ) \times R ^}{R}^{2} d τ^{'}$ Alternatively, we can look at the object in terms of its volume current density $J_{b}$ and surface current density $K_{b}$ , where $J_{b} = \nabla \times M K_{b} = M \times \overset{n}{^}$ $A (r) = \frac{μ _{0}}{4 π} \int_{V} \frac{J _{b} ( r ^{'} )}{R} d τ^{'} + \frac{μ _{0}}{4 π} \oint_{S} \frac{K _{b} ( r ^{'} )}{R} d A^{'}$

This means the potential (and therefore magnetic field) is the same as would be made by some volume current $J_{b}$ throughout the material plus the surface current $K_{b}$ on the boundary.

This means we needn't integrate all the infinitesimal dipoles, but rather just determine the bound currents $J_{b}$ and $K_{b}$ and find the field they produce.

Note: $\nabla \cdot J_{b} = 0$

Chapter 1 - Relevant Mathematics

Reference "Introduction to Electrodynamics" by David Griffiths.

Vectors

We start with a brief review of vector algebra - I'll skip most of it but keep the key things.

Dot product

The dot product is a measure of how "parallel" two vectors are, maximized when parallel, minimized (0) when perpendicular.

$a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = ∣ a ∣∣ b ∣ cos θ$

It is commutative ( $a \cdot b = b \cdot a$ ) and distributive ( $a \cdot (b + c) = a \cdot b + a \cdot c$ ). The result of the dot product is a scalar.

Cross product

The cross product yields a third vector orthogonal to both $a$ and $b$ , maximized in magnitude when $a$ and $b$ are themselves orthogonal to one another.

The cross product is distributive, but not exactly commutative - instead, $(b \times a) = - (a \times b)$ Some other properties:

The result of the cross product is a vector, not a scalar.
The magnitude of the cross product is the area of the parallelogram generated by $a$ and $b$ .
The cross product of a vector with itself is zero.

We can calculate a cross product like so:

$a \times b = \overset{x}{^} a_{1} b_{1} \overset{y}{^} a_{2} b_{2} \overset{z}{^} a_{3} b_{3} = ∣ a ∣∣ b ∣ sin θ \overset{n}{^}$ where $\overset{n}{^}$ is the normal to the plane formed by $a$ and $b$ .

Vector triple product

Triple products are combinations of cross and dot products.

Scalar triple products: the magnitude of this is the volume of the parallelepiped (3D parallelogram) generated by $a$ , $b$ and $c$ . $a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$
Vector triple products: there's no easy geometric interpretation of this, but is useful to reduce complex cross product calculations. It can be memorized by the mnemonic BAC-CAB. $a \times (b \times c) = b (a \cdot c) - c (a \cdot b)$

Spatial vectors

The position vector indicates the position of a point relative to the origin.

In electrodynamics, usually we'll have two points: one for the source charge and one for the test charge. The vector $R$ from one to the other is the useful quantity then.

The Del Operator

Del ( $\nabla$ ) is a vector operator that acts upon functions, and is defined as $\nabla = \frac{d}{d x} \overset{x}{^} + \frac{d}{d y} \overset{y}{^} + \frac{d}{d z} \overset{z}{^}$ $\nabla$ really only is meaningful when applied to functions, and has three (usual) ways of being applied:

To a scalar function ( $\nabla f$ ) to get the gradient of that function.
To a vector function via the dot product ( $\nabla \cdot a$ ) to get that function's divergence.
To a vector function via the cross product ( $\nabla \times a$ ) to get that function's curl.

Gradients

The gradient of some scalar function $f$ gives a vector result that will point in the direction of the max rate of increase of that function.

$\nabla f = \frac{df}{d x} \overset{x}{^} + \frac{df}{d y} \overset{y}{^} + \frac{df}{d z} \overset{z}{^}$ and results in a vector with the derivative of the original function's units.

Fundamental theorem for gradients

Important: The integral of a derivative (here, the gradient) is given by the value of the function at the boundaries $a$ and $b$ . $\int_{a}^{b} (\nabla T) \cdot d l = T (b) - T (a)$

Divergence

The divergence of some vector function $a$ represents how much the function "spreads out" or diverges from a given point. $\nabla \cdot a = \frac{d a _{x}}{d x} + \frac{d a _{y}}{d y} + \frac{d a _{z}}{d z}$ and results in a dimensionless scalar value indicating the rate of divergence from a point.

Green's theorem

Green's theorem is a special application of Stoke's theorem (see below). Geometrically, it relates the sources contained in some surface to the flux emitted by those sources around the boundary of a surface. $\int_{V} (\nabla \cdot v) d τ = \oint_{S} v \cdot d a$ In the case of electrostatics, this might be the flux field generated by some charges in a volume to the total, superposition flux coming out of that entire region - turning one "field of charges" into a single charge.

The integral of a derivative over some region will always be equal to the value of the function at the boundary ( $P$ ) - in this case, the boundary term is an integral (a simple line may just have two endpoints, but a line bounding a volume forms a closed sufrface).

Curl

The curl of some vector function $a$ represents how much the function "swirls around" some given point. Positive curl is given by the right-hand rule (normally counterclockwise). $\nabla \times a = \overset{x}{^} (\frac{d a _{z}}{d y} - \frac{d a _{y}}{d z}) + \overset{y}{^} (\frac{d a _{x}}{d z} - \frac{d a _{z}}{d x}) + \overset{z}{^} (\frac{d a _{y}}{d x} - \frac{d a _{x}}{d y})$

Stokes' Theorem

The integral of the curl over some surface is the "total amount of swirl" on that surface - and mathematically can be distilled into just finding how much the flow is following the boundary of that surface. $\int_{S} (\nabla \times v) \cdot d a = \oint_{P} v \cdot d l$ This last quantity, $\oint_{P} v \cdot d l$ is sometimes called the "circulation" of some vector field $v$ .

A Note on Del Squared ( $\nabla^{2}$ )

Enumerating over the ways we can apply $\nabla$ twice:

Laplacian: the cross product of $\nabla$ with the gradient of a function results in the Laplacian, such that $\nabla \cdot (\nabla f) = \nabla^{2} f = \frac{d ^{2} f}{d x ^{2}} + \frac{d ^{2} f}{d y ^{2}} + \frac{d ^{2} z}{d z ^{2}}$
Curl of gradient: always zero. $\nabla \times (\nabla f) = 0$
Gradient of divergence: not a lot of physical applications, calculable though. $\nabla (\nabla \cdot a)$
Divergence of curl: always zero. $\nabla \cdot (\nabla \times v) = 0$
Curl of curl: most easily defined as the acceleration of the swirl, like a hurricane speeding up. $\nabla \times (\nabla \times a) = \nabla (\nabla \cdot a) - \nabla^{2} a$

Integrals

In electrodynamics, we have line (path) integrals, surface integrals (or flux) and volume integrals.

Line integrals

Iterate over infinitesimally small displacement vectors $d l$ . $\int_{a}^{b} a \cdot d l$ If $a = b$ , the line integral iterates over a closed loop and we can write $\oint a \cdot d l$

In elementary physics, a common example is work done by a force. $W = \int F \cdot d l$

While path taken normally matters (distance), there are some vectors where only displacement matters (for forces, these are called conservative forces, like gravity).

Wikipedia has a really cool animation of line integrals:

Surface integrals

Surface integrals iterate over infinitesimally small areas $d a$ , with norms perpendicular to the surface. $\int_{S} v \cdot d a$ In this case, if the surface is closed (i.e. like a sphere, rather than a hill), then we can write it in closed loop form. $\oint v \cdot d a$

Surface integrals are really useful for things like flux where something is moving through some area - i.e. imagine radio waves passing through a curved gas cloud or radiation falling "through" a planet.

Volume integrals

Volume integrals integrate over infinitesimally small volumes $d τ = d x d y d z$ .

$\int a d τ = \int (a_{x} \overset{x}{^} + a_{y} \overset{y}{^} + a_{z} \overset{z}{^}) d τ$

If some function $f (x, y, z)$ represented density of a substance, then the volume integral over it would be total mass.

Integration by Parts

Also known as the "inverse product rule". $\int u d v = uv - \int v d u$

Curvilinear Coordinates

Taking another look at spherical and cylindrical coordinates, as well as how to convert between them.

Spherical coordinates

Three terms:

$r$ : the distance from the origin (range 0 to $\infty$ )
$θ$ : the polar angle from the $z$ -axis (range 0 to $π$ )
$ϕ$ : the azimuthal angle along the $x y$ -axis (range 0 to $2 π$ )

$x y z = r sin θ cos ϕ = r sin ϕ sin θ = r cos θ$ Alternatively in terms of the unit vectors:

$\overset{x}{^} \overset{y}{^} \overset{z}{^} = sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ cos θ sin ϕ - sin ϕ - sin θ cos θ 0 \overset{r}{^} \hat{θ} \hat{ϕ}$

The interesting thing about the matrix form is that it is orthogonal - according to the invertible matrix theorem, $Q^{- 1} = Q^{T}$ for orthogonal matrices.

We can use this to solve for $\overset{r}{^}$ , $\hat{θ}$ and $\hat{ϕ}$ easily then by just taking the transpose of the above matrix.

$\overset{r}{^} \hat{θ} \hat{ϕ} = sin θ cos ϕ cos θ cos ϕ - sin θ sin θ sin ϕ cos θ sin ϕ cos ϕ cos θ - sin θ 0 \overset{x}{^} \overset{y}{^} \overset{z}{^}$ Our line element in spherical coordinates is

$d l = d r \overset{r}{^} + r d θ \hat{θ} + r sin θ d ϕ \hat{ϕ}$

In terms of triple integrals, the textbook has a good graphic to represent the three displacements $d l_{r}, d l_{θ}, d l_{ϕ}$ .

And a infinitesimal volume element is the product of these three:

$d τ = d l_{r} d l_{θ} d l_{ϕ} = r^{2} sin θ d r d θ d ϕ$

$r$ has a possible range from 0 to $\infty$ , $θ$ from 0 to $π$ and $ϕ$ from 0 to $2 π$ .

The textbook represents the spherical representations of the gradient, divergence, curl and Laplacian in eqs. 1.70-1.73.

Cylindrical coordinates

Three terms:

$s$ : distance from $z$ -axis
$ϕ$ : azimuthal angle along $x y$ -axis
$z$ : height along z-axis

$x y z = s cos ϕ = s sin ϕ = z$ with unit vectors $\overset{x}{^} \overset{y}{^} \overset{z}{^} = cos ϕ sin ϕ 0 - sin ϕ cos ϕ 0 001 \overset{s}{^} \hat{ϕ} \overset{z}{^} \overset{s}{^} \hat{ϕ} \overset{z}{^} = cos ϕ - sin ϕ 0 sin ϕ cos ϕ 0 001 \overset{x}{^} \overset{y}{^} \overset{z}{^}$

For triple integrals, the infinitesimal displacements are $d l_{s} = d s, d l_{ϕ} = s d ϕ, d l_{z} = d z$ our volume element is $d τ = s d s d ϕ d z$ and line displacement is $d l = d s \overset{s}{^} + s d ϕ \hat{ϕ} + d z \overset{z}{^}$

Dirac Delta Function

Imagine we have an infinite point mass at the origin, and it is the only thing in the universe.

The total mass of the universe must be just the mass of the point charge; but the point charge exists only at $(0, 0, 0)$ , and is a point charge - it has no spatial dimension. If we were to integrate over the entire universe, we'd find we have just the mass of the point charge - and can mathematically represent that with the Dirac delta function.

$δ (x) = {0 \infty if x \neq = 0 if x = 0$ and $\int_{- \infty}^{\infty} δ (x) d x = 1$ Therefore, if we have some continuous function $f (x)$ (could be a constant or a function, anything attached to $δ (x)$ ). $f (x) δ (x) = {0 f (0) if x \neq = 0 if x = 0$ which brings us to the neat and nice identity: $\int_{- \infty}^{\infty} f (x) δ (x - a) d x = f (a)$

Dirac Delta in 2D, 3D

$δ^{3} (r) = δ (x) δ (y) δ (z)$

And, unsurprisingly, $\int_{all space} f (r) δ^{3} (r - a) d τ = f (a)$

Note on $R$ : with $R = r - r^{'}$ (the separation vector), $\nabla \cdot (\frac{R}{R ^{2}}) = 4 π δ^{3} (R)$ and $\nabla^{2} \frac{1}{R} = - 4 π δ^{3} (R)$ Reference the textbook equations 1.100-1.102 for derivation.

Chapter 2 - Electrostatics

Reference "Introduction to Electrodynamics" by David Griffiths.

Fundamentally, electrodynamics seeks to model the interaction of some set of charges $q_{1}, q_{2}, q_{3}$ on some other charge, $Q$ . We can do this via superposition, which states the interaction between two charges is unaffected by the presence of others - i.e. $F_{1}$ , the force between $q_{1}$ and $Q$ , is independent of the presence of $q_{2}$ and $q_{3}$ . We can then sum up the forces to get our net force on $Q$ : $F_{net} = F_{1} + F_{2} + F_{3}$ In an electrostatic system (no moving charges as opposed to an electrodynamic system), the forces can be calculated via Coulomb's law.

Coulomb's Law

Electrostatics simply takes into account distance and charge strength. Coulomb's law is

$F = \frac{1}{4 π ϵ _{0}} \frac{qQ}{R ^{2}} \hat{R}$

$ϵ_{0}$ is the permittivity of free space ( $ϵ_{0} = 8.85 \times 1 0^{- 12}$ $C^{2} / N \cdot m^{2})$ and $R = r - r^{'}$ (Griffith's script-r).

in electrostatics, force increases with higher-magnitude charges, and decreases with distance.

If we have several charges, we just sum their individual Coulomb forces as calculated above:

$F = F_{1} + F_{2} + \dots = \frac{Q}{4 π ϵ _{0}} (\frac{q _{1}}{R _{1} ^{2}} \hat{R_{1}} + \frac{q _{2}}{R _{2} ^{2}} \hat{R_{2}} + \dots)$ In shorthand, $F = Q E$ where $E$ , the electric field, is $E = \frac{1}{4 π ϵ _{0}} i = 1 \sum n \frac{q _{i}}{R _{i} ^{2}} \hat{R_{i}}$ where $ϵ_{0} = 8.85 \times 1 0^{- 12} \frac{C ^{2}}{N \cdot m}$ .

The electric field is the force per unit charge - this force can only be imparted by the presence of other charges.

Continuous Charge Distributions

If we have a set of charges $q_{i}$ , then our electric field is generated from a discrete charge distribution. If, however, the charge is distributed continuously over some region, then we have a continuous charge distribution generating the $E$ -field.

The electric field calculated as an integral over that continuous distribution: $E (r) = \frac{1}{4 π ϵ _{0}} \int \frac{1}{R ^{2}} \hat{R} d q$ where $d q$ varies for our continuous distribution type (1D, 2D or 3D), such that:

Line: $d q = λ (r^{'}) d l^{'}$
Area: $d q = σ (r^{'}) d a^{'}$
Volume: $d q = ρ (r^{'}) d τ^{'}$

Divergence and Curl of Electrostatic Fields

Griffiths 2.2.

Field lines are a way to visualize electric fields on a 2D or 3D medium.

Source.

where the density of field lines indicates the strength of the E-field at that location (strongest near the center in the above image). The flux of $E$ through some surface $S$ is $ϕ_{E} = \int_{S} E \cdot d a$ and is a way to measure the total charge contained within a closed surface, while those outside of the surface don't affect it.

This mathematically represented by Gauss's law, which states $\oint E \cdot d a = \frac{1}{ϵ _{0}} Q_{enc}$ There's also a differential version of Gauss's law (found by applying the Divergence Theorem) where $ρ$ is charge density: $\nabla \cdot E = \frac{1}{ϵ _{0}} ρ$

For example: in 3D, if a point charge $q$ is at the origin then the flux of $E$ through a sphere around it is $\oint E \cdot d a = \int \frac{1}{4 π ϵ _{0}} (\frac{1}{r ^{2}} \overset{r}{^}) \cdot (r^{2} sin θ d θ d ϕ \overset{r}{^}) = \frac{1}{ϵ _{0}} q$

We're only evaluating at the surface of the sphere, so $r$ is constant.

Note that Gauss's law is only useful with objects where the E-field is pointing in the same direction as elements $d a$ . This requires both a symmetrical charge distribution within the object, and for the Gaussian surface to be symmetric according to one of the following:

Spherically symmetric (concentric spheres)
Cylindrically symmetric (coaxial cylinders)
Plane symmetric (like a "box" that is cut in half by the plane)

The associated coordinate system is used for each.

Note: flux from external charges is ignored, since the flux entering a Gaussian surface from an external charge will be equal to the flux leaving that surface from the other side - hence, net flux from those external charges is zero.

If the object we're evaluating doesn't have perpendicular surface elements (i.e. a charge at the north pole of a sphere instead of the center), then some problems can be approached by creating another Gaussian element centered over the charge which subtends the same angle, though this can get geometrically complicated.

Divergence of $E$

$\nabla \cdot E = \frac{1}{4 π ϵ _{0}} \int \nabla \cdot (\frac{R ^}{R ^{2}}) ρ (r^{'}) d τ^{'}$ Since $\nabla \cdot (\frac{R ^}{R ^{2}}) = 4 π δ^{3} (R)$ , then this just becomes $\nabla \cdot E = \frac{1}{4 π ϵ _{0}} \int 4 π δ^{3} (r - r^{'}) ρ (r^{'}) d τ^{'} = \frac{1}{ϵ _{0}} ρ (r)$ Or, in shortform, $\nabla \cdot E = \frac{1}{ϵ _{0}} ρ (r)$

Curl of $E$

For any electrostatic charge distribution, the curl of the field is always zero. $\nabla \times E = 0$

Electric Potential $V$

Griffiths 2.3.

Electric potential $V$ is a good indicator of the "strength" of an electric field between two points, and is defined as $V (r) = - \int_{O}^{r} E \cdot d l$ with units $J / C$ .

A "natural" origin $O$ is a point infinitely far from the charge, though only when the charge distribution itself doesn't also extend to infinity.

in differential form, it's the reverse: $E = - \nabla V$ Voltage is path-independent and only displacements matter - hence the electric potential between points $a$ and $b$ is $V (b) - V (a) = - \int_{a}^{b} E \cdot d l$

Poisson & Laplace Equations

Poisson's equation says $\nabla^{2} V = - \frac{ρ}{ϵ _{0}}$ If there is no charge ( $ρ = 0$ ), Poisson's equation turns into Laplace's equation: $\nabla^{2} V = 0$

Voltage equations

For a point charge $q$ $V (r) = \frac{1}{4 π ϵ _{0}} \frac{q}{R}$

$r$ is the reference point from origin, $R$ from the point charge to the reference point $r$ .

For a set of point charges, we can use the principle of superposition: $V (x, y, z) = \frac{1}{4 π ϵ _{0}} [\frac{q}{x ^{2} + y ^{2} + ( z - d ) ^{2}} - \frac{q}{x ^{2} + y ^{2} + ( z + d ) ^{2}}]$

For a continuous distribution, $V (r) = \frac{1}{4 π ϵ _{0}} \int \frac{1}{R} d q$ and for a volume (to compute $V$ when we know $ρ$ ): $V (r) = \frac{1}{4 π ϵ _{0}} \int \frac{ρ ( r ^{'} )}{R} d τ^{'}$

$r^{'}$ from origin to charge.

Note: this is similar to the formula for electric field, but is missing $\hat{R}$ - $V$ is a scalar quantity, directionless.

