The Stern-Gerlach experiment

# The setup

                                   ┌─────────┐
                                   │    N    │
  ┌────────┐                       └─────────┘        ▲ z
  │   Ag   │        Ag beam            ↓   ↓          │
  │  oven  │ ─────────────────────────────────────►   │
  └────────┘                           ↓   ↓
                                   ┌─────────┐
                                   │    S    │
                                   └─────────┘

The Ag atom has an angular momentum solely due to its outer $e^-$ and spin $\implies$ intrinsic.

$$ \therefore \vec\mu_{\text{atom}} \propto \vec S_{\text{electron}} $$

turns out:

$$ \boxed{\vec\mu_{\text{atom}} = \frac{e}{m_e c}\vec S} $$

$F_z$ due to the field:

$$ F_z = \frac{\partial}{\partial z}\left(\vec\mu\cdot\vec B\right) \simeq \mu_z\frac{\partial B_z}{\partial z} $$

ignoring $\vec B$ in other directions.

$\therefore$ a particle with $\mu_z > 0$ ($S_z < 0$) experiences a downward force and $\mu_z < 0$ ($S_z > 0$) upwards.

Then an SG$\hat z$ is measuring the z-component of $\vec\mu$, or the z-component of $\vec S$ upto a proportionality factor.

                  ┌──────────┐   Sz+     ┌──────────┐   Sx+
   Ag  ─────────► │    SG    │ ────────► │    SG    │ ────────►
                  │  z-axis  │ ──► ■     │  x-axis  │ ────────►
                  └──────────┘   Sz-     └──────────┘   Sx-
                              (blocked)

$$ S_z = \pm\frac{\hbar}{2} $$

↳ splitting into two beams $\to$ referred to as space quantization.

# Sequential Stern-Gerlach experiments

               ┌───────┐  Sz+     ┌───────┐  Sx+     ┌───────┐  Sz+
   Ag  ──────► │  SG   │ ───────► │  SG   │ ───────► │  SG   │ ───────►
               │   z   │ ──► ■    │   x   │ ──► ■    │   z   │ ───────►
               └───────┘  Sz-     └───────┘  Sx-     └───────┘  Sz-

The selection of the $S_x+$ beam (or $S_x-$) destroys all information about $S_z$ stored previously. The last SG$\hat z$ splits into $S_z+$ and $S_z-$ again, even though we had blocked $S_z-$ at the start!

For every measurement you make, the other non-commuting observables are in a superposition of all possible states.

# Analogies from polarized light

Drawing analogies from polarized light,

$$ \begin{aligned} |S_x;\pm\rangle &\stackrel{?}{=} \pm\frac{1}{\sqrt{2}}|S_z;+\rangle + \frac{1}{\sqrt{2}}|S_z;-\rangle \\ |S_y;\pm\rangle &\stackrel{?}{=} \frac{1}{\sqrt{2}}|S_z;+\rangle \pm \frac{i}{\sqrt{2}}|S_z;-\rangle \end{aligned} $$

These are vectors in some high dimensional field (the Hilbert space). We’ll actually derive these later in measurements and the S-G experiment.

Next: Kets and the Hilbert space →