Warning: Attempt this problem only if you are familiar with vector calculus. Define the (three-dimensional) probability current by generalization of Problem 1.14:
J≡2miℏ(Ψ∇Ψ∗−Ψ∗∇Ψ). (4.220)
(a) Show that J satisfies the continuity equation
∇⋅J=−∂t∂∣Ψ∣2, (4.221)
which expresses local conservation of probability. It follows (from the divergence theorem) that
∮SJ⋅da=−dtd∫V∣Ψ∣2d3r, (4.222)
where V is a (fixed) volume and S is its boundary surface. In words: The flow of probability out through the surface is equal to the decrease in probability of finding the particle in the volume.
(b) Find J for hydrogen in the state n=2, l=1, m=1. Answer:
64πma5ℏre−r/asinθϕ^.
(c) If we interpret mJ as the flow of mass, the angular momentum is
L=m∫(r×J)d3r.
Use this to calculate Lz for the state ψ211, and comment on the result.