Warning: Attempt this problem only if you are familiar with vector calculus. Define the (three-dimensional) probability current by generalization of Problem 1.14:

(a) Show that J satisfies the continuity equation

which expresses local conservation of probability. It follows (from the divergence theorem) that

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.

(c) If we interpret m j as the flow of mass, the angular momentum is

Use this to calculate L_z for the state Ψ₂₁₁, and comment on the result.

Transcribed Image Text:

J iħ 2m (**). (4.220)