Summary

The three "fundamentals" of electrostatics are $ρ$ , $E$ and $V$ , and are related by the following:

Griffith's Fig. 2.35.

Work and Energy

Griffith's 2.4.

For a conservative force, work is force times displacement. In electrostatics, our equivalent version is $Δ V = \frac{W}{Q} W = Q Δ V$

More extended, $W = \int_{a}^{b} F \cdot d l = - Q \int_{a}^{b} E \cdot d l = Q [V (b) - V (a)]$

where work here is the work it takes, per unit charge, to carry a charge from $a$ to $b$ . If our reference point $a = \infty$ and we let $b = r$ , then $W = Q [V (r) - V (\infty)] \equiv Q V (r)$

Work in a Point Charge Distribution

Imagine we had some charge $q_{1}$ - then brought in a new charge $q_{2}$ to join $q_{1}$ . Our work done would be

It would cost $W_{2} = q_{2} V_{1} (r_{2}) = \frac{1}{4 π ϵ _{0}} q_{2} (\frac{q _{1}}{r _{12}})$ Then, to bring in a third charge $q_{3}$ would act against the superposition of $q_{1}$ and $q_{2}$ : $W_{3} = \frac{1}{4 π ϵ _{0}} q_{3} (\frac{q _{1}}{r _{13}} + \frac{q _{2}}{r _{23}})$ ... etc. To assemble $n$ charges, we'd need a work of

$W = \frac{1}{2} i = 1 \sum n q_{i} j \neq = i \sum n \frac{1}{4 π ϵ _{0}} \frac{q _{j}}{r _{ij}} = \frac{1}{2} i = 1 \sum n q_{i} V (r_{i})$

Work in a Continuous Charge Distribution

For a continuous charge distribution, we use our last equation for our point charge distribution system and replace the sum $\sum$ with an integral $\int$ . $W = \frac{1}{2} \int ρ V d τ$ With a bit of mathematics, this is equivalent to $W = \frac{ϵ _{0}}{2} \int E^{2} \overset{ˊ}{d} τ$

Conductors

Griffith's 2.5.

Conductors are materials in which electrons are free to roam (like cows, on an open ranch 👍). In contrast, insulators have electrons pretty much immobile and packed-together (like Monsanto farms 👎).

We can approximate metals as ideal-case conductors (though perfect conductors don't yet exist, though we're gradually coming closer) with the following attributes:

$E = 0$ inside a conductor - or, the induced field will cancel the external field. If an external field $E_{0}$ is applied to a conductor, free electrons will move toward the E-field until they all sit on the surface, creating a deficit of charges on the opposite surface and positively charging it. Crucially, the net E-field is zero, so the fields must cancel inside: $E_{1} + E_{0} = 0 \to E_{1} = - E_{0}$ Charge will continue to flow until the cancellation is complete. Outside the conductor, $E \neq = 0$ , since the two fields don't tend to cancel.
Charge density is zero $(ρ = 0)$ : Since Gauss's law says $\nabla \cdot E = ρ / ϵ_{0}$ , zero E-field means zero charge density in the conductor.
Net charge resides on surface: positive and negative charges will only sit on the surface after enough time passes.
$E$ is perpendicular to the surface: somewhat obvious, but bears mentioning.

Any dynamical system will try to minimize potential energy - the charges residing on the surface are an extension of this. It might take some time, but will eventually happen.

Induced Charges

If we hold a charge $+ q$ near a conductor, the conductor will move toward the charge - this is because negative charges will accumulate closer to $+ q$ than the "effective" $+$ charges on the far side. Force falls off by $r^{2}$ , so the conductor will be attracted to the $+ q$ charge.

Cavities

Let's say we have a cavity inside our conductive surface. There are two scenarios here:

Empty cavity: If the cavity has no charge, the field within the cavity is zero, regardless of the external fields applied. This is the principle of the Faraday cage.

Non-empty cavity: The charge contained by the cavity will induce an opposite charge $- q$ uniformly distributed on the walls of the cavity - the only information transmitted to an external observer is the distribution of charge on the exterior wall (i.e. the magnitude of the internal E-field, or the amount of net charge contained).

Surface Charge and Force on a Conductor

Through the field inside a conductor is zero, the field immediately outside is $E = - \frac{σ}{ϵ _{0}} \overset{n}{^} where σ = - ϵ_{0} \frac{d V}{d n}$

If we only care about the magnitude of the E-field, then $E = - 1/ ϵ_{0} σ$ .

However, the electric field is discontinuous at a surface charge, so when calculating the force per unit area (or pressure) of an E-field at the surface of a conductor, we average the E-fields above and below the surface, such that $\frac{F}{A} = \frac{1}{2} σ (E_{above} + E_{below}) = \frac{1}{2 ϵ _{0}} σ^{2} \overset{n}{^}$ This is the outward electrostatic pressure on the surface, tending to draw the conductor into a given field, regardless of the sign of $σ$ (squared away).

The pressure at this point, expressed in terms of the field just outside the surface, is $P = \frac{ϵ _{0}}{2} E^{2}$

Capacitors

Put two conductors beside one another, with equal and opposite uniform net charges $+ Q$ on one and $- Q$ on the other:

Since the charge density is uniform on each surface, $σ = Q / A$ (on the left plate) and the field between the two is $E = \frac{σ}{ϵ _{0}} = \frac{1}{ϵ _{0}} \frac{Q}{A}$ with a voltage of $V = \int E d l = E d = \frac{σ}{ϵ _{0}} d = \frac{1}{ϵ _{0}} \frac{Q}{A} d$

$V = E d$ in a uniform E-field.

We'll define a new term to represent the proportionality of the arrangement, capacitance: $C = \frac{Q}{V} \equiv \frac{Q}{E d} \equiv ϵ_{0} \frac{A}{d}$ with units of Farads $(F = \frac{C}{V})$ , usually expressed in $μ F (1 0^{- 6})$ or $p F (1 0^{- 12})$ . The work to go from one side to the other is $W = \frac{1}{2} C V^{2}$

Chapter 3 - Potentials

Reference "Introduction to Electrodynamics" by David Griffiths.

The goal of electrostatics is to find the electric field given some stationary, immobile charge distribution.

Coulomb's Law is the way we do this for simple charge configurations, but for more complex charge configurations it's often easier to work with potential $V$ . For areas of nonzero charge density $ρ \neq = 0$ such as point, surface or volume charges, we use Poisson's equation: $\nabla^{2} V = - \frac{1}{ϵ _{0}} ρ$ Outside of these charge regons (such as in regular space), this reduces to Laplace's equation: $\nabla^{2} V = 0$ $\frac{\partial ^{2} V}{\partial x ^{2}} + \frac{\partial ^{2} V}{\partial y ^{2}} + \frac{\partial ^{2} V}{\partial z ^{2}} = 0$ with solutions called harmonic functions.

Laplace's equation is fundamental to the study of electrostatics according to Griffiths.

Laplace's Equation

In three dimensions, visualization is challenging - but the same two properties apply, with the first this time being

The value of $V (x, y, z)$ is the average $V$ over a spherical surface of radius $R$ centered at that point, with $V (x, y, z) = \frac{1}{4 π R ^{2}} \oint_{sphere} V d a$
$V$ can have no local maxima or minima, with extreme values only permitted at the boundaries (i.e. surface of the sphere).

Point 2 is particularly relevant in each of these circumstances - the value of $V$ at some point is the average of the surrounding $V$ on some surrounding boundary.

Uniqueness Theorems

The proof that a proposed set of boundary conditions will suffice takes the form of a uniqueness theorem (alternatively, a criterion to determine whether a solution to the Laplace or Poisson equations is unique).

Uniqueness Theorem 1: the solution to Laplace's equation $\nabla^{2} V = 0$ in a volume $V$ is unique if $V$ is specified on a boundary surface $S$ enclosing the volume $V$ .

The surface does not have to be Gaussian - it can look totally eldritch and crazy and the theorem would hold.

Uniqueness Theorem 2: given a volume $V$ surrounded by conductors and containing some charge density $ρ$ , the electric field is uniquely determined if the total charge on each conductor is given.

Method of Images

Griffiths 3.2.

Say we have some charge $q$ a distance $d$ above an infinite grounded plane; what is the potential $V$ in the region above the plane? Our boundary conditions state

${V = 0 V \to 0 z = 0 far from the charge$ We could solve Poisson's equation for this region, but a much easier technique is to use the Method of Images. Wikipedia says on the subject:

The method of image charges is used in electrostatics to simply calculate or visualize the distribution of the electric field of a charge in the vicinity of a conducting surface.

It is based on the fact that the tangential component of the electrical field on the surface of a conductor is zero, and that an electric field E in some region is uniquely defined by its normal component over the surface that confines this region (the uniqueness theorem).

Start by removing the conductor, and placing an opposite charge $- q$ at $- d$ :

Then, the potential is easy to calculate: $V (x, y, z) = \frac{1}{4 π ϵ _{0}} [\frac{q}{x ^{2} + y ^{2} + ( z - d ) ^{2}} - \frac{q}{x ^{2} + y ^{2} + ( z + d ) ^{2}}]$ which obeys both of the boundary conditions of the original problem:

$V = 0$ when $z = 0$
$V \to 0$ for $x^{2} + y^{2} + z^{2} ≫ d$

Thus by uniqueness theorem 1, is the solution to our original problem.

Uniqueness theorem 1 means that if a solution satisfies Poisson's equation in the region of interest and assumes the correct value at the boundaries, it must be right.

The Method of Images can be used in any scenario where we have a stationary charge distribution near a grounded conducting plane.

Induced Surface Charges

The surface charge density induced on a conductor is $σ = - ϵ_{0} \frac{\partial V}{\partial n}$ where $\partial V / \partial n$ is the normal derivative of $V$ at the surface. In the above case, this is is in the $z$ direction - if we take this partial derivative of our above calculated voltage, then $σ (x, y) = \frac{- q d}{2 π ( x ^{2} + y ^{2} + d ^{2} ) ^{3/2}}$

As expected from a positive charge, the induced surface charge is negative and greatest at $x = y = 0$ .

The total induced charge is $Q = \int σ d a = \int_{0}^{2 π} \int_{0}^{\infty} - \frac{q d}{2 π ( r ^{2} + d ^{2} ) ^{3/2}} r d r d ϕ = \frac{q d}{r ^{2} + d ^{2}}_{0}^{\infty} = - q$ Yes!

Force & Energy

Since the induced charge on the conductor is $- q$ and our charge is $+ q$ , it is attracted to the plane with a force given by Coulomb's Law: $F = - \frac{1}{4 π ϵ _{0}} \frac{q ^{2}}{( 2 d ) ^{2}} \overset{z}{^}$ While force is the same in our mirror problem, energy is not. For two point charges, $W = \frac{1}{2} \sum_{i = 1}^{n} q_{i} V (r_{i})$ , such that $W = - \frac{1}{4 π ϵ _{0}} \frac{q ^{2}}{2 d}$ But for a single charge and conducting plane (continuous charge distribution), energy is half of this. $W = - \frac{1}{4 π ϵ _{0}} \frac{q ^{2}}{4 d}$

The work to bring two point charges towards one another does work on both of them, while to bring a point charge toward a grounded conductor has us only doing work on one charge - only half the work is necessary.

Separation of Variables

Griffiths' 3.3

Separation of variables is a way to solve ODEs and PDEs by rewriting equations such that each of the two variables occur on different sides of the equation.

Separable equations must be able to be written in the form $\frac{d x}{d y} = g (x) h (y)$ We can rearrange the terms to get $d y \frac{1}{h ( y )} = g (x) d x$ integrate, $\int \frac{1}{h ( y )} d y = \int g (x) d x$ and add some constant term to one side to represent all our constants of integration. $H (y) = G (x) + C$

In the context of electrostatics, separation of variables is very useful when solving 2D Laplace equations, such as $\frac{\partial ^{2} V}{\partial x ^{2}} + \frac{\partial ^{2} V}{\partial y ^{2}} = 0$ We need solutions in the form of $V (x, y) = X (x) Y (y)$ This can be accomplished through some mathematical trickery to find our separated variables ... $\frac{1}{X} \frac{d ^{2} X}{d x ^{2}} + \frac{1}{Y} \frac{d ^{2} Y}{d y ^{2}} = 0$ ... which is of the form $f (x) + g (y) = 0$ . Thus, both $f (x)$ and $g (y)$ must be constant (we can't hold one constant and change the other with this solution still holding).

So, $\frac{1}{X} \frac{d ^{2} X}{d x ^{2}} = C_{1} and \frac{1}{Y} \frac{d ^{2} Y}{d y ^{2}} = C_{2} C_{1} + C_{2} = 0$ Converting each equation into an ODE, $\frac{d ^{2} X}{d x ^{2}} = k^{2} X \frac{d ^{2} Y}{d y ^{2}} = - k^{2} Y$ ... we converted a PDE into two ODEs, which are much easier to solve. Our solutions will be a constant coefficient set: $X (x) = c_{1} e^{k x} + c_{2} e^{- k x}, Y (y) = c_{3} sin (k y) + c_{4} cos (k y)$ We can find our constants based on our boundary conditions now.

Multipole Expansion

Griffiths 3.4

Say you have some charge $Q = Q_{net}$ that you can see in space, far away. $Q$ is almost like an exoplanet in a sense: we know it exists, we know its total mass, and maybe we know stuff like how far away it is.

But, we don't know what the surface looks like; what it's composed of, be it rock or ice or water. Just its net mass.

Similarly, our charge distribution might be crazy complicated, and it's net charge $Q_{net}$ only describes a tiny part of the story, albeit an important one - the multipole expansion is used to describe this fuller story, and is defined in terms of voltage as

$V (r) = \frac{1}{4 π ϵ _{0}} [\frac{1}{r} (monopole moment) + \frac{1}{r ^{2}} (dipole moment) + \frac{1}{r ^{3}} (quadrupole moment) \dots]$

Mathematically, it is $V (r) = \frac{1}{4 π ϵ _{0}} n = 0 \sum \infty \frac{1}{r ^{n + 1}} \int (r^{'})^{n} P_{n} (cos α) ρ (r^{'}) d τ^{'}$ where $α$ is the angle between $r$ and $r^{'}$ , $r$ is the reference point (from origin) and $r^{'}$ the charge (from origin).

Monopole and dipole terms

At large $r$ , a charge distribution just looks like a net charge (like an exoplanet). Thus, the monopole moment $Q$ is just the net charge: $Q \equiv \int ρ d τ V_{mon} (r) = \frac{1}{4 π ϵ _{0}} \frac{Q}{r}$ The dipole moment $p$ describes the individual distribution of charges: $p = i = 1 \sum n q_{i} r_{i}^{'} \equiv \int r^{'} ρ (r^{'}) d τ^{'} V_{dip} (r) = \frac{1}{4 π ϵ _{0}} \frac{p}{r ^{2}}$

$r^{'}$ points from the origin to some charge distribution. $r$ is from the origin to some reference point.

Origin of coordinates

Changing the origin will never change the monopole moment $Q$ , but will change the dipole moment $p$ as long as the total charge $Q \neq = 0$ .

For example: if our system has $+ q$ and $- q$ as its point distribution, $Q = + q + (- q) = 0$ and the dipole moment is origin-independent.

Electric field of moments: the electric field is defined as $E = - \nabla V$ .

To find the electric field i.e. caused by the dipole moment, find the voltage term for that dipole moment, then take the negative gradient of it.

Chapter 1 - Stern-Gerlach Experiments

Reference Quantum Mechanics: A Paradigms Approach by David McIntyre.

First conceptualized in 1922 by Otto Stern and later performed by Walther Gerlach, the SG experiment involved sending a beam of silver atoms through a nonuniform magnetic field and observing their distribution.

Classically, the distribution ought to look like the input beam - since silver atoms are electrically neutral, the nonuniform magnetic field shouldn't affect them, and they should just pass through.

Experimentally, we see the silver beam split at the magnetic field into roughly equal-sized groups, indicating not only that the magnetic field interacts with the silver atoms, but that each electrically-neutral silver atom must have some additional property that interacts with a $B$ -field (i.e. that the $B$ -field puts some force on the silver atoms based on some binary property).

We know $F_{z} = \frac{δ}{δz} (μ \cdot B) = μ_{z} \frac{δ B _{z}}{δz}$ (ref: Wikipedia entry on magnetic fields), where $μ$ is the magnetic moment and $B$ is the magnetic field strength. Classically, $F_{z}$ should be zero for a neutral atom (all atoms hit the center) - however, we see that there must be forces on the atom since our atoms diverge from the beam at the $B$ -field. Further, we have an "upper" and "lower" group with equal distances - so $∣ + F_{z} ∣ \approx ∣ - F_{z} ∣$ . With $\frac{δ B _{z}}{δz}$ known to be constant, then by the results of the experiment we must have two values for $μ_{z}$ : $+ μ_{z}$ and $- μ_{z}$ (the magnetic moment values are quantized).

Enter: Electron Spin

If we imagine (classically) an electron as a charge moving around a loop of current,

then $μ = I A$ . Here, $A = π r^{2}$ and $I = \frac{q}{2 π r / v}$ , so $μ = I A = \frac{q r v}{2}$ Thus, (classically), we can imagine each charge as having some "orbital angular momentum" $L$ (like planets around a star). Angular momentum is $L = m r v$ , so we can rewrite this as $μ = \frac{q}{2 m} L$ Experimentally, we observe that, while normal orbital angular momentum of charged particles still "exists", it's not the whole picture - we also need "spin", where we can write $μ$ as $μ = g \frac{q}{2 m} S$ where $g$ is a dimensionless "gyroscopic ratio" (note: different from planet spin since electron is almost a 1d point). For an electron in the $z$ direction, $μ_{z} = - g \frac{e}{2 m _{e}} S_{z}, F_{z} = - g \frac{e}{2 m _{e}} S_{z} \frac{d B _{z}}{d z}$ and, to achieve the results we observe in the experiment, we can have only two values of $S_{z}$ : $S_{z} = \pm \frac{ℏ}{2}$ where $ℏ$ is a modified Planck's constant $ℏ = h /2 π = 1.0546 \times 1 0^{- 34} J*s$ . This binary value represents a spin-1/2 system, though these aren't the only two possible values.

Quantum States

Postulate 1: the state of a quantum mechanical system includes all information we can know about it. Mathematically, we represent this state by a "ket", $∣ ψ ⟩$ .

In the spin-1/2 system,

$∣ ψ ⟩ = a ∣ + ⟩ + b ∣ - ⟩$ where $∣ + ⟩$ is the quantum state of the atoms that are spin-up $S_{z} = + ℏ/2$ , and $∣ - ⟩$ is the quantum state of the spin-down atoms $S_{z} = - ℏ/2$ .

Stern-Gerlach ran a few experiments using this basic setup to get a better understanding of the quantum nature of spin, and explore this divergence from what is classically expected.

Experiment 1: Analyzing spin in $z$ twice in a row. Spin values were conserved between measurements - if we only take $∣ + ⟩_{z}$ atoms and measure again, we only see $∣ + ⟩_{z}$ atoms. $P_{+} = ∣ ⟨ + ∣ + ⟩ ∣^{2} = 1$ $P_{-} = ∣ ⟨ - ∣ - ⟩ ∣^{2} = 0$
Experiment 2: Analyze spin in $z$ , then in $x$ - $∣ ψ ⟩_{x}$ was found to be totally independent whether we used $∣ + ⟩_{z}$ or $∣ - ⟩_{z}$ . $∣ + ⟩_{x} = \frac{1}{2} [∣ + ⟩ + ∣ - ⟩] P_{+, x} = \frac{1}{2}$ $∣ - ⟩_{x} = \frac{1}{2} [∣ + ⟩ - ∣ - ⟩] P_{-, x} = \frac{1}{2}$ Complicated, though - this is a mixed state, rather than the superposition of experiment 1. We'll investigate options to make it nicer later.
Experiment 3: Analyze spin in $z$ , then $x$ , then $z$ . We expected to see the $z$ spin values conserved - but don't! Instead, by measuring $x$ , we find we "reset" the spin of $z$ .
Experiment 4: Analyze spin in $z$ , then $x$ , then $z$ - however, instead of measuring $x$ , just send both outputs into the next $z$ . Somehow, by not "measuring" $x$ , we don't reset the $z$ spin measurement.

