# 1.104: 88. Related Analysis of the Stern-Gerlach Experiment


Silver atoms are deflected by an inhomogeneous magnetic field because of the two-valued magnetic moment associated with their unpaired 5s electron ([Kr]5s14d10). The beam of silver atoms entering the Stern-Gerlach magnet oriented in the z-direction (SGZ) on the left is unpolarized. This means it is a mixture of randomly polarized Ag atoms. A mixture cannot be represented by a wave function, it requires a density matrix, as will be shown later.

This situation is exactly analogous to the three-polarizer demonstration. Light emerging from an incadescent light bulb is unpolarized, a mixture of all possible polarization angles, so we can't write a wave function for it. The first Stern-Gerlach magnet plays the same role as the first polarizer, it forces the Ag atoms into one of measurement eigenstates - spin-up or spin-down in the z-direction. The only difference is that in the three-polarizer demonstration only one state was created - vertical polarization. Both demonstrations illustrate that the only values that are observed in an experiment are the eigenvalues of the measurement operator.

To continue with the analysis of the Stern-Gerlach demonstration we need vectors to represent the various spin states of the Ag atoms.

## Spin Eigenfunctions

Spin-up in the z-direction:

$\alpha_z = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \nonumber$

Spin-down in the z-direction:

$\beta_z = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \nonumber$

Spin-up in the x-direction:

$\alpha_x = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \nonumber$

Spin-down in the x-direction:

$\beta_x = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \nonumber$

In the next step, the spin-up beam (deflected toward by the magnet's north pole) enters a magnet oriented in the x-direction, SGX. The αz beam splits into αx and βx beams of equal intensity. This is because it is a superposition of the x-direction spin eigenstates as shown below.

$\begin{matrix} \frac{1}{ \sqrt{2}} \left[ \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} + \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \right] \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix} & \frac{1}{ \sqrt{2}} \left( \alpha_x + \beta_x \right) \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{matrix} \nonumber$

Next the αx beam is directed toward a second SGZ magnet and splits into two equal αz and βz beams. This happens because αx is a superposition of the αz and βz spin states.

$\begin{matrix} \frac{1}{ \sqrt{2}} \left[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right] = \begin{pmatrix} 0.707 \\ 0.707 \end{pmatrix} & \frac{1}{ \sqrt{2}} \left( \alpha_z + \beta_z \right) = \begin{pmatrix} 0.707 \\ 0.707 \end{pmatrix} \end{matrix} \nonumber$

## Operators

We can also use the Pauli operators (in units of h/4π) to analyze this experiment.

SGZ operator:

$\text{SGZ} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \nonumber$

SGX operator:

$\text{SGX} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \nonumber$

The probability that an αz Ag atom will emerge spin-up after passing through a SGX magnet:

Probability amplitude:

$\alpha_x^T \text{SGX} \alpha_z = 0.707 \nonumber$

Probability:

$\left( \alpha_x^T \text{SGX} \alpha_z \right)^2 = 0.5 \nonumber$

The probability that an αz Ag atom will emerge spin-down after passing through a SGX magnet:

Probability amplitude:

$\beta_x^T \text{SGX} \alpha_z = -0.707 \nonumber$

Probability:

$\left( \beta_x^T \text{SGX} \alpha_z \right)^2 = 0.5 \nonumber$

The probability that an αx Ag atom will emerge spin-up after passing through a SGZ magnet:

Probability amplitude:

$\alpha_z^T \text{SGX} \alpha_x = 0.707 \nonumber$

Probability:

$\left( \alpha_z^T \text{SGX} \alpha_x \right)^2 = 0.5 \nonumber$

The probability that an αx Ag atom will emerge spin-down after passing through a SGZ magnet:

Probability amplitude:

$\beta_z^T \text{SGX} \alpha_x = 0.707 \nonumber$

Probability:

$\left( \beta_z^T \text{SGX} \alpha_x \right)^2 = 0.5 \nonumber$

In examining the figure above we note that the SGX magnet distroys the entering αz state, creating a superposition of spin-up and spin-down in the x-direction. Again measurement forces the system into one of the eigenstates of the measurement operator.

