Operators

Question 1

Kinetic energy operator

Answer 1

T̂ = -ħ²/2m ∂²/∂x²

Question 2

Potential energy operator

Answer 2

V̂ = V(x)

Question 3

Total energy operator (Hamiltonian)

Answer 3

Ĥ = -ħ²/2m ∂²/∂x² + V(x)
T̂ + V̂

Question 4

Hamiltonian as an eigenvector

Answer 4

Ĥψᵢ(x) = Eᵢψᵢ(x)

Question 5

Eigenvalues and states of an operator, Â

Answer 5

Âψₙ = aₙψₙ
System in the ψₙ state, Â will always give aₙ
System in ψₙ state will stay in ψₙ state
System in mixed state ψ, Â gives one of the {aₙ} values with expectation ⟨A⟩ = ∫ψ * Âψ dx
Once aₙ measured, system left in ψₙ state.

Question 6

Momentum operator

Answer 6

p̂ₓ = -iħ ∂/∂x

Question 7

Position operator

Answer 7

x̂ = x

Question 8

What is the Hermitian conjugate? Define in terms of a pair of wavefunctions Φ and Ψ.

Answer 8

For each  operator, an † operator exists such that:
∫ Φ * † Ψ dx = ∫ (ÂΦ) * Ψ dx

Question 9

What is a Hermitian operator?

Answer 9

An operator for which † = Â

Question 10

Two properties of Hermitian operators?

Answer 10

1. All eigenvalues of a Hermitian operator are real, aka correspond naturally to physical observables.
2. Eigenfunctions of a Hermitian operator with different eigenvalues are orthogonal, meaning ∫ψᵢ * ψⱼ dx = 0 if aᵢ and aⱼ not equal.

Question 11

Destruction operator / lowering operator

Answer 11

â = √(mω/2ħ) x̂ + i/√(2mωħ) p̂ₓ

Question 12

Creation opertor / raising operator

Answer 12

↠= √(mω/2ħ) x̂ - i/√(2mωħ) p̂ₓ

Question 13

2D Hamiltonian

Answer 13

Ĥ = Ĥₓ + Ĥᵧ
Where Ĥₓ = -ħ²/2m ∂²/∂x² + ½ mω²x²
and Ĥᵧ = -ħ²/2m ∂²/∂y² + ½ mω²y²

Question 14

Eigenfunctions of the Hamiltonian

Answer 14

Ĥψₙ(x) = Eₙψₙ(x)
where Eₙ = (n+½)ħω

Question 15

What does separability of an operator tell us about its eigenfunctions?
eg  = B̂ + Ĉ

Answer 15

Products of the eigenfunctions of B̂ and Ĉ are eigenfunctions of Â

Question 16

Angular momentum operator

Answer 16

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

Question 17

Hamiltonian in terms of position and momentum operators.

Answer 17

Ĥ = 1/2m p̂ₓ² + ½ mω²x̂²