Bra-ket notation

Used to represent quantum state vectors, which lie in the Hilbert vector space. The dimension of the Hilbert space is determined by the current system - in our above example, we have only two possible results, so $∣ ψ ⟩ = a ∣ + ⟩ + b ∣ - ⟩$ represents a complete basis with dimensionality 2.

Note: $a$ and $b$ are complex scalar multiples. $⟨ bra ∣ ket ⟩$ Some properties of the bra-ket notation (Dirac's first and only pun):

Each ket has a corresponding bra, such that for some state $ψ$ , $⟨ ψ ∣ = a^{*} ⟨ + ∣ + b^{*} ⟨ - ∣ ∣ ψ ⟩ = a ∣ + ⟩ + b ∣ - ⟩$
Multiplying a bra with a ket represents an inner (dot) product. $⟨ + ∣ + ⟩ = 1 ⟨ ψ ∣ ψ ⟩ = 1$ $⟨ + ∣ - ⟩ = 0 ⟨ - ∣ + ⟩ = 0$ $⟨ + ∣ a ∣ + ⟩ = a ⟨ + ∣ + ⟩$

This means we can multiply $⟨ + ∣$ with $∣ ψ ⟩$ to get each constant (i.e. $a$ here), such that $⟨ + ∣ ψ ⟩ = ⟨ + ∣ (a ∣ + ⟩ + b ∣ - ⟩)$ $= ⟨ + ∣ a ∣ + ⟩ + ⟨ + ∣ b ∣ - ⟩$ $= a ⟨ + ∣ + ⟩ + b ⟨ + ∣ - ⟩$ $= a$

Likewise, $⟨ ψ ∣ + ⟩ = a^{*}$ and $⟨ + ∣ ψ ⟩ = ⟨ ψ ∣ + ⟩^{*}$ .

All quantum state vectors must be normalized, such that $⟨ ψ ∣ ψ ⟩ = ∣ ⟨ + ∣ ψ ⟩ ∣^{2} + ∣ ⟨ - ∣ ψ ⟩ ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} = 1$

If we wanted to normalize some vector $∣ ψ ⟩ = 1 ∣ + ⟩ + 2 i ∣ - ⟩$ , apply some normalization constant $C$ , such that $∣ ψ ⟩ = C [1 ∣ + ⟩ + 2 i ∣ - ⟩]$ and solve for $⟨ ψ ∣ ψ ⟩ = 1$ .

Note: $C$ will be an absval by the end here - but we don't care about it's phase (not physically meaningful), so just make it real and positive.

The complex constants $a$ and $b$ when squared (i.e. $∣ a ∣^{2} = ∣ ⟨ + ∣ ψ ⟩ ∣^{2}$ ) represent probabilities for a given measurement (probability for $+ S_{z}$ or $- S_{z}$ above). The normalization property implies the probabilities must sum up to 1 - helping prove postulate 1.

Matrix form

We can also represent states in matrix form, where $∣ + ⟩ \equiv (10) ∣ - ⟩ \equiv (10)$ and $∣ ψ ⟩ = (⟨ + ∣ ψ ⟩ ⟨ - ∣ ψ ⟩) = (a b)$ with the corresponding bra represented by the row vector $⟨ ψ ∣ = (a^{*} b^{*})$ so

$⟨ ψ ∣ ψ ⟩ = (a^{*} b^{*}) (a b) = ∣ a ∣^{2} + ∣ b ∣^{2}$

General quantum systems

For some general quantum system, where we might not only have 2 results (i.e. not only spin-1/2), such as

then, generally speaking,

$⟨ a_{i} ∣ a_{j} ⟩ = δ_{ij}$ $∣ ψ ⟩ = i \sum ⟨ a_{i} ∣ ψ ⟩ ∣ a_{i} ⟩$

Note: $⟨ a_{i} ∣ ψ ⟩$ represents each corresponding complex scalar multiple, and $δ_{ij}$ is the Kronecker delta, which is 1 if $i = j$ and 0 if $i \neq = j$ .

Converting kets between axes

We can represent up-down spin measurements in $x$ and $y$ in terms of our $z$ -spin kets, such that

$∣ + ⟩_{z} ∣ + ⟩_{x} = \frac{1}{2} [∣ + ⟩ + ∣ - ⟩] ∣ + ⟩_{y} = \frac{1}{2} [∣ + ⟩ + i ∣ - ⟩]$ $∣ - ⟩_{z} ∣ - ⟩_{x} = \frac{1}{2} [∣ + ⟩ - ∣ - ⟩] ∣ - ⟩_{y} = \frac{1}{2} [∣ + ⟩ - i ∣ - ⟩]$ In matrix form, $∣ + ⟩_{z} \equiv (10) ∣ + ⟩_{x} \equiv \frac{1}{2} (11) ∣ + ⟩_{y} \equiv \frac{1}{2} (1 i)$ $∣ - ⟩_{z} \equiv (01) ∣ - ⟩_{x} \equiv \frac{1}{2} (1 - 1) ∣ - ⟩_{y} \equiv \frac{1}{2} (1 - i)$

The Postulates of Quantum Mechanics

Postulate 1: the state of a quantum mechanical system, $∣ ψ ⟩$ , contains everything we can mathematically know about it - experimentally, the hidden variables theory (regardless of validity) is irrelevant (since the variables are by definition hidden.)

Postulate 2: Physical observables (such as spin) are represented mathematically by some operator $A$ that operates on kets.

Postulate 3: The only results possible from a measurement are contained in the set of eigenvalues $[a_{1}, a_{2} \dots a_{n}]$ of the operator $A$ .

Postulate 4: The possibility of obtaining some eigenvalue $a_{n}$ in a measurement of the observable $A$ on the system $∣ ψ ⟩$ is g $P_{a_{n}} = ∣ ⟨ a_{n} ∣ ψ ⟩ ∣^{2}$

Postulate 5: After a measurement of $A$ (such as spin) that yields some result $a_{n}$ (such as $+$ ), the quantum system is in a new state that is a normalized projection of the original system ket $∣ ψ ⟩$ onto the ket (or kets) corresponding to the result of the measurement $∣ ψ^{'} ⟩$ $∣ ψ^{'} ⟩ = \frac{P _{n} ∣ ψ ⟩}{⟨ ψ ∣ P _{n} ∣ ψ ⟩}$

Postulate 6: The time evolution of a quantum system is determined by the Hamiltonian, or total energy operator $H (t)$ via the Schrödinger equation $i ℏ \frac{d}{d t} ∣ ψ (t) ⟩ = H (t) ∣ ψ (t) ⟩$

Note: only postulates 1 and 2 were covered in this chapter.

Chapter 2 - Operators & Measurement

Reference Quantum Mechanics: A Paradigms Approach by David McIntyre.

Operators are mathematical objects that operate on kets to turn them into new kets; $S_{z}$ is one such example. $A ∣ ψ ⟩ = ∣ ϕ ⟩$ If a ket is not changed by the application of some operator, then the ket is known as an eigenvector and the associated constants are eigenvalues. $A ∣ ψ ⟩ = a ∣ ψ ⟩$ Above, $∣ ψ ⟩$ is the eigenvector and $a$ is an eigenvalue. Both eigenvectors and eigenvalues are properties of the operator $A$ - if it changes, so does the eigenstuff. For the operator $S_{z}$ , the eigenvectors / values are: $S_{z} ∣ + ⟩ = + \frac{ℏ}{2} ∣ + ⟩ S_{z} ∣ - ⟩ = - \frac{ℏ}{2} ∣ - ⟩$

Postulate 3: The only possible result of a measurement of an observable (like $S_{z}$ ) is one of the eigenvalues $a_{n}$ of the corresponding operator $A$ .

This postulate implies that, if we have the eigenvectors and values of $A$ , we can actually reconstruct $A$ from them.

Let $S_{z} \equiv (a c b d)$ . $S_{z} ∣ + ⟩ = + \frac{ℏ}{2} ∣ + ⟩ (a c b d) (10) = + \frac{ℏ}{2} (10)$ $S_{z} ∣ - ⟩ = - \frac{ℏ}{2} ∣ - ⟩ (a c b d) (01) = - \frac{ℏ}{2} (01)$ Which leads us to $(a c) = + \frac{ℏ}{2} (10) (b d) = - \frac{ℏ}{2} (01)$ Now, plugging these solved values in, $S_{z} \equiv (ℏ/2 0 0 - ℏ/2) = \frac{ℏ}{2} (10 0 - 1)$

Operators are diagonal in their own basis (i.e. they only have elements along the main diagonal).
The elements along the diagonal are the eigenvalues of the operator.
- The eigenvectors themselves are not included in the operator matrix.

Note: an observable is some physical quantity that can be measured - such as spin. In that context, the operator $S_{z}$ represents that observable.

$S_{z} = \frac{ℏ}{2} (10 0 - 1) ∣ + ⟩_{z} = (10) ∣ - ⟩_{z} = (01) >$ $S_{x} = \frac{ℏ}{2} (0110) ∣ + ⟩_{x} = \frac{1}{2} (11) ∣ - ⟩_{x} = \frac{1}{2} (1 - 1)$ $S_{y} = \frac{ℏ}{2} (0 i - i 0) ∣ + ⟩_{y} = \frac{1}{2} (1 i) ∣ - ⟩_{y} = \frac{1}{2} (1 - i)$

Other representations of matrix elements

We can also represent an operator matrix $A$ in terms of its elements. Let the operator $A$ describe some two-dimensional spin-1/2 system (like $S_{z}$ does).

$A \equiv (a c b d) = (⟨ + ∣ A ∣ + ⟩ ⟨ - ∣ A ∣ + ⟩ ⟨ + ∣ A ∣ - ⟩ ⟨ - ∣ A ∣ - ⟩)$ Where $⟨ + ∣ A ∣ + ⟩ = (10) (a c b d) (10) = a$

Finding eigenvectors / values from an observable

If we know the operator $A$ , and need to find the possible results of a measurement of the corresponding observable, we can start from the general eigenvalue equation and work from there.

$A ∣ λ ⟩ = λ ∣ λ ⟩$ where $λ$ is the eigenvalue and $∣ λ ⟩$ is the corresponding eigenvector. Eigenvalues can be found then by solving the secular equation, $det ∣ A - λ I ∣ = 0$ Note: for a $2 \times 2$ operator (i.e. a spin-1/2 system), this is $a d - b c = 0$ .

After finding the eigenvalues, we now know $A$ and $λ$ - all that remains is to find $∣ λ ⟩$ by solving for it in $A ∣ λ ⟩ = λ ∣ λ ⟩$ .

The process of finding the eigenvectors and eigenvalues of a matrix is known as diagonalization.

Hermitian operators

To find the associated bra operator to some operator acting on a ket, we use the $^{†}$ operator

$⟨ ϕ ∣ = ⟨ ψ ∣ A^{†} for ∣ ϕ ⟩ = A ∣ ψ ⟩$ where $A^{†}$ is the Hermitian adjoint of $A$ , found by transposing and taking the complex conjugate of $A$ . $A^{†} = A_{T}^{*}$ If $A = A^{†}$ , then $A$ is said to be Hermitian, and $⟨ ϕ ∣ = ⟨ ψ ∣ A$ for $∣ ϕ ⟩ = A ∣ ψ ⟩$ .

This is not normally the case - if $A$ is not Hermitian, the corresponding bra-state operator will be different from $A$ - i.e. $⟨ ϵ ∣ = ⟨ ψ ∣ A$ will not be the bra $⟨ ϕ ∣$ corresponding to $∣ ϕ ⟩$ .

Hermitian operators have some properties to be aware of:

They will always have real eigenvalues which ensures the results from a measurement are always real.
The eigenvectors of a Hermitian operator form a complete set of basis states, ensuring we can use the eigenvectors of any observable as a valid basis.

New operators

Let's say we wanted to find the spin component in some general direction $\overset{n}{^}$ .

where $\overset{n}{^} = sin θ cos ϕ \hat{i} + sin θ sin ϕ \hat{j} + cos θ \hat{k}$ . Projecting the spin vector onto $\overset{n}{^}$ such that $S \cdot \overset{n}{^} = S_{x} sin θ cos ϕ + S_{y} sin θ sin ϕ + S_{z} cos θ$ then $S_{\overset{n}{^}} = \frac{ℏ}{2} (cos θ sin θ e^{i ϕ} sin θ e^{- i ϕ} - cos θ)$ and diagonalizing the vector above, we find the eigenvectors for $S_{\overset{n}{^}} = \pm \frac{ℏ}{2}$ to be $∣ + ⟩_{\overset{n}{^}} = cos \frac{θ}{2} ∣ + ⟩ + sin \frac{θ}{2} e^{i ϕ} ∣ - ⟩$ $∣ - ⟩_{\overset{n}{^}} = sin \frac{θ}{2} ∣ + ⟩ - cos \frac{θ}{2} e^{i ϕ} ∣ - ⟩$ with the state vector $∣ ψ ⟩ = ∣ a ∣ ∣ + ⟩_{\overset{n}{^}} + 1 - ∣ a ∣^{2} e^{i ϕ} ∣ - ⟩_{\overset{n}{^}}$

Note: the coefficient calculation attached to $∣ - ⟩_{\overset{n}{^}}$ is from total probability being equal to 1.

Outer product

Instead of taking the inner product between two state vectors, we can also find the outer product ( $∣ + ⟩ ⟨ + ∣$ ). For example, we can rewrite $∣ ψ ⟩ = ⟨ + ∣ ψ ⟩ ∣ + ⟩ + ⟨ - ∣ ψ ⟩ ∣ - ⟩$ as $∣ ψ ⟩ = (∣ + ⟩ ⟨ + ∣ + ∣ - ⟩ ⟨ - ∣) ∣ ψ ⟩$ $∣ + ⟩ ⟨ + ∣$ and $∣ - ⟩ ⟨ - ∣$ are called projection operators: $P_{+} = ∣ + ⟩ ⟨ + ∣ = (10) (10) = (1000)$ $P_{-} = ∣ - ⟩ ⟨ - ∣ = (01) (01) = (0001)$ $P_{+} + P_{-} = ∣ + ⟩ ⟨ + ∣ + ∣ - ⟩ ⟨ - ∣ = (1001)$

This last formula is known as a closure or completeness relation due to the equivalency to the identity operator $I$ , meaning that these basis states form a complete set of states.

When our projection operator acts on some state $∣ ψ ⟩$ ,

$P_{+} ∣ + ⟩ = ∣ + ⟩ ⟨ + ∣ ψ ⟩ = (⟨ + ∣ ψ ⟩) ∣ + ⟩$ $P_{-} ∣ ψ ⟩ = ∣ - ⟩ ⟨ - ∣ ψ ⟩ = (⟨ - ∣ ψ ⟩) ∣ - ⟩$

Note: $⟨ \pm ∣ ψ ⟩$ is equivalently the $a$ or $b$ coefficients.

and the probability of observing some state is then $P_{\pm} = ∣ ⟨ ψ ∣ P_{\pm} ∣ ψ ⟩ ∣^{2}$ This allows us to write the fifth postulate

Postulate 5: After a measurement of some observable operator $A$ that yields the result $a_{n}$ , the quantum system is in a new state that is the normalized projection of the original system ket $∣ ψ ⟩$ onto the new ket $∣ ψ^{'} ⟩$ corresponding to the result of the measurement $∣ ψ^{'} ⟩ = \frac{P _{n} ∣ ψ ⟩}{⟨ ψ ∣ P _{n} ∣ ψ ⟩}$

Returning to Stern-Gerlach 3 & 4

We were able to look at experiments 1 and partially 2 in chapter 1, but not experiments 3 and 4, which required expounding on operator products a bit further.

Experiment 3

Experiment 3 measured $z$ , then measured $x$ , then $z$ again, observing a "reset" with the other measurement.

The probability that an atom stays $∣ + ⟩$ throughout the three measurements is the product of each probability - such that

$P_{+++} = ∣ ⟨ + ∣ + ⟩_{x} ∣^{2} ∣_{x} ⟨ + ∣ + ⟩ ∣^{2} ∣ ⟨ + ∣ + ⟩ ∣^{2}$ Likewise for it to be $S_{z} = + ℏ/2$ , then $S_{x} = + ℏ/2$ , then $S_{z} = - ℏ/2$ is $P_{++-} = ∣ ⟨ - ∣ + ⟩_{x} ∣^{2} ∣_{x} ⟨ + ∣ + ⟩ ∣^{2} ∣ ⟨ + ∣ + ⟩ ∣^{2}$

Tradition holds QM magnitudes / probabilities to be read from right to left.

Experiment 4

Experiment 4 measured $z$ , then put the atoms through an $x$ analyzer (but didn't measure them), then measured $z$ again, finding the "measurement" step to be an important one.

The results in this one are interesting but simple - representing them with postulate 5,

$∣ ψ_{2} ⟩ = \frac{( P _{+ x} + P _{- x} ∣ ψ _{1} ⟩ )}{⟨ ψ _{1} ∣ ( P _{+ x} + P _{- x} ) ∣ + ⟩} = \frac{( P _{+ x} + P _{- x} ∣ + ⟩}{⟨ + ∣ ( P _{+ x} + P _{- x} ) ∣ + ⟩}$

This can be expanded out (and is done so in McIntyre 2.2.4), but will not be included here for the sake of brevity.

Mean and Standard Deviation

The expected mean (or expectation value) of some measurement is represented by $⟨ S_{z} ⟩$ , such that $⟨ S_{z} ⟩ = (+ \frac{ℏ}{2}) P_{+} + (- \frac{ℏ}{2}) P_{-}$ and is a sum of each possible result multiplied by the probability of that result.

Probability can be represented by both outer products and inner products - inner products are more readable, but require squared absvals $∣ ∣^{2}$ , while outer products require more terms overall. $P_{+} = ∣ ⟨ + ∣ ψ ⟩ ∣^{2} = ⟨ ψ ∣ + ⟩ ⟨ + ∣ ψ ⟩$

We can also write the expectation value for some operator $A$ as $⟨ A ⟩ = ⟨ ψ ∣ A ∣ ψ ⟩ = n \sum a_{n} P_{a_{n}}$ For $S_{z}$ , this can be written out as

$⟨ S_{z} ⟩ \Rightarrow ⟨ + ∣ S_{z} ∣ + ⟩ \Rightarrow ⟨ + ∣ \frac{ℏ}{2} ∣ + ⟩ \Rightarrow \frac{ℏ}{2} ⟨ + ∣ + ⟩ \Rightarrow \frac{ℏ}{2}$ Which makes sense, given $ℏ/2$ is the only possible result of a measurement of $S_{z}$ for the $∣ + ⟩$ state. For some system prepared $∣ + ⟩_{z}$ .

