course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Quantum Mechanics |
IB |
67 |
|
Paper 4, Section I, B |
2016 |
(a) Define the quantum orbital angular momentum operator $\hat{\boldsymbol{L}}=\left(\hat{L}{1}, \hat{L}{2}, \hat{L}_{3}\right)$ in three dimensions, in terms of the position and momentum operators.
(b) Show that $\left[\hat{L}{1}, \hat{L}{2}\right]=i \hbar \hat{L}_{3}$. [You may assume that the position and momentum operators satisfy the canonical commutation relations.]
(c) Let $\hat{L}^{2}=\hat{L}{1}^{2}+\hat{L}{2}^{2}+\hat{L}{3}^{2}$. Show that $\hat{L}{1}$ commutes with
[In this part of the question you may additionally assume without proof the permuted relations $\left[\hat{L}{2}, \hat{L}{3}\right]=i \hbar \hat{L}{1}$ and $\left.\left[\hat{L}{3}, \hat{L}{1}\right]=i \hbar \hat{L}{2} .\right]$
[Hint: It may be useful to consider the expression
(d) Suppose that
with