Introduction to Quantum Mechanics · Problem 4.73

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · stern_gerlach_ehrenfest

Problem

Example 4.4, couched in terms of forces, was a quasi-classical explanation for the Stern–Gerlach effect. Starting from the Hamiltonian for a neutral, spin-1/2 particle traveling through the magnetic field given by Equation 4.169, H=p22m−γ B⋅S,H = \frac{p^2}{2m} - \gamma\,\mathbf{B} \cdot \mathbf{S}, use the generalized Ehrenfest theorem (Equation 3.73) to show that md2dt2⟨z⟩=γ α ⟨Sz⟩.m\frac{d^2}{dt^2}\langle z\rangle = \gamma\,\alpha\,\langle S_z\rangle. Comment: Equation 4.170 is therefore a correct quantum-mechanical statement, with the understanding that the quantities refer to expectation values.

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