$⟨ S_{z} ⟩ =_{x} ⟨ + ∣ S_{z} ∣ + ⟩_{x} = \frac{1}{2} (11) \frac{ℏ}{2} (10 0 - 1) \frac{1}{2} (11)$ $= \frac{ℏ}{4} (11) (1 - 1) = 0ℏ$

Standard Deviation

The root-mean-square deviation (r.m.s. deviation) is

$Δ A = ⟨ (A - ⟨ A ⟩)^{2} ⟩ \equiv ⟨ A^{2} ⟩ - ⟨ A ⟩^{2} \equiv ⟨ ψ ∣ A^{2} ∣ ψ ⟩ - ⟨ A ⟩^{2}$ In the rightmost version, the second term is the squared expectation value. The first term, in the case of the $S_{z} ∣ + ⟩$ initial state, can be expanded into $⟨ S_{z}^{2} ⟩ = ⟨ + ∣ S_{z}^{2} ∣ + ⟩ = ⟨ + ∣ S_{z} S_{z} ∣ + ⟩ = (10) \frac{ℏ}{2} (10 0 - 1) \frac{ℏ}{2} (10 0 - 1) (10) = \frac{ℏ}{2} \frac{ℏ}{2} = (\frac{ℏ}{2})^{2}$ and $Δ S_{z} = ⟨ S_{z}^{2} ⟩ - ⟨ S_{z} ⟩^{2} = (\frac{ℏ}{2})^{2} - (\frac{ℏ}{2})^{2} = 0ℏ$ Expected, since we only have one possible result for measuring $∣ + ⟩$ - so no spread of possible results.

Commuting Observables

Note: eigenstates are equivalent to eigenvectors.

The commutator for two operators is defined as $[A, B] = A B - B A$ If the commutator is zero, then the two operators (observables) commute. If nonzero, they don't commute. Logical consequences can be determined by just this statement alone - the first being: $[A, B] = 0 A B - B A = 0 A B = B A$ The order of operation doesn't matter for commuting observables. Now, let $∣ a ⟩$ be an eigenstate (eigenvector) for some operator $A$ with eigenvalue $a$ : $A ∣ a ⟩ = a ∣ a ⟩$ Let's apply $B$ to both sides of this. Using the commutability of $A$ and $B$ , $B A ∣ a ⟩ = B a ∣ a ⟩$ $A B ∣ a ⟩ = a B ∣ a ⟩$ $A (B ∣ a ⟩) = a (B ∣ a ⟩)$ This last equation says $B ∣ a ⟩$ is also an eigenvector of $A$ corresponding to the same eigenvalue $a$ . Thus, $B ∣ a ⟩$ must be a scalar multiple of $∣ a ⟩$ (let's say this scalar multiple is $b$ ) - and we can write $B ∣ a ⟩ = b ∣ a ⟩$ Commuting operators will therefore share common eigenvectors - this means the two operators (that represent observables) are compatible, and we can measure one without erasing our knowledge of the results of the other observable.

This is often phrased as knowing the results of these observables simultaneously, though realistically we're still measuring each sequentially.

If two operators do not commute, then they are incompatible. This is the case for our orthogonal spin operators - and the commutations for $S_{x}$ , $S_{y}$ and $S_{z}$ are

$[S_{x}, S_{y}] = i ℏ S_{z}$ $[S_{y}, S_{z}] = i ℏ S_{x}$ $[S_{z}, S_{x}] = i ℏ S_{y}$

The Uncertainty Principle

We can relate the product of two standard deviations (i.e. uncertainties) of two observables with the commutator by $Δ A Δ B \geq \frac{1}{2} ∣ ⟨ [A, B] ⟩ ∣$ $Δ A Δ B \geq \frac{1}{2} ∣ ⟨ A B - B A ⟩ ∣$

The Uncertainty Principle: the product of two uncertainties will always be greater than or equal to the absolute expected value of the two.

For the $S_{x}$ and $S_{y}$ spin components,

$Δ S_{x} Δ S_{y} \geq \frac{1}{2} ∣ ⟨ [S_{x}, S_{y}] ⟩ ∣$ $Δ S_{x} Δ S_{y} \geq \frac{1}{2} ∣ ⟨ i ℏ S_{z} ⟩ ∣$ $Δ S_{x} Δ S_{y} \geq \frac{ℏ}{2} ∣ ⟨ S_{z} ⟩ ∣$ $Δ S_{x} Δ S_{y} \geq (\frac{ℏ}{2})^{2} \neq = 0$

Implying $Δ S_{x} \neq = 0$ and $Δ S_{y} \neq = 0$ - while we can know one spin component, we can never know the other two (they are incompatible observables).

$S^{2}$ Operator

$S^{2} = S_{x}^{2} + S_{y}^{2} + S_{z}^{2}$ and is represented in matrix form as

$S^{2} = \frac{3}{4} ℏ^{2} (1001)$ Thus, since the $S^{2}$ operator is proportional to the identity operator, it must therefore commute with all the other operators $S_{x}, S_{y}$ and $S_{z}$ - this implies all states are eigenstates of the $S^{2}$ operator, and we can write $S^{2} ∣ ψ ⟩ = \frac{3}{4} ℏ^{2} ∣ ψ ⟩$ for any state $ψ$ in the spin-1/2 system. The $S^{2}$ vector has an expectation value of

$⟨ S^{2} ⟩ = \frac{3}{4} ℏ^{2}$ and a "length" of $∣ S ∣ = ⟨ S^{2} ⟩ = 3 \frac{ℏ}{2}$

This is longer than the "normal" measured component of $ℏ/2$ along any axis - implying the spin vector can never be fully aligned along any one axis, since there will be always components in other axes.

This is often called "quantum fuzziness".

Regarding Photons

Photons also have spin - but due to moving at $c$ (ultra-relativistic), it can never have a spin of 0 - must be either $S_{P} = \pm 1$ .

Spin-1 Systems

Fermions are particles with spins of multiples of 1/2 (i.e. $1/2$ , $3/2$ , $5/2$ ), while bosons are those with full-integer spins (i.e. $0$ , $1$ , $2$ ). Spin-1 bosons have spin states of $∣ 1 ⟩$ , $∣ 0 ⟩$ or $∣ - 1 ⟩$

where

$∣ 1 ⟩ = 100 ∣ 0 ⟩ = 010 ∣ - 1 ⟩ = 001$ $S_{z} ∣ 1 ⟩ = ℏ ∣ 1 ⟩ S_{z} ∣ 0 ⟩ = 0ℏ ∣ 0 ⟩ S_{z} ∣ - 1 ⟩ = - ℏ ∣ - 1 ⟩$ $S_{z} \equiv ℏ 100000 00 - 1$

The eigenvalues $1ℏ$ , $0ℏ$ and $- 1ℏ$ are on the main diagonal of $S_{z}$ .

For $S_{x}$ , $∣ 1 ⟩_{x} = \frac{1}{2} \frac{1}{2} \frac{1}{2} ∣ 0 ⟩_{x} = \frac{1}{2} 0 - \frac{1}{2} ∣ - 1 ⟩_{x} = \frac{1}{2} - \frac{1}{2} \frac{1}{2}$

$S_{x} \equiv \frac{ℏ}{2} 010101010$ and for $S_{y}$ ,

$∣ 1 ⟩_{y} = \frac{1}{2} i \frac{1}{2} - \frac{1}{2} ∣ 0 ⟩_{y} = \frac{1}{2} 0 \frac{1}{2} ∣ - 1 ⟩_{y} = \frac{1}{2} - i \frac{1}{2} - \frac{1}{2}$

$S_{y} \equiv \frac{ℏ}{2} 0 i 0 - i 0 i 0 - i 0$

Note: The Stern-Gerlach experiments have conceptually the same results - but note that experiment 2 differs (not all 1/3 - rather, one is 1/2, the other two 1/4).

General Quantum Systems

Let $s$ denote the spin of some $s$ -spin system with number of beams $2 s + 1$ (i.e. for a spin-1 system, this would be 3, spin-1/2 system, 2). Let the possible values for spin on the $z$ axis be labeled by $m$ - then,

$S^{2} ∣ s m ⟩ = s (s + 1) ℏ^{2} ∣ s m ⟩$ $S_{z} ∣ s m ⟩ = m ℏ ∣ s m ⟩$

$s$ is known as the spin (angular momentum) quantum number and $m$ is the spin component quantum number (or magnetic quantum number).

In spin-1/2, $∣ \frac{1}{2}, \frac{1}{2} ⟩ = ∣ + ⟩$ $∣ \frac{1}{2}, - \frac{1}{2} ⟩ = ∣ + ⟩$ and in spin-1,

$∣ 11 ⟩ = ∣ 1 ⟩$ $∣ 10 ⟩ = ∣ 0 ⟩$ $∣ 1, - 1 ⟩ = ∣ - 1 ⟩$

Chapter 3 - Schrödinger Time Evolution

Reference Quantum Mechanics: A Paradigms Approach by David McIntyre.

The time evolution of a quantum system is governed by the differential equation

$i ℏ \frac{d}{d t} ∣ ψ (t) ⟩ = H (t) ∣ ψ (t) ⟩$

where $H$ corresponds to the total energy of the system (the Hamiltonian operator, different from the Hermitian operator - though the Hamiltonian is still a Hermitian operator).

Postulate 6 The time evolution of a quantum system is determined by the Hamiltonian (total energy operator) $H (t)$ , through the Schrödinger equation: $i ℏ \frac{d}{d t} ∣ ψ (t) ⟩ = H (t) ∣ ψ (t) ⟩$

The eigenvalues $E_{n}$ of the Hamiltonian are the allowed energies of the quantum system, and the eigenvectors (eigenstates) $∣ E_{n} ⟩$ are the energy eigenvectors of the system, such that $H ∣ E_{n} ⟩ = E_{n} ∣ E_{n} ⟩$

The Energy Basis

Let's say we've already diagonalized $H$ and found values for the allowed energies $E_{n}$ and $∣ E_{n} ⟩$ . General state vectors can be written in terms of these energy eigenstates: $∣ ψ (t) ⟩ = n \sum c_{n} (t) ∣ E_{n} ⟩$ and, since the energy eigenvectors are orthonormal, we can write the energy basis as $⟨ E_{k} ∣ E_{n} ⟩ = δ_{kn}$ Assuming the Hamiltonian $H (t)$ in this context is time-independent, such that $H (t_{0}) = H (t)$ for all $t$ , then the time-dependent Schrödinger equation can be written $∣ ψ (t) ⟩ = n \sum c_{n} e^{- i E_{n} t /ℏ} ∣ E_{n} ⟩ = n \sum c_{n} e^{- i ω_{n} t} ∣ E_{n} ⟩$ where $ω_{n} = E_{n} /ℏ$ , an angular frequency.

Stationary States

Let's start with the simplest possible situation, where the quantum system is in one single energy eigenstate: $∣ ψ (0) ⟩ = ∣ E_{1} ⟩$ After some time $t$ , this system will be in the state $∣ ψ (t) ⟩ = e^{- i ω_{1} t} ∣ E_{1} ⟩$ where $ω_{1} = E_{1} /ℏ$ . This state, however, differs from our initial state only by the phase factor $- i ω_{1} t$ , and since phase changes will never affect the probability of measurements, the probability of observing some eigenvalue $a_{j}$ for an observable $A$ will be $P_{aj} = ∣ ⟨ a_{j} ∣ ψ (t) ⟩ ∣^{2} = ∣ ⟨ a_{j} ∣ e^{- i ω_{1} t} ∣ E_{1} ⟩ ∣ ∣^{2} = ∣ ⟨ a_{1} ∣ E_{1} ⟩ ∣^{2}$ This probability is time-independent and equal to the probability at $t_{0}$ . There is no measurable time evolution for this state, and the energy eigenstates are called stationary states - if a system begins in some energy eigenstate, it will remain in that state.

This same idea goes for multiple energy eigenstates: $∣ ψ (0) ⟩ = c_{1} ∣ E_{1} ⟩ + c_{2} ∣ E_{2} ⟩$ where $P_{E_{1}} = ∣ ⟨ E_{1} ∣ ψ (t) ⟩ ∣^{2} \Rightarrow ∣ c_{1} ∣^{2}$

Non-commuting Observables

If an observable $A$ commutes with $H$ , then $A$ and $H$ have common eigenstates - so measuring $A$ is equivalent to measuring $H$ . However, if $A$ does not commute with $H$ , then the two observables will not share common eigenstates, and the eigenvalues of $A$ will be some superposition of energy eigenstates.

Let the eigenstates of $A$ corresponding to $∣ a_{1} ⟩$ be represented by $∣ a_{1} ⟩ = α_{1} ∣ E_{1} ⟩ + α_{2} ∣ E_{2} ⟩$ and the probability of measuring $a_{1}$ would be $P_{a_{1}} = ∣ ⟨ a_{1} ∣ ψ (t) ⟩ ∣^{2} = ∣ [α_{1}^{*} ⟨ E_{1} ∣ + α_{2}^{*} ⟨ E_{2} ∣] [c_{1} e^{- i ω_{1} t} ∣ E_{1} ⟩ + c_{2} e^{- i ω_{2} t} ∣ E_{2} ⟩] ∣^{2} = ∣ a_{1}^{*} c_{1} e^{- i ω_{1} t} + a_{2}^{*} c_{2} e^{- i ω_{2} t} ∣^{2}$ The overall phase drops out - only the relative phase between $ω_{1}$ and $ω_{2}$ remains. $ω_{21} = \frac{E _{2} - E _{1}}{ℏ}$ which is the also Bohr frequency.

Summary

In a time-dependent quantum system $∣ ψ (t) ⟩$ with a time independent Hamiltonian $H (t_{0}) = H (t)$ , the probability of measuring $a_{j}$ of some observable $A$ at time $t$ can be found through the following process:

Diagonalize $H$ to find the eigenvalues $E_{n}$ and eigenvectors $∣ E_{n} ⟩$ .
Write $∣ ψ (0) ⟩$ in terms of the energy eigenstates $∣ E_{n} ⟩$ .
Multiply each eigenvector coefficient by $e^{- i E_{n} t /ℏ} = e^{- i ω_{n} t}$ to get $∣ ψ (t) ⟩$ for some arbitrary $t$ .
Calculate the probability $P_{a_{j}} = ∣ ⟨ a_{j} ∣ ψ (t) ⟩ ∣^{2}$ .

Time Evolution of Spin

Let's apply Schrödinger time evolution to the spin-1/2 system.

The Hamiltonian operator represents the total energy of a system. In time-dependent solutions, only energy differences are important - thus, the Hamiltonian is just the magnetic potential energy of a spin $S$ system $H = - μ \cdot B = - g \frac{q}{2 m _{e}} S \cdot B$ If we say the gyromagnetic ratio $g \approx 2$ and $q = - e$ (an electron), then $H = \frac{e}{m _{e}} S \cdot B$

$z$ -direction

In the $z$ direction, $B = B_{0} \overset{z}{^}$ , and

$H = \frac{e B _{0}}{m _{e}} S_{z} \equiv ω_{0} S_{z} = ω_{0} \frac{ℏ}{2} (10 0 - 1)$ where $ω_{0} = \frac{e B _{0}}{m _{e}}$ , and

$H ∣ + ⟩ H ∣ - ⟩ = ω_{0} S_{z} ∣ + ⟩ = ω_{0} \frac{ℏ}{2} ∣ + ⟩ = E_{+} ∣ + ⟩ = ω_{0} S_{z} ∣ - ⟩ = - ω_{0} \frac{ℏ}{2} ∣ - ⟩ = E_{-} ∣ - ⟩$ with eigenstuff $E_{+} ∣ E_{+} ⟩ = ω_{0} \frac{ℏ}{2} E_{-} = - ω_{0} \frac{ℏ}{2} = ∣ + ⟩ ∣ E_{-} ⟩ = ∣ - ⟩$ Applying the Schrödinger time evolution to a general state in direction $θ$ with phase shift $ϕ$ in $∣ ψ (0) ⟩ = ∣ + ⟩_{n} = cos \frac{θ}{2} ∣ + ⟩ + sin \frac{θ}{2} e^{i ϕ} ∣ - ⟩$ : $∣ ψ (t) ⟩ = e^{- i ω_{0} t /2} (cos (θ /2) e^{i (θ + ω_{0} t)} sin (θ /2))$ If we were to calculate the probability of measuring $+$ along the $z$ axis for this arbitrary direction $θ$ and phase $ϕ$ , $P_{+} = ∣ ⟨ + ∣ ψ (t) ⟩ ∣^{2} = (10) e^{- i ω_{0} t /2} (cos (θ /2) e^{i (ϕ + ω_{0} t)} sin (θ /2))^{2} = ∣ e^{- i ω_{0} t /2} cos (θ /2) ∣^{2} = cos^{2} (θ /2)$

Probability here is time-independent because the $S_{z}$ eigenstates are also energy eigenstates for this problem (i.e. $H$ and $S_{z}$ commute).

$x$ -direction

The probability of measuring spin along the $x$ direction instead, where instead of $(10)$ we have $\frac{1}{2} (11)$ is:

$P_{+ x} = ∣_{x} ⟨ + ∣ ψ (t) ⟩ ∣^{2} = \frac{1}{2} (11) e^{- i ω_{0} t /2} (cos (θ /2) e^{i (ϕ + ω_{0} t)} sin (θ /2))^{2} = \frac{1}{2} [1 + sin θ cos (ϕ + ω_{0} t)]$

Probability in the $x$ axis is time-dependent because the $S_{x}$ operator does not commute with $H$ .

For $y$ -direction, we'd use the same equation as above, except with $\frac{1}{2} (1 i)$ instead.

Expectation values

$⟨ S_{z} ⟩ ⟨ S_{y} ⟩ ⟨ S_{x} ⟩ = ⟨ ψ (t) ∣ S_{z} ∣ ψ (t) ⟩ = \frac{ℏ}{2} cos θ = ⟨ ψ (t) ∣ S_{y} ∣ ψ (t) ⟩ = \frac{ℏ}{2} sin θ sin (ϕ + ω_{0} t) = ⟨ ψ (t) ∣ S_{x} ∣ ψ (t) ⟩ = \frac{ℏ}{2} sin θ cos (ϕ + ω_{0} t)$ Irrespective of the three axis is the total spin vector's expectation value $⟨ S ⟩$ , the precession of which is known as Larmor precession, with frequency known as the Larmor frequency. The expectation value of the spin vector precesses in a uniform magnetic field, as visualized below.

Ehrenfest's theorem states that quantum mechanical expectation values obey classical laws - as visualized above, the precession of the spin vector makes it clear the system has nonzero angular momentum (as opposed to just a magnetic dipole moment).

Magnetic field in a general direction

Let's say we have a magnetic field with both an $x$ and $z$ component, such that $B = B_{0} \overset{z}{^} + B_{1} \overset{x}{^}$ This field will be oriented in the $x z$ -plane with some angle $θ$ w.r.t the $z$ -axis. First, we'll define Larmor frequencies with each field component: $ω_{0} = \frac{e B _{0}}{m _{e}} ω_{1} = \frac{e B _{1}}{m _{e}}$ And the Hamiltonian is then defined as $H = - μ \cdot B = ω_{0} S_{z} + ω_{1} S_{x} = \frac{ℏ}{2} (ω_{0} ω_{1} ω_{1} - ω_{0})$ This Hamiltonian isn't diagonal - so we need to diagonalize it. This leads to $λ = \pm \frac{ℏ}{2} ω_{0}^{2} + ω_{1}^{2}$

When $ω_{1} = 0$ , the energy eigenvalues are just $\pm ℏ \frac{ω _{0}}{2}$ , same as $S_{z}$ .

