Autumn · project 9 of 20

The Stern–Gerlach experiment

Topic: Spin and measurement

Goal: Study the trajectories of a spin-1/2 particle in an inhomogeneous magnetic field \(\mathbf{B} = \nabla \Phi(x, y, z)\) using the Pauli equation \[i\frac{\partial \psi}{\partial t} = \left(\frac{(\widehat{\boldsymbol{P}} - q\boldsymbol{A})^2}{2m} - \frac{q}{2m} \boldsymbol{\sigma} \cdot \boldsymbol{B}\right)\psi\] and Ehrenfest’s theorem, and analyze how the initial spin, the inhomogeneity of the field and the initial parameters of the particle affect its motion.

  1. Equations of motion: Use Ehrenfest’s theorem to compute the mean values of the coordinates and momentum: \[\frac{d\langle \mathbf{r} \rangle}{dt} = \frac{\langle \mathbf{p} \rangle}{m}, \quad \frac{d\langle \mathbf{p} \rangle}{dt} = \langle \nabla (\boldsymbol{\mu} \cdot \boldsymbol{B}) \rangle,\] where \(\boldsymbol{\mu} = \frac{q}{2m} \boldsymbol{\sigma}\) is the magnetic moment of the particle (the Lorentz force is neglected here).

  2. Initial conditions:

    1. initial coordinates: \(\langle \mathbf{r}(0) \rangle = (0, 0, 0)\);

    2. initial momentum: \(\langle \mathbf{p}(0) \rangle = (p, 0, 0)\);

    3. initial spin: along the \(+z\) axis (\(\sigma_z = +1\)) or the \(-z\) axis (\(\sigma_z = -1\)).

  3. Magnetic field: Consider an inhomogeneous magnetic field given, for example, by the scalar potential \(\Phi(x, y, z)\): \[\mathbf{B} = \nabla \Phi(x, y, z), \quad \Phi(x, y, z) = B_0 z + \frac{B_1}{2} (x^2 - y^2).\] This field approximately models a real Stern–Gerlach apparatus.

  4. Trajectories: Solve numerically the equations of motion for the mean coordinates \(\langle x(t) \rangle\), \(\langle y(t) \rangle\), \(\langle z(t) \rangle\). Plot the trajectories of particles with spin along \(+z\) and \(-z\).

  5. Plots:

    1. the trajectories \(\langle x(t) \rangle\) and \(\langle z(t) \rangle\) for different values of the initial momentum \(p\), the field strength \(B_0\) and the gradient \(B_1\);

    2. a comparison of the deflections of particles with different initial spins (\(+z\) and \(-z\)).

  6. Analysis: Explain how the inhomogeneity of the field (\(B_1\)) and the parameters of the particle (\(p\), \(m\), initial spin) affect the trajectories. Explain why the Lorentz force does not affect the separation of the trajectories in this experiment.

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