Introduction to Quantum Mechanics · Problem 4.74

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · stern_gerlach_schrodinger

Problem

Neither Example 4.4 nor Problem 4.73 actually solved the Schrödinger equation for the Stern–Gerlach experiment. In this problem we will see how to set up that calculation. The Hamiltonian for a neutral, spin-1/2 particle traveling through a Stern–Gerlach device is H=p22m−γ B⋅SH = \frac{p^2}{2m} - \gamma\,\mathbf{B} \cdot \mathbf{S} where B\mathbf{B} is given by Equation 4.169. The most general wave function for a spin-1/2 particle—including both spatial and spin degrees of freedom—is Ψ(r,t)=Ψ+(r,t) χ++Ψ−(r,t) χ−.\Psi(\mathbf{r}, t) = \Psi_+(\mathbf{r}, t)\,\chi_+ + \Psi_-(\mathbf{r}, t)\,\chi_-. (a) Put Ψ(r,t)\Psi(\mathbf{r}, t) into the Schrödinger equation H Ψ=iℏ∂∂tΨH\,\Psi = i\hbar\frac{\partial}{\partial t}\Psi to obtain a pair of coupled equations for Ψ±\Psi_\pm. Partial answer: −ℏ22m∇2Ψ+−ℏ2γ(B0+αz)Ψ++ℏ2γαx Ψ−=iℏ∂∂tΨ+.-\frac{\hbar^2}{2m}\nabla^2\Psi_+ - \frac{\hbar}{2}\gamma(B_0 + \alpha z)\Psi_+ + \frac{\hbar}{2}\gamma\alpha x\,\Psi_- = i\hbar\frac{\partial}{\partial t}\Psi_+. (b) We know from Example 4.3 that the spin will precess in a uniform field B0k^B_0\hat{k}. We can factor this behavior out of our solution—with no loss of generality—by writing Ψ±(r,t)=e±iγB0t/2 Ψ~(r,t).\Psi_\pm(\mathbf{r}, t) = e^{\pm i\gamma B_0 t/2}\,\tilde{\Psi}(\mathbf{r}, t). Find the coupled equations for Ψ~±\tilde{\Psi}_\pm. Partial answer: −ℏ22m∇2Ψ~+−ℏ2γαz Ψ~++ℏ2γαx e−iγB0tΨ~−=iℏ∂∂tΨ~+.-\frac{\hbar^2}{2m}\nabla^2\tilde{\Psi}_+ - \frac{\hbar}{2}\gamma\alpha z\,\tilde{\Psi}_+ + \frac{\hbar}{2}\gamma\alpha x\,e^{-i\gamma B_0 t}\tilde{\Psi}_- = i\hbar\frac{\partial}{\partial t}\tilde{\Psi}_+. (c) If one ignores the oscillatory term in the solution to (b)—on the grounds that it averages to zero (see discussion in Example 4.4)—one obtains uncoupled equations of the form −ℏ22m∇2Ψ~±+V± Ψ~±=iℏ∂∂tΨ~±.-\frac{\hbar^2}{2m}\nabla^2\tilde{\Psi}_\pm + V_\pm\,\tilde{\Psi}_\pm = i\hbar\frac{\partial}{\partial t}\tilde{\Psi}_\pm. Based upon the motion you would expect for a particle in the "potential" V±V_\pm, explain the Stern–Gerlach experiment.

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