From the above figure, we know $tan θ = \frac{B _{1}}{B _{0}} = \frac{ω _{1}}{ω _{0}}$ and $H = \frac{ℏ}{2} ω_{0}^{2} + ω_{1}^{2} (cos θ sin θ sin θ - cos θ) = ω_{0}^{2} + ω_{1}^{2} S_{n}$ where $S_{n}$ is the spin operator in some direction $\overset{n}{^}$ , with vectors (found in chapter 2) $∣ + ⟩_{\overset{n}{^}} ∣ - ⟩_{\overset{n}{^}} = cos \frac{θ}{2} ∣ + ⟩ + sin \frac{θ}{2} e^{i ϕ} ∣ - ⟩ = sin \frac{θ}{2} ∣ + ⟩ - cos \frac{θ}{2} e^{i ϕ} ∣ - ⟩$

Spin flip

The probability that some initial state $∣ + ⟩$ later evolves into $∣ - ⟩$ (known as spin flip) is $P_{+ \to -} = \frac{ω _{1}^{2}}{ω _{0}^{2} + ω _{1}^{2}} sin^{2} (\frac{ω _{0}^{2} + ω _{1}^{2}}{2} t)$

This is called Rabi's formula - see the Rabi cycle.

Neutrino Oscillations

Neutrinos are produced via decay processes, such as $n p \to p + e^{-} + \overset{ν_{e}}{ˉ} \to n + e^{+} + ν_{e}$ where $ν_{e}$ is an electron neutrino and $\overline{ν_{e}}$ is an electron antineutrino. Neutrinos interact with normal matter via the weak nuclear force (the weakest of the four fundamental forces), making them extremely hard to detect. Other flavors of neutrinos are produced via processes such as the pion-muon decay and muon-electron decay: $π^{+} μ^{-} \to μ^{+} + ν_{μ} \to e^{-} + ν_{μ} + \overline{ν_{e}}$ where $ν_{μ}$ is a muon electron.

Electrons, muons and tau ( $τ$ ) particles, as well as their neutrino byproducts, are known as leptons.

There is no theoretical basis for conservation of lepton flavors, so reactions of the type $ν_{e} \leftrightarrow ν_{μ}$ are possible, where an electron neutrino changes flavor to a muon neutrino or visa versa - these reactions are known as neutrino mixing or neutrino oscillations.

When neutrinos are interacting with the weak nuclear force, they have some energy within that interaction, and the quantum states $∣ ν_{e} ⟩$ and $∣ ν_{μ} ⟩$ are eigenstates of the Hamiltonian describing the weak nuclear force - in free space, with no weak nuclear force present, the only relevant Hamiltonian (energy description operator) is the relativistic energy of the particle (which due to $E = m c^{2}$ includes the rest masses and momenta) - this relativistic Hamiltonian has mass eigenstates.

If $m_{ν_{e}} \neq = m_{ν_{μ}}$ , then the mass eigenstates do not coincide with the weak interaction eigenstates - allowing for flavor-changing processes in the first place (somehow?).

Let $∣ ν_{e} ⟩ ∣ ν_{μ} ⟩ = cos \frac{θ}{2} ∣ ν_{1} ⟩ + sin \frac{θ}{2} ∣ ν_{2} ⟩ = cos \frac{θ}{2} ∣ ν_{1} ⟩ - sin \frac{θ}{2} ∣ ν_{2} ⟩$ where $θ /2$ is the mixing angle between the two. If this angle is small, then the small angle hypothesis applies and $∣ ν_{e} ⟩ ∣ ν_{μ} ⟩ \approx ∣ ν_{1} ⟩ \approx ∣ ν_{2} ⟩$

Example

Assume an electron is created by a weak-interaction process and propagates through free space to a detector. We detect it by seeing the signature of flavor mixing, rather than the neutrino itself. Let our initial neutrino be an electron neutrino: $∣ ψ (0) ⟩ = ∣ ν_{e} ⟩ = cos \frac{θ}{2} ∣ ν_{1} ⟩ + sin \frac{θ}{2} ∣ ν_{2} ⟩$ Since this neutrino's now propagating through free space, the energy eigenstates are just its mass eigenstates (no weak interaction) - so the Schrödinger time evolution is $∣ ψ (t) ⟩ = cos \frac{θ}{2} e^{- i E_{1} t /ℏ} ∣ ν_{1} ⟩ + sin \frac{θ}{2} e^{- i E_{2} t /ℏ}$ where $E_{i}$ represents the relativistic energy eigenvalues, determined by the rest mass and momenta of each. $E_{i} = (p c)^{2} + (m_{i} c^{2})^{2}$ Assuming the neutrinos are highly relativistic, then $(m c^{2} << p c)$ and $E_{i} = p c + \frac{( m _{i} c ^{2} ) ^{2}}{2 p c}$ The probability of the neutrino mixing is $P_{ν_{e} \to ν_{μ}} = ∣ ⟨ ν_{μ} ∣ ψ (t) ⟩ ∣^{2} = sin^{2} θ sin^{2} (\frac{( E _{1} - E _{2} ) t}{2ℏ})$ The energy difference is just proportional to the difference in masses (see $E_{i}$ above), and, with $t = L / c$ where $L$ is the distance from the source from the detector and momentum $p = E / c$ : $P_{ν_{e} \to ν_{μ}} = sin^{2} θ sin^{2} (\frac{( m _{1}^{2} - m _{2}^{2} ) L c ^{3}}{4 E ℏ})$ The probability oscillates from 0 to a maximum value of $sin^{2} θ$ ... hence the term neutrino oscillation.

Experimental measurements have this mass difference is approximately $m_{1}^{2} - m_{2}^{2} = 8 \times 1 0^{- 5} e V^{2} / c^{4}$ and $θ \approx 6 9^{\circ}$ .

Time-dependent Hamiltonians

Thus far, we've only looked at time-independent Hamiltonians. Time-dependent Hamiltonians are to them what a salmon in the ocean is to nigiri - complicated.

Generalizing Rabi's formula with the change $ω_{0} \to - Δ ω$ , $P_{+ \to -} = \frac{ω _{1}^{2}}{Δ ω ^{2} + ω _{1}^{2}} sin^{2} (\frac{Δ ω ^{2} + ω _{1}^{2}}{2} t) = \frac{ω _{1}^{2}}{( ω - ω _{0} ) ^{2} + ω _{1}^{2}} sin^{2} (\frac{( ω - ω _{0} ) ^{2} + ω _{1}^{2}}{2} t)$ where $ω$ is the frequency of incident light (i.e. incident light is not static, but oscillates with time). For the case of $ω = 0$ (zero-frequency light), the applied field is static and we can use our earlier definition of Rabi's equation.

The static field case ( $ω = 0$ ) is referred to as spin procession, the rotating field case ( $ω > 0$ ) as Rabi flopping. In the static field case, the spin precession is a natural oscillation of a quantum system operating in a superposition of energy eigenstates. In the rotating field case, the Rabi flopping represents transitions between energy eigenstates, and there is an exchange of energy between the particle system and the applied rotating field - since the Hamiltonian is time-dependent, energy is not static within the system, but moving into and out of the applied field.

Time-dependent Hamiltonian

Let's look at the scenario where some incident photons collide with some matter, increasing the energy levels of local electrons.

In a quantum sense, the oscillating electric field of the light wave interacts with the electric dipole of the atom, and the energy exchange between the field and the atom corresponds to the absorption and emission of photons.

The probability of a Rabi spin flip changes/oscillates with the angular frequency $(ω - ω_{0})^{2} + ω_{1}^{2}$ (typically referred to as the generalized Rabi frequency - though the Rabi frequency itself usually refers to $ω_{1}$ .

The Hamiltonian of this system is $H = \frac{ℏ}{2} (ω_{0} ω_{1} e^{iω t} ω_{1} e^{- iω t} - ω_{0})$

Resonant frequency

If $ω = ω_{0}$ , the frequency $ω$ of the rotating field is equal to the Larmor precession frequency $ω_{0}$ , and the probability of a spin flip just becomes $P_{+ \to -} = sin^{2} (\frac{ω _{1}^{2}}{2} t)$

Spin is flipped with 100% probability with some frequency $ω_{1}$ .

Non-resonant frequency

At values $ω \neq = ω_{0}$ , the spin flip oscillates with an amplitude smaller than 1.

The FWHM of the above curve is $2 ω_{1}$ .

Chapter 4 - Quantum Spookiness

Reference Quantum Mechanics: A Paradigms Approach by David McIntyre.

In the 20th century, as quantum mechanics was first developing widespread acknowledgement, various theories were put forth to try and explain the probabilistic (rather than deterministic) nature that surprised a lot of classical physicists.

Einstein-Podolsky-Rosen Paradox (EPR)

Some, such as Albert "God does not play dice" Einstein, thought the probabilistic nature of QM (especially when juxtaposed with the more-deterministic classical mechanics) was just because we weren't seeing some variables - we thought it probabilistic only because there was some backstage action we weren't seeing (an incomplete description of reality).

The experiment begins with an unstable particle with spin 0, which then decays into two spin $\pm 1/2$ particles (consv. of angular momentum) traveling in opposite directions (consv. of linear momentum).

Because each spin is opposite the other, if observer 1 sees spin +1/2, observer 2 must see spin -1/2, and visa versa. After measurement, each observer always knows what the other observer sees.

This would be later called an entangled state.

There are two ways of looking at these particles:

Each exists as part of a state $∣ ψ ⟩$ that is in superposition, so neither particle has a defined spin until measurement.
Both particles have spin values defined at the moment of decay - particle 1 always had spin $+$ , particle 2 always spin $-$ .

For example, imagine a particle decays on Earth, and the resulting 1/2 particles are kept (unmeasured) in two chambers (i.e. the quantum state $∣ ψ ⟩$ describing both is in superposition by view 1). One 1/2 particle is sent with a ship to Mars, and another is sent on a ship to Alpha Centauri.

If Mars makes the measurement $∣ S_{2 z} ⟩ = + ℏ/2$ , according to idea 1, the state $∣ ψ ⟩$ instantaneously collapses into the state $∣ - ⟩_{1} ∣ + ⟩_{2}$ and Alpha Centauri will measure $∣ S_{1 z} ⟩ = - ℏ/2$ , regardless of the distance between the two. Idea 2 says that Mars always had $∣ S_{2 z} ⟩ = + ℏ/2$ , while perspective 1 maintains that the superposition state must have collapsed then.

The EPR paradox argues in favor of the second: the spin is a "real" property of the particle - a variable (invisible to our instruments then) describing reality. Instead of being in a superposition state pre-measurement, both particles had their spin states specified at the moment of decay and always did - Einstein's local hidden variables theory (localized to each particle).

For a while, it was thought impossible to know whether theory 1 or theory 2 was the correct interpretation ... that is, until 1964. Enter John Bell.

Bell's Theorem

Bell's theorem sets up a mathematical inequality that provides different results for a setup assuming HVT and a setup assuming the Copenhagen interpretation.

In the hidden variables theory (HVT) interpretation, this equality is $∣ ⟨ A_{0} B_{0} ⟩ + ⟨ A_{0} B_{1} ⟩ + ⟨ A_{1} B_{0} ⟩ - ⟨ A_{1} B_{1} ⟩ ∣ \leq 2$ where $A_{0, 1}$ and $B_{0, 1}$ represent the measurements taken by two observers $A$ and $B$ and $⟨ X ⟩$ denotes the expectation (or average) value of some measurement. This inequality is also called the Clauser-Horne-Shimony-Holt (CHSH) inequality.

Now, assume this was set up with qubits instead - see the Wikipedia article for more details. Summing the expectation values yields $⟨ A_{0} \otimes B_{0} ⟩ + ⟨ A_{0} \otimes B_{1} ⟩ + ⟨ A_{1} \otimes B_{0} ⟩ - ⟨ A_{1} \otimes B_{1} ⟩ = 22$ which is greater than the maximum of 2 allowed by the CHSH inequality, implying hidden variables cannot exist.

Schrödinger's Cat

Schrödinger, in a fit of psychosis, proposed putting a cat in a box. Never a good idea - but further, he decided that box should include both unstable radioactive isotopes and a bottle of cyanide! If an isotope decay is detected, boom. Dead cat. Now, this is a dumb experiment for so many reasons (animal cruelty not the least of which), but it's animal cruelty for science ... so ... yeah. That doesn't even justify it. Whatever. Don't do this experiment at home or anywhere else for that matter.

The atom has a half life of 1 hour. After 1 hour, the state of the life-deciding decaying atom is described by $∣ ψ_{nucleus} ⟩ = \frac{1}{2} (∣ ψ_{decayed} ⟩ + ∣ ψ_{undecayed} ⟩)$ ... with the quantum state of the cat implied to be $∣ ψ_{cat} ⟩ = \frac{1}{2} (∣ ψ_{dead} ⟩ + ∣ ψ_{alive} ⟩)$ ... implying the cat is in a superposition state between dead and alive. Purgatory's got nothing on Schrödinger's pets. This (thankfully only) thought experiment raises two questions:

Can macroscopic (cat) states be described quantum mechanically?
What qualifies as a measurement? I.e. what causes the collapse of the wave function?

The Copenhagen interpretation states that, no, cats (nor any other macroscopic thing for that matter) cannot be represented by quantum functions - only classically, like with ... normal things, like what color it is, food, how fast it's going, etc. Normal things, like, by Jove, physicists are weird.

More human-centric folks have argued human consciousness causes the collapse, others that there is no collapse - rather, just a bifurcation into separate universes (the multiverse).

Feynman had the non-answer to answer this:

Shut up and calculate.

Chapter 5 - Quantized Energies

Reference Quantum Mechanics: A Paradigms Approach by David McIntyre.

Spectroscopy

Atoms are differentiated by their specific atomic structure - how many neutrons, protons and electrons they have form the basis for elements in the periodic table. Electrons form the bulk of ways we can identify one element from another - when an atom absorbs a photon, the atom responds by raising an electron up one energy level, then dropping it (emitting a photon).

Since the energy of a photon is $Δ E = h f$ and we can only have specific values of energies, we can identify individual atoms by their frequencies - i.e. with quantized values for energy come quantized values for frequencies.

The lowest energy state ( $n = 1$ for hydrogen) is called the ground state, with higher levels called excited states. The set of quantized energy states is referred to as the energy spectrum of a system.

From a quantum perspective, we can visualize the energy spectrum like this:

For a system prepared in some initial state $∣ ψ ⟩$ , the probability of measuring some energy shift $E_{i}$ is $P_{E_{i}} = ∣ ⟨ E_{i} ∣ ψ ⟩ ∣^{2}$ We can find the energy levels $E_{i}$ and their corresponding eigenstates $∣ E_{i} ⟩$ by applying the Hamiltonian in the energy eigenvalue equation, such that $\hat{H} ∣ E_{i} ⟩ = E_{i} ∣ E_{i} ⟩$

Energy Eigenvalue Equation

Also known as the time-independent Schrödinger equation since it can be derived from the S.E. by separating the time and space components.

To find the equation describing energy eigenvalues,

Find the classical form of the system's energy
Replace each physical observable (position, momentum etc) with their quantum mechanical operators.

For a simple moving particle, the energy is the sum of kinetic and potential energy. $E = K (x) + V (x) \equiv \frac{1}{2} m v^{2} + V (x) \equiv \frac{1}{2} \frac{ρ _{x}^{2}}{m} + V (x)$ where $ρ_{x} = m v_{x}$ is the momentum of the particle in the $x$ direction and $V (x)$ is the potential energy.

In quantum mechanics, our primary physical observables are usually position $x$ and momentum $p$ , so our energy operator $\hat{H}$ is just the quantum version of $E$ using $\overset{x}{^}$ and $\overset{p}{^}$ : $\hat{H} = \frac{p _{x} ^ ^{2}}{2 m} + V (\overset{x}{^})$ $\overset{x}{^} \equiv x \overset{p}{^} \equiv - i ℏ \frac{d}{d x}$ where $- i ℏ$ is used to fix dimensions and ensure measurable results are real (non-imaginary).

Our quantum variables $\overset{x}{^}$ and $\overset{p}{^}$ are both used in wave functions, which are really just alternate representations of quantum states: $∣ ψ ⟩ \equiv ψ (x)$

This is called position representation since we're using the position eigenstates as the preferred basis.

The wave function representing the energy eigenstates in position representation is $∣ E_{i} ⟩ \equiv ψ_{E_{i}} (x)$ so our energy eigenvalue equation $\hat{H} ∣ E_{i} ⟩ = E_{i} ∣ E_{i} ⟩$ becomes $\hat{H} ψ_{E_{i}} (x) = E_{i} ψ_{E_{i}} (x)$

Right. Let's combine everything. Using $\hat{H}$ with our quantum variables: $(\frac{1}{2 m} (- i ℏ \frac{d}{d x})^{2} + V (x)) ψ_{E_{i}} (x) = E_{i} ψ_{E_{i}} (x)$ simplifying, $(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + V (x)) ψ_{E} (x) = E ψ_{E} (x)$

$V (\overset{x}{^})$ is the potential energy function, $ψ (x)$ is the wave function representing energy and $E_{i}$ is the energy eigenvalue.

The big thing that happens with wave functions: operator equations turn into differential equations.

The Wave Function $ψ (x)$

The wave function $ψ (x) = ⟨ x ∣ ψ ⟩$ is the probability amplitude for the quantum state $∣ ψ ⟩$ to be measured in the position eigenstate $∣ x ⟩$ , with the actual probability of measuring some value $x$ being $P (x) = ∣ ψ (x) ∣^{2}$

Probability density $P (x)$

Since the probabilities of all measurements still sum to unity, for a continuous probability density function $P (x)$ , all of the following are equivalent: $\int_{- \infty}^{\infty} P (x) d x = \int_{- \infty}^{\infty} ∣ ψ (x) ∣^{2} d x = 1$ $⟨ ψ ∣ ψ ⟩ = \int_{- \infty}^{\infty} ψ^{*} (x) ψ (x) d x = 1$

Similarly, if we wanted to know the probability a particle would be found between $x = a$ and $x = b$ , $P_{a < x < b} = \int_{a}^{b} ∣ ψ (x) ∣^{2} d x$

Note: the above curves of $P (x)$ represent probability density: $\frac{probability}{unit length}$ By multiplying by $d x$ , we get rid of the unit length and end up with probability.

Generally, to translate bra-ket formulae to wave function versions:

Replace kets with wave function $∣ ψ ⟩ \to ψ (x)$
Replace bras with wave function conjugate $⟨ ψ ∣ \to ψ^{*} (x)$
Replace braket with integral over all space $⟨ ∣ ⟩ \to \int_{- \infty}^{\infty} d x$
Replace operator with position representation $\hat{A} \to A (x)$

So, to convert some probability amplitude $⟨ ϕ ∣ ψ ⟩$ : $⟨ ϕ ∣ ψ ⟩ = \int_{- \infty}^{\infty} ϕ^{*} (x) ψ (x) d x$ with probability $P = ∣ ⟨ ϕ ∣ ψ ⟩ ∣^{2} = \int_{- \infty}^{\infty} ϕ^{*} (x) ψ (x) d x^{2}$ To transform expectation values, $⟨ \overset{x}{^} ⟩ = ⟨ ψ ∣ \overset{x}{^} ∣ ψ ⟩ = \int_{- \infty}^{\infty} ψ^{*} (x) x ψ (x) d x = \int_{- \infty}^{\infty} x ∣ ψ (x) ∣^{2} d x$

Energy Wells

Our energy eigenvalue equation is $(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + V (x)) ψ_{E} (x) = E ψ_{E} (x)$ Solutions to this equation depend on what our potential energy $V (x)$ is - and $V (x)$ is dependent on context. Often, potential energy will resemble an energy well, such as seen below:

Some notes on vocabulary:

Classically-forbidden region: kinetic energy can't be negative. Since $T = K E = E_{1} - V (x)$ , any region where $V (x) > E_{1}$ is called classically forbidden, since it would imply a negative kinetic energy. The edges of the well by each side are called classical turning points.
Classical turning points: locations where $E_{1} = V (x)$ . The particle has only potential energy (no kinetic), and must "turn around" to go back into the well.
Particles within the well are in bound states, while those outside the well are in unbound states.

