Franz Schwabl Quantum Mechanics 4th Edit · Problem 6.13

Motion in an Electromagnetic Field

Physics — Quantum Mechanics · lenz-vector-properties

Problem

The Hermitian vector operator A\boldsymbol{A} corresponding to the Lenz vector is A=12m(p×L−L×p)−Ze2rr.\boldsymbol{A} = \frac{1}{2m}(\boldsymbol{p} \times \boldsymbol{L} - \boldsymbol{L} \times \boldsymbol{p}) - \frac{Ze^2}{r}\boldsymbol{r} \quad. Show that A†=A\boldsymbol{A}^\dagger = \boldsymbol{A} and that A\boldsymbol{A}, as in classical theory, is a constant of motion and is normal to L\boldsymbol{L}, i.e., [A,H]=0,A⋅L=L⋅A=0,[\boldsymbol{A}, H] = 0 \quad, \quad \boldsymbol{A} \cdot \boldsymbol{L} = \boldsymbol{L} \cdot \boldsymbol{A} = 0 \quad, where HH is the Hamiltonian of the hydrogen atom.

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