Commutators

Question 1

Define the commutator of  and B̂

Answer 1

[Â, B̂] = ÂB̂ - B̂Â
aka difference between measuring B then A, and A then B

Question 2

If  and B̂ commute?

Answer 2

[Â, B̂] = 0
ÂB̂Ψ = B̂ÂΨ for any Ψ
 and B̂ can be measured in either order and give the same answer.
 and B̂ share eigenfunctions and are therefore compatible.

Question 3

If  and B̂ dont commute?

Answer 3

[Â, B̂] ≠ 0
The order of measurement matters.
Both cannot be known simultaneously.
 and B̂ do not share eigenfunctions and are therefore not compatible.

Question 4

Important result: [x̂, p̂ₓ]
(also practice proving)

Answer 4

[x̂, p̂ₓ] = x̂p̂ₓ - p̂ₓx̂
= iħ

Question 5

[p̂ₓ²,x̂]

Answer 5

-2iħp̂ₓ

Question 6

[x̂², p̂ₓ]

Answer 6

2iħx̂

Question 7

[Ĥ, â†]

Answer 7

Use [p̂ₓ²,x̂] = -2iħp̂ₓ and [x̂², p̂ₓ] = 2iħx̂
[Ĥ, â†] = -ħωâ†

Question 8

[Ĥ, â]

Answer 8

Use [p̂ₓ²,x̂] = -2iħp̂ₓ and [x̂², p̂ₓ] = 2iħx̂
[Ĥ, â] = -ħωâ

Question 9

[L̂ᶻ, Ĥ]

Answer 9

0