The extent of the allowed and forbidden regions depends on our total energy $E$ used for a particular bound state. $E_{2}$ might be less than $E_{1}$ , and $E_{1}$ might change from particle to particle.

Infinite Square Well

The classical model for a particle well is a ball bouncing between two perfectly elastic walls, like the old bouncing DVD screensaver. A simple model for a bound particle follows the same rules:

The ball flies freely between the walls
The ball is reflected perfectly at each bounce
The ball remains in the box regardless how fast it is

Let's find the energy eigenstates & eigenvalues using the energy eigenvalue equation. Outside the box, $V (x) = \infty$ , so $(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + \infty) ψ_{E} (x) = E ψ_{E} (x)$ which is satisfied only if $E = \infty$ (a nice thought) or if $ψ_{E} (x) = 0$ everywhere outside the box. Inside the box, $V (x) = 0$ , so $- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} ψ_{E} (x) = E ψ_{E} (x)$ We already know $m$ , $ℏ$ and is confined to a box of size $L$ . We need to find our total energy $E$ and the wave function $ψ_{E} (x)$ . Rewriting our EEQ, $\frac{d ^{2}}{d x ^{2}} ψ_{E} (x) = - \frac{2 m E}{ℏ}^{2} ψ_{E} (x) = - k^{2} ψ_{E} (x)$

Above, $k^{2} = \frac{2 m E}{ℏ ^{2}}$ and is defined as the wave vector.

The solution to this equation is $ψ_{E} (x) = A^{'} e^{ik x} + B^{'} e^{- ik x} = A sin (k x) + B cos (k x)$ There are three unknowns: $A$ , $B$ and $k$ , which contains our total energy $E$ . We can find two of them with our boundary conditions: $x = 0$ and $x = L$ . At $x = 0$ , $ψ_{E} (0) = A sin (0) + B cos (0) = 0$ so $B = 0$ . At $x = L$ with $B = 0$ , $ψ_{E} (L) = A sin (k L) = 0$ which is zero if $A = 0$ (implying a full-zero wave function, not very useful, or when $k L = nπ$ for $n = 1, 2, 3 \dots$ , allowing us to define yet another quantity $k_{n}$ : $k_{n} = n \frac{π}{L} n = 1, 2, 3 \dots$

$n$ is called the quantum number, while $k_{n}$ is the quantization condition.

If we use this quantization condition as the wave vector $k$ , then we can solve for our total energy $E$ as $E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m L ^{2}} n = 1, 2, 3 \dots$ with allowed energy eigenstate wave functions $ψ_{E} (x) = A sin \frac{nπ x}{L} n = 1, 2, 3 \dots$ Our final constant $A$ can be solved for by normalizing the wave function to unity, such that $1 = ⟨ E_{n} ∣ E_{n} ⟩ = \int_{- \infty}^{\infty} ψ_{n}^{*} (x) ψ_{n} (x) d x = \int_{- \infty}^{\infty} ∣ ψ_{n} (x) ∣^{2} d x = \int_{0}^{L} ∣ A ∣^{2} sin^{2} (k_{n} x) d x = ∣ A ∣^{2} \frac{L}{2}$ leading us to the final definition of our wave equation, $ψ_{n} (x) = \frac{2}{L} sin \frac{nπ x}{L}, n = 1, 2, 3 \dots$ Since probability density can be calculated from some wave equation (which represents probability amplitude) via $P (x) = ∣ ψ (x) ∣^{2}$ , then our quantized probability density becomes $P_{n} (x) = ∣ ψ_{n} (x) ∣^{2} = \frac{2}{L} sin^{2} \frac{nπ x}{L}$ We can visualize these wave functions for different quantum numbers (and hence energy levels) like this:

Finite Square Well

Now, our energy eigenvalue equation outside the well becomes $(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + V_{0}) ψ_{E} (x) = E ψ_{E} (x)$ while inside the well it remains $- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} ψ_{E} (x) = E ψ_{E} (x)$ When we were forced to always have our particle sitting in the well, we only needed to worry about our wave number $k$ - let's also now set up $q$ to represent the case when the particle is outside the well. $k = \frac{2 m E}{ℏ ^{2}} q = \frac{2 m}{ℏ ^{2}} (V_{0} - E)$ For bound states (i.e. when $E < V_{0}$ both inside and outside the well), $k$ and $q$ are real, so our EEVs are $\frac{d ^{2} ψ _{E} ( x )}{d x ^{2}} = - k^{2} ψ_{E} (x) inside box$ $\frac{d ^{2} ψ _{E} ( x )}{d x ^{2}} = - q^{2} ψ_{E} (x) outside box$ with general solutions $ψ_{even} (x) = ⎩ ⎨ ⎧ A e^{q x}, D cos (k x), A e^{- q x}, x < - a - a < x < a x > a ψ_{odd} (x) = ⎩ ⎨ ⎧ A e^{q x}, C sin (k x), - A e^{- q x}, x < - a - a < x < a x > a$ where the even, odd values are applied to the energy levels (i.e. $E_{1}, E_{3}, E_{5} \dots$ are odd).

Let's start with the even constants. At each boundary $x = a$ and $x = - a$ , the solutions are the same since we positioned our "zero point" between them. $ψ_{e v e n} (a) \frac{d}{d x} ψ_{e v e n} (a) D cos (ka) = A e^{- q a}_{x = a} - k D sin (ka) = - q A e^{- q a}$ with the normalization constant providing the third equation for all three unknowns ( $A$ , $D$ , $E$ ). Dividing the two above equations gives $k tan (ka) = q$ which is equivalently (by substituting in our $k$ and $q$ formulae) $\frac{2 m}{ℏ ^{2}} E tan (\frac{2 m}{ℏ ^{2}} E a) = \frac{2 m}{ℏ ^{2}} (V_{0} - E)$ Similarly for the odd solutions: $- k cot (ka) = q$

We can simplify all of this by creating some new constants $z$ , $z_{0}$ and $q a$ , such that $z = ka = \frac{2 m E a ^{2}}{ℏ ^{2}} z_{0} = \frac{2 m V _{0} a ^{2}}{ℏ ^{2}} q a = \frac{2 m ( V _{0} - E ) a ^{2}}{ℏ ^{2}}$ so $ka tan (ka) = q a - ka cot (ka) = q a \to z t an (z) = z_{0}^{2} + z^{2} \to - z cot (z) = z_{0}^{2} - z^{2}$

Chapter 6 - Unbound States

Reference Quantum Mechanics: A Paradigms Approach by David McIntyre.

In bound states, the energy levels of particles are quantized - they are restricted to specific values. In unbound states, energy is no longer quantized, and operates on a continuum.

We still, however, will use the energy eigenvalue equation, its associated Hamiltonian, and the energy eigenvalue wave differential equation. $\hat{H} ∣ E ⟩ = E ∣ E ⟩$ $\hat{H} = \frac{p ^ ^{2}}{2 m} + V (\overset{x}{^})$ $(- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + V (x)) ψ_{E} (x) = E ψ_{E} (x)$

Free particle eigenstates

For a free particle, $V (x)$ is zero everywhere, so the EEV differential equation becomes $\frac{d ^{2}}{d x ^{2}} ψ_{E} (x) = - k^{2} ψ_{E} (x)$ where $k^{2} = \frac{2 m E}{ℏ ^{2}}$ . This has the general solution of $ψ_{E} (x) = A e^{ik x} + B e^{- ik x}$ Since we're no longer in a bound state, we no longer have constraints on extrema (except for the normalization condition) - and hence continuous energy, no longer quantized. We can apply the Schrödinger time evolution to this by multiplying the energy basis (above) by a phase factor, such that $ψ_{E} (x, t) = ψ_{E} (x) e^{- i Et /ℏ} = A e^{ik x} + B e^{- ik x}$ Using the Planck energy relation $E = h f = ℏ ω$ , $ψ_{E} (x, t) = A e^{i (k x - ω t)} + B e^{- i (k x + ω t)}$ This wave function represents a wave that retains its shape as it moves, with a speed determined by $∣ v ∣ = ω / k$ , the phase velocity.

Thus, $A e^{i (k x - ω t)}$ represents the part moving in the positive direction, while $B e^{- i (k x + ω t)}$ represents the part moving in the negative direction.

We can also use $ψ_{k} (x) = A e^{ik x}$ to represent the wave vector eigenstates, with the sign of $ik x$ indicating the direction of motion.

Note: we need both positive and negative $k$ values to make a general energy eigenstate.

Momentum Eigenstates & de Broglie Relation

The momentum eigenvalue equation is $\overset{p}{^} ψ_{p} (x) = p ψ_{p} (x)$

This is equivalently $\overset{p}{^} ∣ p ⟩ = p ∣ p ⟩$ in braket notation.

Applying $\overset{p}{^}$ to $ψ_{k} (x)$ above, $\overset{p}{^} ψ_{k} (x) = (- i ℏ \frac{d}{d x}) A e^{ik x} = ℏ k ψ_{k} (x)$ implying that $p = ℏ k$ is the momentum eigenvalue with an associated eigenstate $ψ_{p} (x) = A e^{i p x /ℏ}$

Note: this is a function of position $x$ , not momentum $p$ : $x$ is the independent variable while $p$ is the particular momentum eigenvalue.

Since $k = 2 π / λ$ , we can use our momentum eigenvalue $p$ to create the de Broglie relation: $λ_{de Broglie} = \frac{h}{p}$ The momentum eigenstates are also energy eigenstates for our free particle, with energy $E = \frac{p ^{2}}{2 m}$

This means that the momentum and energy operators commute. Any given momentum eigenstate will have some energy given by the above equation, but some energy state doesn't necessarily have a definite momentum, since a single energy state usually corresponds to 2+ momentum states - meaning the energy state is degenerate.

Momentum W.E.

The time-dependent wave equation governing a momentum eigenstate is $ψ_{p} (x, t) = ψ_{p} (x) e^{- i E_{p} t /ℏ} = A e^{i p /ℏ (x - pt /2 m)}$ with a probability density given by $P (x) = ∣ ψ_{p} (x) ∣^{2} = ∣ A ∣^{2}$ Unfortunately, this momentum eigenstate is constant regardless of position - it is spread out over all space unto $\infty$ . This makes the momentum eigenstate impossible to normalize - unless we create a superposition of momentum eigenstates to create wave packets.

Review on Basis States:

All basis state should demonstrate

$⟨ a_{i} ∣ a_{j \neq = i} ⟩ = 0 ⟨ a_{i} ∣ a_{i} ⟩ = 1 i \sum ∣ a_{i} ⟩ ⟨ a_{i} ∣ = 1 orthogonality normalization completeness$ with the orthogonality and normalization conditions able to be written in the form of the Kronecker delta: $⟨ a_{i} ∣ a_{j} ⟩ = δ ij = {01 if i \neq = j, if i = j$ The Dirac delta function is the Kronecker delta used for continuous, rather than discrete sets, where $δ (x - x_{0})$ is a function that is zero for all $x$ , except at $x = x_{0}$ , where it is infinite.

Thus, the orthonormality condition for the momentum eigenstate can be expressed in Dirac notation like this: $⟨ p^{''} ∣ p^{'} ⟩ = δ (p^{''} - p^{'})$ which can be translated into wave function notation using the rules from Chapter 5: $\int_{- \infty}^{\infty} ψ_{p^{''}}^{*} (x) ψ_{p^{'}} (x) d x = δ (p^{''} - p^{'})$ If we define our normalization constant $A = \frac{1}{2 π ℏ}$ then our normalized momentum eigenstates are $ψ_{p} (x) = \frac{1}{2 π ℏ} e^{i p x /ℏ}$

Momentum P.D.

If we wanted to find the probability amplitude for some general state $∣ ψ ⟩$ to have momentum $p$ (i.e the projection of the general state $∣ ψ ⟩$ onto momentum basis $∣ p ⟩$ ), we'd want to find $⟨ p ∣ ψ ⟩ = ψ (p)$ ... however, we're using $p$ in place of $x$ in our wave-function.

... but wait! $ψ (x)$ and $ψ (p)$ are not the same functions. They're the general state operating in different bases.

So ... let's just represent this momentum wave function with a different symbol instead. $ϕ (p) = ⟨ p ∣ ψ ⟩$ This is known as the momentum space wave function, a continuous wave function that represents the quantum state vector in terms of the momentum eigenstates.

The sinusoidal waves that make up this momentum eigenstate can be combined using the Fourier transform of $ϕ (p)$ to form a combined "wave packet" with a location in the $x$ basis, given by $ψ (x) = \frac{1}{2 π ℏ} \int_{- \infty}^{\infty} ϕ (p) e^{i p x /ℏ} d p$ ... which we can also write in the $p$ basis with an inverse Fourier transform: $ϕ (p) = \frac{1}{2 π ℏ} \int_{- \infty}^{\infty} ψ (x) e^{- i p x /ℏ} d x$

Wave Packets

Wave packets are localized superpositions of smaller waves that obey constructive and destructive interference.

The wave packet envelope and carrier often move at different speeds. There is much more to this chapter, but I think I'll call it here for now and write up a study guide for the imminent final.

Chapter 16 - Quantum Computing

In the 1980s, Feynman asked "Can a classical computer reliably model a quantum mechanical system?" - initially, the answer was no, since the Hilbert space scales based on the number $n$ of particles involved by a factor of $2^{n}$ ...

... except ... nature does this anyway, regardless of the complexity involved.

Thus, enter the quantum computer - at least, the idea of it.

Qubits - Quantum Bits

Classical computing uses binary digits (bits) to store information: 0 or 1. These numbers are strung together to represent larger numbers (1000101), but fundamentally use transistors that are either on or off (binary states).

In quantum information systems, information is stored in quantum bits (qubits) - each a two-state system: $∣ 0 ⟩ = ∣ + ⟩ ∣ 1 ⟩ = ∣ - ⟩$ The key difference is that qubits can exist in superposition states - where the state of the processor isn't only 0/1, but rather $∣ ψ ⟩ = c_{0} ∣ 0 ⟩ + c_{1} ∣ 1 ⟩ = (c_{0} c_{1})$ with probabilities of measuring each state as $∣ c_{0} ∣^{2}$ or $∣ c_{1} ∣^{2}$ , as opposed to classically 100% either 0 or 1. As we include more qubits, we exponentially increase the information storage capacity of the system: $∣ 00 ⟩ ∣ 01 ⟩ ∣ 10 ⟩ ∣ 11 ⟩ = ∣ 0 ⟩_{A} ∣ 0 ⟩_{B} = ∣ 0 ⟩_{A} ∣ 1 ⟩_{B} = ∣ 1 ⟩_{A} ∣ 0 ⟩_{B} = ∣ 1 ⟩_{A} ∣ 1 ⟩_{B}$ and the general superposition state is $∣ ψ ⟩ = c_{00} ∣ 00 ⟩ + c_{01} ∣ 01 ⟩ + c_{10} ∣ 10 ⟩ + c_{11} ∣ 11 ⟩$ For an $N$ -qubit system, a single superposition state contains $2^{N}$ coefficients to represent information - compared to a classical system, where though $N$ bits has $2^{N}$ possible states, each state contains only $N$ bits of information.

However - if we measure a state, the system state vector collapses from superposition onto the measured state vector - so we can only extract $N$ pieces of information from our system. We can get around this with quantum parallelism, as a consequence of quantum entanglement.

Entangled States

The EPR state $∣ ψ ⟩ = \frac{1}{2} (∣ + ⟩_{1} ∣ - ⟩_{2} - ∣ - ⟩_{1} ∣ + ⟩_{2})$ is entangled because measurements on one spin are perfectly anti-correlated with measurements on the other spin. It's the fourth of a set of Bell states, the complete set being $∣ β_{00} ⟩ ∣ β_{01} ⟩ ∣ β_{10} ⟩ ∣ β_{11} ⟩ = \frac{1}{2} (∣ 00 ⟩ + ∣ 11 ⟩) = \frac{1}{2} (∣ 01 ⟩ + ∣ 10 ⟩) = \frac{1}{2} (∣ 00 ⟩ - ∣ 11 ⟩) = \frac{1}{2} (∣ 01 ⟩ - ∣ 10 ⟩)$ This set is known as the Bell basis. Measuring one qubit in an entangled state affects the other (regardless of its location) instantaneously, allowing us to achieve the aforementioned parallel measurements - as long as we're clever about data organization.

We can also have 2-qubit product states which are expressed as a product of 1-qubit states $∣ ψ ⟩ = (a_{0} ∣ 0 ⟩_{A} + a_{1} ∣ 1 ⟩_{A}) (b_{0} ∣ 0 ⟩_{B} + b_{1} ∣ 1 ⟩_{B})$

Quantum algorithms are not immune to the probabilistic nature of quantum mechanics - if the same program is run twice on a quantum computer the result may not be the same twice over. However, it allows us to produce answers in many, many fewer steps than a classical computer.

This is why the idea of a binary-quantum processor is popular - both have strengths and weaknesses that each "half" makes up for.

Extinction

This is how we quantify how a star's magnitude changes with the airmass between the observer and the observed star.

Unfortunately, extinction varies with wavelength, since the sky absorbs some wavelengths more intensely than others ... so we'll see different values of $k$ for each filter used.

To correct for this, let $k = k^{'} + k^{''} C$ , where $C$ is a color (i.e. $B - V$ ) - then, $m_{0} = m - k^{'} X - k^{''} CX$ 2nd-order $k^{''}$ extinction is normally quite small (~0.04 magnitudes), but still important to include for high-precision photometry & blue stars. It can be determined with a close pair of different-color stars by fitting the slope of $Δ m$ vs $X Δ C$ (since $Δ C$ is a constant change-in-color for the two) - we end up with $V = v_{0} + μ_{v} (b_{0} - v_{0}) + C_{V}$ $B = b_{0} + μ_{b} (b_{0} - v_{0}) + C_{B}$

Image Reduction Pipeline

Use several Jupyter notebooks, rather than a single "The Master Notebook" - it can get long and big and slow.

Take calibration frames (biases, flats (dome and/or sky) and darks, if necessary)
Observe targets (science, standard stars and airmass-constraint stars (extinction stars))
Remove instrumental signature from observations using calibration frames.
Measure stellar fluxes (i.e. with aperture photometry or PSF photometry)
Solve for extinction coefficients / the space magnitude

Charge-Coupled Devices (CCDs)

The CCD process:

Light hits some detector array, triggering a tiny detector (typically 10-15 $μ m$ in size)
Detector translates energy of the incident photon to some digital value
Software confirms the receipt and reconstructs an array from the values.

High quantum efficiency (electrons per incident photon), and also generally linear (though not as we approach blue, where QE falls off sharply - though total quantum yield (number of electrons extracted per incident photon) increases).

Additional Reference: see the Hubble Space Telescope docs.

CCDs vs CMOS

Note: each element of a camera array is a CCD or a CMOS chip.

CCDs do the photon-to-electron conversion at the CCD chip - then the entire array is converted at some analogue-digital (A/D) converter at the end.
CMOS chips (such as in phones) both detect electrons and have built-in A/D converters on every CMOS array element - so each CMOS chip is able to report its current receipt value.

