3D Particle on a Sphere

Question 1

General equation

Answer 1

(-(ℏ^2)/2m)(∇^2Ψ) + V(x)Ψ =EΨ
(If no charge no potential energy term)

Question 2

Laplace for a particle in 3D

Answer 2

(remember these are partial derivatives)
∇^2= d^2/dx^2 +d^2/dy^2 +d^2/d^z^2
For polar coordinates in 3d (r,ϕ and θ)
∇^2 =d^2/dr^2+(1/r)d/dr +1/r^2((1/sin^2(θ))(d^2/dϕ^2)+(1/sin(θ))(d/dθ)(sin(θ)(d^2/dθ^2))
For constant r (on a sphere not within a sphere):
∇^2 = 1/r^2((1/sin^2(θ))(d^2/dϕ^2)+(1/sin(θ))(d/dθ)(sin(θ)(d^2/dθ^2))

Question 3

Boundary condition

Answer 3

Must be cyclic around the sphere for constructive interference of wavefunction

Question 4

Consequence of boundary condition

Answer 4

Since we change 2 polar coordinates we require 2 quantum numbers, l and m_l

Question 5

Wavefunction

Answer 5

More complex as we have 2 variables:
Ψ(ϕ,θ) =Θ(θ)Φ(ϕ)

Question 6

Energy of a sphere:

Answer 6

E = l(l+1)ℏ^2/2I, where l = 0,1,2,3….(n-1)

Question 7

Angular momentum

Answer 7

E =J^2/2I
J = ( (l(l+1))^1/2) ℏ
J = m_l ℏ where m_l = 0,±1,±2,….,±l

Question 8

Orientation per orbital

Answer 8

From angular momentum equation we can see for every orbital l we have 2l+1 orientation
for example, l=2, we have 5 degenerate orientations