Introduction to Quantum Mechanics · Problem 4.72

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · charged_particle_magnetic_field

Problem

Consider a particle with charge qq, mass mm, and spin ss, in a uniform magnetic field B0\mathbf{B}_0. The vector potential can be chosen as A=−12r×B0.\mathbf{A} = -\frac{1}{2}\mathbf{r} \times \mathbf{B}_0. (a) Verify that this vector potential produces a uniform magnetic field B0\mathbf{B}_0. (b) Show that the Hamiltonian can be written H=p22m+q φ−B0⋅(γo L+γ S)+q28m[r2B02−(r⋅B0)2],(4.230)H = \frac{p^2}{2m} + q\,\varphi - \mathbf{B}_0 \cdot (\gamma_o\,\mathbf{L} + \gamma\,\mathbf{S}) + \frac{q^2}{8m}\left[r^2 B_0^2 - (\mathbf{r} \cdot \mathbf{B}_0)^2\right], \qquad (4.230) where γo=q/2m\gamma_o = q/2m is the gyromagnetic ratio for orbital motion. Note: The term linear in B0\mathbf{B}_0 makes it energetically favorable for the magnetic moments (orbital and spin) to align with the magnetic field; this is the origin of paramagnetism in materials. The term quadratic in B0\mathbf{B}_0 leads to the opposite effect: diamagnetism.

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