The problem is CMOS chips heat up quickly, which causes thermal noise on the image. CCDs also generate heat in the A/D chip, but to a much smaller extent which we can correct for a bit using a cooling device.

CCD Noise

Random noise from:

Read noise (incl. A/D 'digitization' noise)
Poisson noise - originates from the quantization of electric charges.

$k$ is the number of photons detected, $P (x = k)$ is ... what?

Systematic noise from:

Bias level - like ISO. Applied before the A/D converter to ensure the read noise is interpretable. Since the received photons will usually follow a Poisson distribution and look like a Gaussian curve, we "shift it" to avoid below-zero counts - corrected for with bias frames.
Dark current - originates from the camera's electrical construction itself, from the random generation of electrons and holes within the depletion region of the camera itself - corrected for with dark frames.
Multiplicative noise - non-uniform illumination, other sources, hot pixels, etc (often called "fixed pattern noise") - removed by using flat frames.

Characterization

We can characterize the noise and linearity of detectors using Photon-Transfer-Curves (PTCs):

Time

Definitions:

Solar time: noon is defined as when the sun crossed the meridian.
Mean solar time: allows each day to have the same length.
- In 1884, London decreed the Mean Solar Time would become Greenwich Mean Time (GMT) ... which then became UTC.
- ... which was then modified to become independent from GMT, now known as UT, as the time when stars cross the meridian (transit).
  - UT Has versions - UT1 is most popular, UT0 or UT2 are also used (rarely).
  - Standard stars used to determine UT.
Transit telescope: slews along the meridian (north / south) to tell when a star moves across - does not slew east-west.
Julian date: count of days since noon on January 1, 4713 BC.
- Created in 1582 by Joseph Scaliger to help make history line up.
- Synchronized with UT (i.e. J2000 is Julian 2000 w.r.t UT)
- MJD is Modified Julian Date - first 2,400,000.5 days subtracted to make the number shorter.
  - i.e. measured since 1857 CE rather than 4713 BCE.

Days:

Sidereal (or stellar days) are measured with respect to the stars. Solar days are measured with respect to the sun - slightly longer because the Earth moves approximately 1 degree around it's orbit.

LST and Hour Angle

Local Sidereal Time (LST) and Hour Angle are the way we navigate using equatorial coordinates.

Spectrographs

Spectrographs:

Take light from a source (into the telescope)
Pass it through a slit
Collimate it (force all beams to pass in a single direction)
Push it through a dispersing element (like a prism to separate it)
Pass it through a camera (to reform it)
Send to the observer

Diffraction gratings are often used as the dispersing element (in place of a prism) because prisms can get really expensive.

Sometimes, folks will sandwich prisms, gratings and another prism, to form a GRISM - this is the case at APO's KOSMOS.

Taking spectra

We don't need great seeing for spectra - so they can be useful to take on nights with low seeing. However, spectra require long exposure times for reasonable S/N ratio in every pixel.

Guiding can be done by a "slit view" camera (i.e. spectrograph in center, normal otherwise) or via an off-axis guider (like at MRO).

Resolving power

For spectra, resolution is defined as

$R \equiv \frac{λ}{Δ λ}$ which can be $\geq 40, 000$ for high-resolution spectra. $R$ is generally limited by slit-width.

Resolution can also be related to Doppler shift via

$R \equiv \frac{λ}{Δ λ} \approx \frac{c}{v}$

Wavelength calibration

Find a lamp spectrum (i.e. using a dome light)
Extract an ID spectrum using the same aperture and trace
Using known lines and their pixel values, determine a dispersion solution
Apply it to the 1-D spectrum

Checklist

Arrival

Log the level of the water tank. The water tank is located down the stairs heading toward the bedrooms, in the utility room on the right. The light to this room is (most inconveniently!) located on the left side of the doorway. Measure the water level by opening the valve to the gauge marked "in. H2O", which is located just above the water pump. The clear tubing will also match the level in the tank.
- Measure each day - refill if under 1/4 full.
Prepare the water system.
- Turn on the water pump with circuit breaker 8 on panel M.
- Flush both toilets, which helps avoid the dreaded sewer backup!
- Check the level of the water tank on a daily basis so as to avoid the (also dreaded) low water alarm.
  - Do not turn on the water heater.
Prepare heat and air conditioning.
1. If heat is needed - flip switch from 'Off' to 'Heat' on per-room basis.
2. If AC needed, uncover the vents outside just north of the main entrance and turn on the AC circuit breaker.
Add potable water to foot pump - in kitchen, put the kitchen tube in one of the big blue water tanks.
Initialize camera - on Sleipnir the Slow, open Evora (http://localhost/ on Sleip, or http://72.233.250.83/ elsewhere on the network).
1. Initialize and set temperature to -82° C.

See Telescope Setup & Calibration for information on preparing the telescope for observation.

Departure

Put away all media items (books, games, cables) in living room & observing room.
Kitchen:
1. Wash dishes & clean up kitchen - wipe the sink down too.
2. Collect trash & recycling to bring back.
3. Refill kitchen and restroom soap dispensers w/ big soap bottle (add water if foaming dispenser).
Restrooms - clean restrooms, wash toilets & sinks, sweep the floors.
Cleaning the floors - vacuum floor upstairs, sweep floor downstairs.
Water cleanup:
1. Measure water level and log it in the Google Doc.
2. Remove tube from water jug under sink & seal it.
Electricity:
1. Turn off water pump circuit breaker.
2. Unplug appliances & turn off power strips (don't turn off observing room computers).
3. Turn off lights.
4. Turn all thermostats off.
Groceries - take pictures of inventories - note anything the next group should pick up from the store.
Water jugs - put empty blue water jugs and empty clear plastic gallon jugs (if you can fit them) in car.
Finalization - close living room shutters & all interior doors, and lock them.

This guide is shortened from the official version, available at https://sites.google.com/a/uw.edu/mro/checklist.

Telescope Setup & Calibration

Initialize camera - on Sleipnir the Slow, open Evora (http://localhost/ on Sleip, or http://72.233.250.83/ elsewhere on the network).
1. Initialize and set temperature to -82° C.
2. Press 'Home' on the filter wheel.
Initialize telescope - open Bifrost on Heimdall.
1. 'Initialize telescope systems'
2. Make sure control paddle is set to set not slew
3. Turn telescope on with key on RA skirt in dome
  1. Confirm 4 green lights in / around mount
Slew to zenith - use bubble level and hand paddle to bring scope to zenith.
1. 'Load Zenith Coordinates' on Bifrost once this is done, then 'Update Telescope Coordinates'

Wait till sunset now.

Open dome
1. Unclip dome power cables
2. Plug in 3-prong lower shutter door motor, and round upper-shutter motor cable (note the upper-shutter cable has a specific orientation to follow).
3. Raise upper shutter until the lower shutter motor turns on. Continue raising upper shutter, but you can now open the lower shutter.
4. Once both are fully open, re-clip power cables where they were.
Remove telescope covers
1. Make sure telescope is at zenith with correct RA/DEC in Bifrost
2. 'Slew to Cover Position' in Bifrost - watch telescope cables. Emergency stop if it looks like anything's going to get caught.
3. Use the ladder to remove the main telescope cover & the 6" finder cover. Store both on south table.
Take bias exposure
1. It should look similar to this example bias.
Take twilight flats
1. Move the telescope ~20 degrees east of zenith (west for morning flats)
Calibrate pointing
1. Load pointing_stars.txt in Bifrost and select a bright start with a low airmass.
2. Turn on tracking, slew to the star and then try to center the star in the finderscope manually with the Xbox remote. Click 'Update Pointing to Target' once it's lined up.
3. Check another bright star nearby to be sure. Fine-tune to get to center-ish of FoV.
Calibrate focus - use a dimmer star for this one
1. Goal is to minimize star PSF. Use focus helper in Evora for this - large leaps in focus increments on Bifrost are okay.
2. Remember: focus is a relative measurement on the telescope, not absolute - "zero" focus is physically wherever the focus was when Bifrost booted up.

End-of-night

If dome flats needed: close dome, turn on dome flat light and point telescope at a relatively matte / flat area. Take flats as needed, then follow procedure below.

Shutdown Evora.
Attach covers
1. 'Slew to Cover Position' in Bifrost, then replace covers. Pay attention to cables.
2. 'Park Telescope' to slew back to zenith
Turn off the telescope with the key, then close Bifrost.
Close dome
1. Rotate dome so cables are near ports
2. Plug cables in - close smaller shutter (3-prong plug) first, followed by upper shutter
3. After dome is closed, unplug and reclip cables.

Calibration Frames

Frame type	Details	Exptime	Num	Periodicity
Bias	Readout noise on CCD - 'static' you see on low exposure times.	0	50-100	Few weeks
Flat	Help mitigate effect of dust and lens vingetting on lights - also acts as per-pixel correction factor.	1-50	~50	Every run
Dark	Try to capture noise from temp and ISO over length of exposure time.	LIGHTs	At least 5, ~20/exp	Every new exp

Filter notes:

Bias frames are filter-irrelevant.
Flat frames are filter-specific - 1 every 5 exptime steps should work.
Rerun darks every once in a while - build a library with ISO, temp and exptime.
For sky flats, take them when sky is as bright (twilight) as possible with least sensitive filter, tracking on, with some offset between each exposure ... OR ... no tracking and allow startrails.
- Allows for variations in star intensities (i.e. frame 1 has a bright star, frame 2 only dark stars)

Telescope notes:

Bifrost does NOT record airmass in FITs header ... nor UTC or RA or DEC.
The MRO telescope will start to create "footballs" at ~12-15 minutes due to imperfect tracking - we could better this with a "real" pointing module.

Resources & Info

Resources, links and reference information for Manastash Ridge Observatory.

Location: 46.9511°N 120.7245°W

Resources

Evora Client - frontend website for controlling the MRO CCD camera.

Evora Server - backend to work with the Andor SDK 2.

Git cheat sheet - really lovely design, covers most use cases

Exoplanet transit calculator (NASA) <-- this should automatically set up for MRO

Exoplanet transit calculator (TESS)

Camera Info

The CCD is an Oxford Instruments Andor iKon-M 934.

Links

Datasheet

Fri. May 24 - Sat. May 25

José, Dylan, Eliz, Parker, Matt, Yasin, Ishan

Arrival

Acquired copious amounts of soda from Fred Meyer due to a bizarre buy-two-get-three deal. Arrived at MRO around 7:00 PM, skies clearing up from earlier rain, mid-50s temperature - hoping for clear skies tonight.

Checklist notes: Water level in tank after turning on the pump at 20% - refill next time.

Night - Fri. May 24

Notes:

Always use the target list to start slewing. Make sure telescope is able to slew from Vega to Arcturus.
Started the night aligning the finder scope with the 0.8m.
Started observations around 3:30 AM - see problems log below.
Nonlabeled flats are sky flats - domeflats are indicated such in header
- Wrote bash script to take domeflats automatically

Object	Exptime	Filter	File	Notes
M57	10	V	ecam-81
M57	50	Ha	ecam-82
M57	30	B	ecam-83	DUST
M57	10	B	ecam-84	DUST :(
GJ 1243	35	r	ecam-(85:105)

Problems:

Focus issues early on that we worked on until midnight - emergency stop of Bifrost software meant we lost track of current focus position. Eventually reset.
Slewing / connection issues with Bifrost
DUST - see ecam-83
Same dust particles different filters? ... filter may not have moved or dust on CCDs.

Master filters

Sat. May 25 - Day

Forgot to disable tracking while doing domeflats - telescope slewed entire night. Spent morning re-acquiring domeflats.

Pulled off CCD to see if dust effects were on it.

Dusty darling	Better

Wish List

Resupply

AA batteries
Recycling bin
Sponges 🙏

Projects

Web interface to interface with both evora-server and Bifrost
- Dome control in web interface
Weather details + webcam in web interface
Autofocus mechanism - APO-style
Multi-filter series exposure planning
- Bash script already set up
Old raspberry pi w/ wifi to act as an audio server (bluetooth and airplay, spot maybe)

Fri. June 21 - Sat. June 22

Oliver, Carter, Bruce, Maggie, Daniel, Thomas, Ruby, Parker, Anika

ASTR 481 Trip - Orientation

Fri. June 21

Arrived to observatory around 3:30 PM - after tour we made (slightly undercooked :( ) baked potatoes on the grill as well as beans / guac - ended up acceptably tasty.

Set up audio server to allow AirPlay, Spotify Play and Bluetooth - see below.

Around 11:00 PM started observatory setup.

Sleipnir didn't boot for a long time - see details below.
After Sleipnir woke up, we ran into issues with telescopepi not starting - the mounted telescopepi is a Pi 2 with an old version of the filter control software. The new version did not work with evora-server. Fixed corrupted Pi 2 with the old filter control software + restarted, no troubles after.
- Old filter control vs. new?

Acquired Vega and Arcturus, until clouds moved in making further observation difficult. Trip snoozed around 3 AM.

Grocery List

Out of white sugar, almost out of brown sugar
Ginger powder as a spice
Paper towels
Aluminum foil
Coffee creamer
Plastic forks, napkins, paper plate

Inventory

Sleipnir

Sleipnir was acting slower than usual - ironically named. Boot hung up on something to do with SGX - apparently okay to ignore, but boot logs also showed some issue with an md5sum failed verification on a portable disk image (paraphrased). StackExchange answers claimed it was probably something to do with Mint, no solutions provided. Booting takes ~3 minutes as a result, in addition to the slowness of ubuntu.

To test whether the bottleneck was software or hardware, installed another distro alongside ubuntu - shrunk ubuntu disk 100 gigabytes and installed manjaro on it. Note - boot drive backup made at /home/mrouser/.backup just in case.

Results: Manjaro booted much faster, but still some of the same slow speeds persisted, even with no services running - so I bet it's hardware. Hard drive is a Toshiba MQ01ACF050 - 7200 rpm, Pentium 4 processor.

Audio Server

Set up an audio server connected to the 3.5mm jack on the vinyl-amp-thing - it supports connections via Bluetooth, AirPlay and Spotify Play, and can be further configured at http://audio.

As it runs on a Raspberry Pi, make sure the little black box with the fan is powered on and shows a red / green lights - if you run into any issues just restart it.

Thurs. Jul 11 - Mon. Jul 15

José, Naim, Parker, Ryan, Anika, Daniel

ASTR 481 Trip - Team 3 Trip 1

Week Goal(s)

Characterize Evora, the iKon-M 934 camera that MRO uses as a CCD imaging camera, by obtaining extinction and transformation coefficients.

While here, our plan is:

At the start of each observing run, take 10 bias frames to act as "per-night calibration frames".
- Thurs. 11
- Fri. 12
- Sat. 13
- Sun. 14
Create a set of <20 dome flats to find the shortest accurate shutter speed without seeing the remnants of the shutter.
- Done Sat. 13
Create a set of <15 sky flats to show linearity from minimum to CCD saturation.
- Done Sat. 13
Take a few darks to estimate Evora's dark current in counts / min.
- Done Fri. 12 & Sat. 13
Take 5 dome flats in $B$ and $V$ filters at around 80% of the point that nonlinearity begins (around 40,000?)
- Done Sat. 13
Take 5 sky flats in $B$ and $V$ filters around 80% of the nonlinearity point (similar to dome flats)
- Done Sat. 13
Observe an extinction star field over the course of several airmasses
- Thurs. 11 & Fri. 12
Observe standard stars at the lowest possible airmass
- Done Fri. 12

... and to observe some DSOs of interest.

Possible Interesting Objects

	Veil Nebula	Iris Nebula	Eagle Nebula	North American Nebula	Ring Nebula	Dumbbell Nebula
Alt. Name	NGC 6960	NGC 7023	M16	NGC 7000	M57	M27

Thurs. June 11

Stopped by Uwijimaya and Fred Meyer along the way to collect supplies for 5 nights' stay. It's nice out today - the skies are clear, temperatures high 70s. No bugs onsight in the observatory which was a welcome change. Started our epic dinner saga off by making sushi rolls!

After sunset, we started by calibrating the telescope. An issue that came up early in the night was that, for some reason after updating pointing to a target (i.e. once we got lined up with Vega), the telescope would stop tracking (without indicating such on Bifrost). This was resolved by just stopping and starting tracking again - still, curious.

Spent a few hours focusing the telescope and making sure it was pointing alright. Spent the remainder of the night taking pictures of GD 336.

Fri. June 12

Quiet day today. Folks started waking up around noon - we added the spare darkroom monitor to the observing room to try getting some more real-estate on Sleipnir.

Garlic brown butter tilapia for dinner - Ryan made some rice and Anika a salad to accompany our meal. Observations started tonight with little fuss - we continued with our extinction field GD 336 initially, then stars in our standard field SA38.

We noticed around 2:00 AM that the red lights in the dome might've been visible in the images themselves (tested with on and off) - decided to keep them on since they'd been on for the whole night thus far. More testing needed.

Left: SA38 with lights on. Right: SA38 with lights off.

To end the night, grabbed some pictures of the Dumbbell Nebula (M27) and the Ring Nebula (M57).

Sat. Jul 13

Waffles for breakfast - started a Smash Bros Subspace Emissary campaign today within which we led a campaign of carnage (on normal mode).

Later, using SIRIL, I processed our pictures of M57 in false color ( $H α$ to red, $B$ to blue and $G$ to green):

And the dumbbell nebula using a similar process

We hadn't applied our calibration frames yet, so I wonder how different they'd look if we did - let's try it once we get the master frames set up. Also each of these was only a combination of the 3 exposures - I wonder if we did a whole bunch of exposures whether we'd get a much higher SNR.

Wishlist item: a non-relative focusing mechanism. The focus seems to "drift" overnight and it can be hard to know when we're on target.

At sunset, we took our sky flats, then some dome flats and darks to finish up our first project report. Since we still had an hour or so until astronomical twilight, we tried training the telescope on the Moon - and saw this!

Since the Evora control software was modified to allow <1s exposure times, we were able to take ~0.3s exposures of the lunar surface.

After this, we sat down and watched the first Alien (1979).

Fun fact, the actor who played Bilbo Baggins in the Lord of the Rings (Ian Holm) was the same who played the science officer, Ash.

Sun. Jul 14

Cinnamon rolls in the AM, then more Brawl. Swamp is a hard level. Cleaned up some in the afternoon, then went on a hike in the evening. I think it was called Umpdenum Falls? When we returned, Ryan made some hotdogs with pickled onion - everyone was famished, so the meal was extra good.

Tried to take a look at some variable stars that needed observation on AAVSO, but we chose one that was a bit too dim (mag 16) to be distinguished easily - it was visible with a ~200 second exposure, but the SNR was so high that it was almost indistinct - so we packed it in for the evening, anticipating an early day on Monday.

Mon. Jul 15

Hoping to return to Seattle with some time to spare, we woke up around 8:00 AM and started cleaning, grabbing breakfast while we ran around getting everything back in order.

Inventory notes made in the class Doc and checklist followed, we packed our gear up and headed out around 8:45 AM, grabbing coffee in Ellensburg on the way back from Jenikka's.

Using the PIC16F88 as an I2C device

Final project for PHYS 335A: Digital Electronics, taught Spring of 2024 by Dr. David Pengra at the University of Washington.

The PIC16F88 is a 16-bit, lightweight microcontroller from Microchip devices which is useful for a variety of low-cost and low-power, high speed embedded systems operations.

Here I describe some details of how I try to set up an I $^{2}$ C implementation on the PIC16F88, and design a program using MPASM Assembly language to operate it as an I2C target device (slave), to be controlled by higher-level controller (master) devices such as Arduinos or Raspberry Pis.