## Density Operator (Matrix) Approach

A more general analysis is based on the concept of the density operator (matrix), in general given by the following outer product |Ψ >< Ψ|. It is especially important because it can be used to represent mixtures, which cannot be represented by wave functions as noted above.

For example, the probability that an αz spin system will emerge in the αx channel of a SGX magnet is equal to the trace of the product of the density matrices representing the αz and αx states as shown below.

$\left| \left \langle \alpha_x | \alpha_z \right \rangle \right|^2 = \left \langle \alpha_z | \alpha_x \right \rangle\left \langle \alpha_x | \alpha_z \right \rangle = \sum_i \left \langle \alpha_z | i \right \rangle \left \langle i | \alpha_x \right \rangle \left \langle \alpha_x | \alpha_z \right \rangle = \sum_i \left \langle i | \alpha_z \right \rangle \left \langle \alpha_x | \alpha_z \right \rangle \left \langle \alpha_z | i \right \rangle = Tr \left( | \alpha_x \rangle \langle \alpha_x | \alpha_z \rangle \langle \alpha_z | \right) = Tr \left( \widehat{ \rho_{ \alpha x}} \widehat{ \rho_{ \alpha z}} \right) \nonumber$

where the completeness relation $$\sum_{i} |i \rangle \langle i | =1$$ has been employed.

Density matrices for spin-up and spin-down in the z-direction:

$\begin{matrix} \rho_{ \alpha z} = \alpha_z \alpha_z^T & \rho_{ \beta z} = \beta_z \beta_z^T \end{matrix} \nonumber$

Density matrices for spin-up and spin-down in the x-direction:

$\begin{matrix} \rho_{ \alpha x} = \alpha_x \alpha_x^T & \rho_{ \beta x} = \beta_x \beta_x^T \end{matrix} \nonumber$

An unpolarized spin system can be represented by a 50-50 mixture of any two orthogonal spin density matrices. Below it is shown that using the z-direction and the x-direction give the same answer.

$\rho_{mix} = \frac{1}{2} \rho_{ \alpha z} + \frac{1}{2} \rho_{ \beta z} = \begin{pmatrix} 0.5 & 0 \\ 0 & 0.5 \end{pmatrix} \nonumber$

Now we re-analyze the Stern-Gerlach experiment using the density operator (matrix) approach.

The probability that an unpolarized spin system will emerge in the αz channel of a SGZ magnet is 0.5:

$\text{tr} \left( \rho_{ \alpha z} \rho_{ \text{mix}} \right) = 0.5 \nonumber$

The probability that the αz beam will emerge in the αx channel of a SGX magnet is 0.5:

$\text{tr} \left( \rho_{ \alpha x} \rho_{ \alpha z} \right) = 0.5 \nonumber$

The probability that the αx beam will emerge in the αz channel of the final SGZ magnet is 0.5:

$\text{tr} \left( \rho_{ \alpha z} \rho_{ \alpha x} \right) = 0.5 \nonumber$

The probability that the αx beam will emerge in the βz channel of the final SGZ magnet is 0.5:

$\text{tr} \left( \rho_{ \beta x} \rho_{ \alpha z} \right) = 0.5 \nonumber$

After the final SGZ magnet, 1/8 of the original Ag atoms emerge in the αz channel and 1/8 in the βz channel.

$\begin{matrix} \text{tr} \left( \rho_{ \alpha z} \rho_{ \alpha x} \right) \text{tr} \left( \rho_{ \alpha x} \rho_{ \alpha z} \right) \text{tr} \left( \rho_{ \alpha z} \rho_{ \text{mix}} \right) = 0.125 & \text{tr} \left( \rho_{ \beta z} \rho_{ \alpha x} \right) \text{tr} \left( \rho_{ \alpha x} \rho_{ \alpha z} \right) \text{tr} \left( \rho_{ \alpha z} \rho_{ \text{mix}} \right) = 0.125 \end{matrix} \nonumber$

1.104: 88. Related Analysis of the Stern-Gerlach Experiment is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.