Question

The wave function of a particle in a spherically symmetric potential is 

Ψ(x, y, z) = C (x y + yz + zx)e^−αr2 

Show that the probability is zero for the angular momentum l = 0 and l = 1 and that it is unity for l = 2

Answer 1

In spherical coordinates

The above expression does not contain Y₀₀ corresponding to l = 0, nor Y₁₀ and Y_1±1 corresponding to l = 1. All the terms belong to l = 2, and the probability for finding l = 2 and therefore L₂ = l(l + 1)ℏ = 6ℏ² is unity.