Note: MPASMx is no longer supported as of MPLAB X v5.40 - use >v5.35 if you're intending to use it here. See the footnote¹ for more details on this & using OSx.

Introduction to I2C

I $^{2}$ C (or inter-integrated circuit) is, like so many things in electrical engineering, a complicated-looking protocol that's actually pretty simple in a clever sort of way.

When compared to its closest serial communications cousin SPI (or serial peripheral interface), I $^{2}$ C only requires 2 datalines (SDA and SCL) compared to SPI's 4 (MISO, MOSI, SCLK and CS), and allows for far more devices to be connected due to the use of peripheral addressing rather than SPI's chip select.

Typical I2C implementation diagram The master device / controller, in my case an Arduino Mega 2560, specifies some baudrate to operate the clock frequency on. When no data transmission is happening, both SDA and SCL are pulled to Vdd through the pullup resistors ( $R$ above - typical choices range from 4.7k $Ω$ ) to 10k $Ω$ ).

Since this implementation of I $^{2}$ C is open-drain, the controller and target oscillate lines by just directly grounding them - the pullup resistors avoid shorting the circuit out this way. Note this means that the longer SCL=0 or SDA=0, the more power the circuit will use by extension.

Pullup resistor choice is important if you're trying to optimize - small PU resistors have high power consumption but also high speeds, while large PU resistors have low power consumption but also create capacitance delays. Texas Instruments has a good description of this.

By my understanding, I $^{2}$ C (in 7-bit addressing space²) generally operates like this:

Controller device sends a START bit by holding SDA low with SCL high.
Controller sends a 7-bit address, one bit per clock cycle.
At the 8th bit (8th time SDA goes HIGH), target devices compare the sent address with their own. If the two match, continue. Otherwise, ignore - this stream isn't meant for that device.
Controller and specified target communicate.
END bit set by either to indicate the conversation ifs over.

Each 'communication sequence' happens in 8-bit intervals - 7 bits for data, the 8th allowing time for each device to tell the other what to expect next. The PIC16F88 SSP section of the datasheet has a handy visualization of the target transmitting data back to the host - what we're planning on doing here.

So - let's talk about the project.

I2C Rangefinder Project

We'll use the PIC16F88 in conjunction with the HC-SR04 ultrasonic rangefinder module to create a rangefinding device, retrieving the rangefinder values from the PIC by communicating over I $^{2}$ C.

For our controller, we'll use an Arduino Mega 2560 - chosen because it allows serial console monitoring & has 5V I $^{2}$ C logic by default (the PIC16F88 uses 5V, and thus has 5V I $^{2}$ C outputs), and has a number of high-level logic libraries for interfacing with I $^{2}$ C (see the Wire library for more information).

It is possible to use an I $^{2}$ C controller with 3.3V logic (such as a Raspberry Pi) with 5V target devices or visa-versa, however you'll need to convert logic levels between the two, either with a dedicated logic-level conversion device or by using N-channel MOSFETs and pull-up resistors. Refer to NXP note AN10441 for more information.

Hardware

Making a circuit diagram for this,

For my pullup resistors I decided to use 4.7k $Ω$ , as I'm not particularly worried about power consumption on a proof-of-concept. Note that RA<x> refers to pins PORTA<x>, and similarly RB<x> refer to pins PORTB<x>.

Note: it's quite important that the controller and target devices share a common ground - otherwise, the "reference ground voltage" for I $^{2}$ C pulses may not be the same and a circuit may not function as hoped!

Here's the physical implementation of this circuit.

Software

First, I made sure code for the rangefinder was working before fiddling with I $^{2}$ C. I made use of some of the code provided in Lab 8 - Sonic Ranger with Interrupts & Hardware Control, ensuring the rangefinder worked by tinkering with the debugger and observing variable changes in response to changed echo distances from the rangefinder. This could have also been done by reusing our Lab 7 code, but I thought the Lab 8 code looked much neater.

Now - onto the software side of I $^{2}$ C. The SSP section on I $^{2}$ C in the PIC16F88 datasheet is unfortunately quite lacking with regards to exact implementation methods, so I was forced to scrounge around a bit through sources official and otherwise for information.

In brief, the PIC16F88 doesn't easily support a controller-mode implementation of I $^{2}$ C out of the box, lacking the MSSP register found in other PICs that would allow easier implementation.

Instead, the 'F88 supports a variety of both SPI and I $^{2}$ C target device implementations that may be used instead - refer to Register 10.2 in the datasheet SSP section. If you're looking to implement master-mode on a PIC, look to other devices such as the PIC16F1508 which have MSSP registers.

Using the PIC16F88 datasheet section on SSP (paying particular attention to section 10.3.1) as well as the section on interrupts and the section on configuring PORTB, here's the process I followed to implement "I $^{2}$ C Slave mode, 7-bit address with Start and Stop bit interrupts enabled".

Clear ANSEL to enable digital (instead of analog) inputs.
Set TRISB<1> (SDA) and TRISB<4> (SCL) pins, turning them into inputs.
- "But wait! I $^{2}$ C is bidirectional!" you might argue - this is true. The SSP (synchronous serial port) module in the PIC16F88 will automatically set and clear TRISB<1>/<4> in response to SDA and SCL events.
Set bits in SSPCON (SSP control register):
- SSPCON<5> - SSPEN, enables SSP
- SSPCON<4> - CKP, clock polarity, allowing controller to oscillate SCL
- SSPCON<3:0>=1110 - SSPM<3:0>, "SSP mode select". See Register 10.2 in the SSP section of the datasheet for other options - by using 1110, we're setting our SSP mode to I $^{2}$ C target mode with 7 bit addressing, start/stop interrupts enabled.
Choose an address for your device. It must not be 0x00 or 0x01, other values allowed up to 0xFF. I chose 0x77 arbitarily. Write it into SSPADD.
- Since only bits SSPSR<7:1> are compared to SSPADD, run an RLF instruction (rotate-left-through-carry) on SSPADD to make sure your address matches the checked one.
Enable interrupts. Set bits:
- INTCON<7> - GIE, enable global interrupts
- INTCON<6> - PEIE, enable peripheral interrupts
- PIE1<3> - SSPIE, enable SSP interrupts
Create interrupt functions in your ISR space
- Check if PIR<3> (SSPIF) is set to make sure the interrupt was caused by I $^{2}$ C, if there are multiple possible interrupts in your code.
- Function to Save STATUS and W as you normally would.
- WriteData - to send data, write (less than 8 bits total) to SSPBUF, then set SSPCON<4> to indicate to the controller that you (the target) are ready to transmit. SSPBUF will now start automatically sending.
- Function to Load as you normally would
- Clear PIR1<3> (SSPIF) to reset the SSP ...
... and RETFIE.

My ISR implementation following 6-7 above looks like this:

; Interrupt Service Routine ----------------------------------------------------

	ORG 0x0004 ; ISR beginning

SaveState
	MOVWF SAVE_W ; Save W register
	SWAPF STATUS, W ; Save STATUS reg
	MOVWF SAVE_STAT ; ... into temporary reg

LoadAndSend
	MOVFW TimerCounts ; Load last pulse period into W
	MOVWF SSPBUF ; Load SSPBUF with W (will be sent)
	BSF SSPCON,CKP ; Set CKP bit to indicate our buffer is ready

LoadState
	SWAPF SAVE_STAT,W ; Load STATUS
	MOVWF STATUS
	SWAPF SAVE_W, F ; Load W
	SWAPF SAVE_W, W ; Load W into W
	
	BCF PIR1,SSPIF ; Clear serial interrupt flag
	RETFIE ; ... and return to program execution
	
; End ISR ----------------------------------------------------------------------

and relevant I $^{2}$ C code blocks like this:

; I2C initialization subroutine ------------------------------------------------

SetI2C
    BANKSEL	ANSEL           ; Bank 1
    CLRF	ANSEL           ; Set to all-digital inputs
    BSF		TRISB, TRISB1	; Set PORTB<1> (SDA) as an input
    BSF		TRISB, TRISB4	; Set PORTB<4> (SCL) as an input
    CLRF	SSPSTAT		; Reset SSPSTAT
    
    BANKSEL	SSPCON		; Bank 0
    BSF		SSPCON, SSPEN	; Turn on SSP
    BSF		SSPCON, CKP	; Enable clock (if 0, holds clock low)
    BSF		SSPCON, SSPM3	; I2C 7-bit slave-mode w/ int is SSPM<3:0>=1110
    BSF		SSPCON, SSPM2
    BSF		SSPCON, SSPM1
    BCF		SSPCON, SSPM0
    
    BANKSEL	SSPADD		; Bank 1
    MOVLW	I2C_ADDR	; Load I2C address into W
    MOVWF	SSPADD		; Load SSPADD with I2C_ADDR (0xIC random)
    RLF		SSPADD,F	; Left-shift since SSPSR<7:1> compared
    
    ; Interrupt configuration
    BANKSEL	INTCON		; Bank 0
    BSF		INTCON, GIE	; Enable global interrupts - SSP interrupt on START
    BSF		INTCON, PEIE	; Enable peripheral interrupts
    BANKSEL	PIE1		; Select PIE register
    BSF		PIE1, SSPIE	; Enable SSP interrupts in peripheral interrupt reg
	
; End subroutine ---------------------------------------------------------------

Results

Using the Arduino IDE with the Wire library, I set up a basic script that, after initializing I $^{2}$ C and the serial monitor both at a baudrate of 9600, would send an I $^{2}$ C "read" query down SDA to address 0x77 (our PIC) to try and retrieve our TimerCounts value from our interrupt function LoadAndSend shown above.

#include <Wire.h>

// Notes: D21 is SCL, D20 is SDA for MEGA 2560

void setup() {
  // Initialize I2C
  Wire.begin();
  
  // Start serial output for console monitoring
  Serial.begin(9600);  
  Serial.println("Setup complete. Starting loop ...");
}

// Poll 0x77 address - then sleep for two seconds - then poll.
void loop() {
  // Request 2 bits from 0x77
  Wire.requestFrom(0x77, 1);

  // Send results (if any) to serial monitor
  while (Wire.available()) {
    char c = Wire.read();
    Serial.print("Distance: ");
    Serial.print(c, DEC);
    Serial.println(" cm");
  }

  delay(2000);
}

And, after some tinkering, checking the serial monitor revealed ...

Voilà!!

Notes

I definitely didn't get this first try - I wrote the Arduino script pretty early on to monitor the PIC16F88 while I figured out the maze of interrupt flags necessary to work with I $^{2}$ C.

Something I should note - I've wasted at least a few hours during this project because I wasn't operating on the correct bank while trying to operate on some register. Such problems won't often be immediately apparent, only becoming clear when you step through your code with a debugger and realize a value isn't changing when it should. BANKSEL is a friend!

Code

The full program I used for this is written below.

;*******************************************************************************
;                                                                             
;    Filename: Final Project I2C Ranger                                       
;    Date: 5/31/2024                                                           
;    File Version: 1.0                                                            
;    Author: Parker Lamb                                                                   
;    Description: Sets PIC16F88 up as I2C-ready ultrasonic rangefinding device.
;                                                                              
;*******************************************************************************

;*******************************************************************************
;
;                Procesor initial setup
;
;*******************************************************************************
    
    list       F=inhx8m, P=16F88, R=hex, N=0 ; File format, chip, and default radix

#include p16f88.inc ;   PIC 16f88 specific register definitions

    __config _CONFIG1, _MCLR_ON & _FOSC_INTOSCCLK & _WDT_OFF & _LVP_OFF & _PWRTE_OFF & _BODEN_ON & _LVP_OFF & _CPD_OFF & _WRT_PROTECT_OFF & _CCP1_RB0 & _CP_OFF
    __config _CONFIG2 , _IESO_OFF & _FCMEN_OFF

    Errorlevel -302 ; switches off msg [302]: Register in operand not in bank 0.

;*******************************************************************************
;
;                Constants and variables
;
;*******************************************************************************

; Program vars
TimerCounts EQU     h'20'	; Saving timer counts
 
;  vars
SAVE_W	    EQU	    h'21'	; Interrupt temporary W storage
SAVE_STAT   EQU	    h'22'	; Interrupt temporary STATUS FSR storage
   
; I2C status
I2C_STAT    EQU	    h'23'	; Check if I2C is connected
I2C_ADDR    EQU	    0x77	; I2C address (const)
	    
; Delay count registers
DInd1	    EQU	    h'24'
DInd2	    EQU	    h'25'
	    
; Delay times
DTime1	    EQU	    .199	; 60 ms delay - outer loop
DTime2	    EQU	    .60		; Nested loop runs for 59,941 cycles
	    
; SCL and SDA locations
#define	    _SDA    PORTB,RB1	; Make _SDA easier to check
#define	    _SCL    PORTB,RB4	; Make _SCL easier to check
	    
;*******************************************************************************
;
;                Memory init & interrupts
;
;*******************************************************************************
 
	    ORG	    0x00
	    GOTO    Init
	    
; Interrupt Service Routine ----------------------------------------------------
            ORG     0x0004               ; ISR beginning
		    
SaveState
    MOVWF	SAVE_W		; Save W register
    SWAPF	STATUS, W	; Save STATUS reg
    MOVWF	SAVE_STAT	; ... into temporary reg
    
LoadAndSend
    MOVFW	TimerCounts	; Load last pulse period into W
    MOVWF	SSPBUF		; Load SSPBUF with W (will be sent)
    BSF		SSPCON,CKP	; Set CKP bit to indicate our buffer is ready
    
LoadState
    SWAPF	SAVE_STAT,W	; Load STATUS
    MOVWF	STATUS
    SWAPF	SAVE_W, F	; Load W
    SWAPF	SAVE_W, W	; Load W into W
    
    BCF		PIR1,SSPIF
    RETFIE			; ... and return to program execution

;*******************************************************************************
;
;                Program start
;
;*******************************************************************************
Init	    ORG	    0x0020
	    
; Set up oscillator to 4 MHz
SetOsc_4MHz 
    BANKSEL	OSCCON
    CLRF	OSCCON
    BSF		OSCCON, IRCF1   ; Bit 5
    BSF		OSCCON, IRCF2   ; Bit 6

    ; Tuned value from Lab 7 - adjust as needed
    MOVLW	0x16
    MOVWF	OSCTUNE

; Reset IO to all digital, all outputs, clear latches
ResetIO
    BANKSEL	PORTA   ; Clear data latch registers
    CLRF	PORTA
    CLRF	PORTB
    BANKSEL	TRISA   ; Data direction on PORTA / PORTB
    CLRF	TRISA   ; Set PORTA to all output
    CLRF	TRISB   ; Set PORTB to all output
    BANKSEL	ANSEL   ; Select digital / analogue register
    CLRF	ANSEL   ; And set to all-digital inputs
	  
; ... and set what we need to
SetIO
    BANKSEL	TRISA
    MOVLW	0xFF
    MOVWF	TRISA		; Set PORTA direction to all inputs
    BCF		TRISA, RA0	; ... except PORTA<0>, an output
 
SetTMR0
    ; TMR0 duration between pulses indicates our radar distance
    BANKSEL	OPTION_REG	; TMR0 operation controlled via OPTION_REG
    CLRF	OPTION_REG	; Clear it - sets PSA to TMR0, low-hi edge
    BSF		OPTION_REG, 0x0
    BSF		OPTION_REG, 0x2	; Set prescaler rate to 1:64 w.r.t oscillator (101)
    BANKSEL	INTCON		; Select interrupt control register
    BCF		INTCON,TMR0IE	; Disable TMR0 OF interrupt (not using here)
    BCF		INTCON,TMR0IF	; Clear flag (maybe unnecessary - do anyway)
    
;*******************************************************************************
;
;                I2C configuration
;   We're using this device in 7-bit I2C slave-mode with stop + start
;   interrupts enabled. 
;
;*******************************************************************************
    
SetI2C
    BANKSEL	TRISB		; Bank 1
    BSF		TRISB, TRISB1	; Set PORTB<1> (SDA) as an input
    BSF		TRISB, TRISB4	; Set PORTB<4> (SCL) as an input
    
;    BANKSEL	SSPSTAT		; Bank 1
    CLRF	SSPSTAT		; Reset SSPSTAT
    
    BANKSEL	SSPCON		; Bank 0
    BSF		SSPCON, SSPEN	; Turn on SSP
    BSF		SSPCON, CKP	    ; Enable clock (if 0, holds clock low)
    BSF		SSPCON, SSPM3	; I2C 7-bit slave-mode w/ int is SSPM<3:0>=1110
    BSF		SSPCON, SSPM2
    BSF		SSPCON, SSPM1
    BCF		SSPCON, SSPM0
    
    BANKSEL	SSPADD		; Bank 1
    MOVLW	I2C_ADDR	; Load I2C address into W
    MOVWF	SSPADD		; Load SSPADD with I2C_ADDR (0xIC random)
    RLF		SSPADD,F	; Left-shift since SSPSR<7:1> compared
    
    ; Interrupt configuration
    BANKSEL	INTCON		; Bank 0
    BSF		INTCON, GIE	; Enable global interrupts - SSP interrupt on START
    BSF		INTCON, PEIE	; Enable peripheral interrupts
    BANKSEL	PIE1		; Select PIE register
    BSF		PIE1, SSPIE	; Enable SSP interrupts in peripheral interrupt reg

;*******************************************************************************
;
;                Main program loop
;   Cause the sonic module to pulse every 10 microseconds. Wait for a response,
;   then save time it took into a register. Repeatedly do this.
;
;   Once I2C interrupt is detected, send information via I2C. 
;
;   TODO only start this loop if START condition detected, stop with I2C STOP.
;
;*******************************************************************************
    ; Reset to bank 0
    BCF	    STATUS, RP0
    BCF	    STATUS, RP1	    ; All of the action is in Bank 0 now
	    
MainLoop    
; Send sonic device a 10-microsecond pulse
Pulse
    BSF		PORTA, RA0	; Set PORTA<0> output to high
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    BCF		PORTA, RA0	; Set PORTA<0> output low

; Loop until PORTA<1> goes HI, indicating a response
WaitForResp
    BTFSS	PORTA, RA1	; Check PORTA<1>
    GOTO	WaitForResp	; ... and loop if it's still zero

    ; Start of our response
    CLRF	TMR0		; Start timer from now
WaitUntilLow
    BTFSC	PORTA, RA1	; Check PORTA<1> to see if response finished
    GOTO	WaitUntilLow	; ... loop if still not LOW
    
    ; We have a response! Store timer value in a variable
    MOVFW	TMR0
    MOVWF	TimerCounts
    
    ; Delay for a bit so we aren't constantly polling
    CALL	Delay
    
    ; Return to MainLoop
    GOTO	MainLoop

;*******************************************************************************
;
;                Subroutines
;
;*******************************************************************************
; Loop from Template_for_Ranger_with_interrupts.asm, course lab template
Delay
		MOVLW   DTime2
		MOVWF   DInd2
Loop1		MOVLW   DTime1
		MOVWF   DInd1
Subloop1	NOP
		NOP
		DECFSZ  DInd1,F
		GOTO    Subloop1
		DECFSZ  DInd2,F
		GOTO    Loop1
	RETURN
    
Finish
    END

The code for the Mega 2560 is written in Results.

With version v5.40 of MPLAB X, Microchip's dedicated compiler/IDE, Microchip upgraded all their binaries to 64-bit. However, since mpasmx was not updated from 32 bits, and Mac OSx does not support 32-bit applications since Mojave, Mac users will need to either use Windows or a VM.

See Wikipedia's page for addressing structure for more details.

UW Notes & Docs