Angular momentum

Question 1

What is degeneracy? 2 different definitions.

Answer 1

The degeneracy of an energy level is the number of different states that have that energy.
Degeneracy is the concept that different eigenfunctions of an operator can have the same eigenvalue.

Question 2

Energy of the superposition of degenerate states?

Answer 2

If states are eigenstates of the same energy, then any linear superposition/ combination of them is also an eigenstate of the same energy.

Question 3

Angular momentum operator

Answer 3

L̂ = r̂ x p̂
L̂ᶻ = x̂p̂ᵧ - ŷp̂ₓ
= (-iħ)(x ∂/∂y - y ∂/∂x)
Using plane polar:
L̂ᶻ = -iħ ∂/∂φ

Question 4

State and solve the eigenfunction equation for L̂ᶻ

Answer 4

L̂ = r̂ x p̂
L̂ᶻΦ(φ) = LᶻΦ(φ)
-iħ d/dφ Φ(φ) = LᶻΦ(φ)

dΦ/Φ = iLᶻ/ħ dφ
Φ(φ) = exp(iLᶻφ/ħ)

Question 5

Φ(φ) = exp(iLᶻφ/ħ)
What do boundary conditions show?

Answer 5

φ and φ+2π refer to the same point
Φ(φ) = Φ(φ+2π)
exp(iLᶻ(φ+2π)/ħ) = exp(iLᶻφ/ħ)
exp(2πiLᶻ/ħ)exp(iLᶻφ/ħ) = exp(iLᶻφ/ħ)
exp(2πiLᶻ/ħ) = 1
cos(2πLᶻ/ħ) + isin(2πLᶻ/ħ) = 1
Therefore wavefunction is only single-valued if Lᶻ/ħ is an integer.
Lᶻ = mħ, m = 0, ±1, ±2, ±3…
Proves that angular momentum is quantized in units of ħ

Question 6

Are there QHO solutions that are eigenfunctions of L̂ᶻ?

Answer 6

Yes - by assuming a linear combination of QHO solutions for each degeneracy level.
By substituting into the L̂ᶻ eigenfunction and comparing coefficients, it can be shown that n linear combinations exist for E=nħω.
example: for ψ₂,₀, ψ₁,₁, and ψ₀,₂, 3 linear combinations exist with E=3ħω and Lᶻ = 2, 0, -2.
example:
4 linear combinations exist with E=4ħω and Lᶻ = 3, 1, -1, -3.

Question 7

What is the good quantum number, n,for the QHO?

Answer 7

n = nₓ + nᵧ
n defines the energy of the state, where E = (n+½)ħω

Question 8

What is the good quantum number, m?

Answer 8

m describes the orbital angular momentum of a state, where Lᶻ = mħ

Question 9

What is meant by ‘good’ quantum numbers?

Answer 9

They can be known simultaneously/ correspond to operators that commute.
They fully specify the state of the system. eg an (n, m) pair uniquely describes a state with definite energy and orbital angular momentum.

Question 10

Relation between n and m quantum numbersfor the 2D QHO?

Answer 10

For fixed n (n = 0, 1, 2, 3…):
Values of m range from -n to n in steps of 2
There are n+1 m values for each n, meaning the degeneracy of the nth excited state is n+1.