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 $F_z=\frac{\delta }{\delta z}(\vec{\mu }\cdot \vec{B})={\mu }_z\frac{\delta B_z}{\delta z}$ (ref: Wikipedia entry on magnetic fields), where $\vec{\mu }$ is the magnetic moment and $\vec{B}$ 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 $\frac{\delta B_z}{\delta z}$ known to be constant, then by the results of the experiment we must have two values for ${\mu }_z$: $+{\mu }_z$ and $-{\mu }_z$ (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 $\mu =IA$. Here, $A=\pi r^2$ and $I=\frac{q}{2\pi r/v}$, so $$\mu =IA=\frac{qrv}{2}$$ Thus, (classically), we can imagine each charge as having some "orbital angular momentum" $L$ (like planets around a star). Angular momentum is $L=mrv$, so we can rewrite this as $$\mu =\frac{q}{2m}L$$ 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 $\mu$ as $$\vec{\mu }=g\frac{q}{2m}\vec{S}$$ 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, $${\mu }_z=-g\frac{e}{2m_e}{S}_z,F_z=-g\frac{e}{2m_e}{S}_z\frac{d{B}_z}{dz}$$ and, to achieve the results we observe in the experiment, we can have only two values of $S_z$: $$S_z=\pm \frac{\hslash }{2}$$ where $\hslash$ is a modified Planck's constant $\hslash =h/2\pi =1.0546\times 10^{-34}\text{J*s}$. 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", $|\psi ⟩$.

In the spin-1/2 system,

$$|\psi ⟩=a|+⟩+b|-⟩$$ where $|+⟩$ is the quantum state of the atoms that are spin-up $S_z=+\hslash /2$, and $|-⟩$ is the quantum state of the spin-down atoms $S_z=-\hslash /2$.

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.

1. 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. $${\mathcal{P}}_{+}=|⟨+|+⟩{|}^2=1$$ $${\mathcal{P}}_{-}=|⟨-|-⟩{|}^2=0$$

2. Experiment 2: Analyze spin in $z$, then in $x$ - ${|\psi ⟩}_x$ was found to be totally independent whether we used ${|+⟩}_z$ or ${|-⟩}_z$. $${|+⟩}_x=\frac{1}{\sqrt{2}}\left[|+⟩+|-⟩\right]{\mathcal{P}}_{+,x}=\frac{1}{2}$$ $${|-⟩}_x=\frac{1}{\sqrt{2}}\left[|+⟩-|-⟩\right]{\mathcal{P}}_{-,x}=\frac{1}{2}$$ Complicated, though - this is a mixed state, rather than the superposition of experiment 1. We'll investigate options to make it nicer later.

3. 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 $z$.

4. 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 $|\psi ⟩=a|+⟩+b|-⟩$ represents a complete basis with dimensionality 2.

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

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

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

Likewise, $⟨\psi |+⟩=a^{\ast }$ and $⟨+|\psi ⟩={⟨\psi |+⟩}^{\ast }$.

3. All quantum state vectors must be normalized, such that $$⟨\psi |\psi ⟩=|⟨+|\psi ⟩{|}^2+|⟨-|\psi ⟩{|}^2=|a{|}^2+|b{|}^2=1$$

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

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.

4. The complex constants $a$ and $b$ when squared (i.e. $|a{|}^2=|⟨+|\psi ⟩{|}^2$ ) 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 $$|+⟩\equiv \left(\begin{array}{c}1\\ 0\end{array}\right)|-⟩\equiv \left(\begin{array}{c}1\\ 0\end{array}\right)$$ and $$|\psi ⟩=\left(\begin{array}{c}⟨+|\psi ⟩\\ ⟨-|\psi ⟩\end{array}\right)=\left(\begin{array}{c}a\\ b\end{array}\right)$$ with the corresponding bra represented by the row vector $$⟨\psi |=\left(\begin{array}{cc}{a}^{\ast }& b^{\ast }\end{array}\right)$$ so

$$⟨\psi |\psi ⟩=\left(\begin{array}{cc}{a}^{\ast }& b^{\ast }\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=|a{|}^2+|b{|}^2$$

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,

$$⟨a_i|a_j⟩={\delta }_{ij}$$ $$|\psi ⟩=\sum _i⟨a_i|\psi ⟩|a_i⟩$$

Note: $⟨a_i|\psi ⟩$ represents each corresponding complex scalar multiple, and ${\delta }_{ij}$ is the Kronecker delta, which is 1 if $i=j$ and 0 if $i\ne j$.

Converting kets between axes

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

$${|+⟩}_z{|+⟩}_x=\frac{1}{\sqrt{2}}\left[|+⟩+|-⟩\right]{|+⟩}_y=\frac{1}{\sqrt{2}}[|+⟩+i|-⟩]$$ $${|-⟩}_z{|-⟩}_x=\frac{1}{\sqrt{2}}\left[|+⟩-|-⟩\right]{|-⟩}_y=\frac{1}{\sqrt{2}}[|+⟩-i|-⟩]$$ In matrix form, $${|+⟩}_z\equiv \left(\begin{array}{c}1\\ 0\end{array}\right){|+⟩}_x\equiv \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right){|+⟩}_y\equiv \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ i\end{array}\right)$$ $${|-⟩}_z\equiv \left(\begin{array}{c}0\\ 1\end{array}\right){|-⟩}_x\equiv \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\end{array}\right){|-⟩}_y\equiv \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -i\end{array}\right)$$

The Postulates of Quantum Mechanics

Postulate 1: the state of a quantum mechanical system, $|\psi ⟩$, 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 $|\psi ⟩$ is g $${\mathcal{P}}_{a_n}=|⟨a_n|\psi ⟩{|}^2$$

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 $|\psi ⟩$ onto the ket (or kets) corresponding to the result of the measurement $|{\psi }^{\prime }⟩$ $$|{\psi }^{\prime }⟩=\frac{P_n|\psi ⟩}{\sqrt{⟨\psi |P_n|\psi ⟩}}$$

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

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