A quantum particle with charge q, mass m, and momentum p in a magnetic field B(r) = ∇ × A(r) has velocity ν(r) = p/m − (q/m)A(r). This means that a charge distribution ρ(r) generates a “diamagnetic current” j (r) = −(q/m)ρ(r)A(r) when it is placed in a magnetic field.

(a) Show that A(r) = 1/ 2 B × r is a legitimate vector potential for a uniform magnetic field B.
(b) Let ρ(r) = ρ(r) be the spherically symmetric charge distribution associated with the electrons of an atom. Expose the atom to a uniform magnetic field B and show that the ensuing diamagnetic current induces a vector potential 

(c) Expand Aind(r) for small values of r and show that the diamagnetic field at the atomic nucleus can be written in terms of ϕ(0), the electrostatic potential at the nucleus produced by ρ(r):

This formula was obtained by Willis Lamb in 1941. He had been asked by Isidore Rabi to determine
whether B_ind(0) could be ignored when nuclear magnetic moments were extracted from molecular beam data.

Transcribed Image Text:

Non-18+ [1 ][ = Ho eB xr Aind (r) = 4π 6m d³p' P(r') r'r