2015-70.md

course	course_year	question_number	tags	title	year
Quantum Mechanics	IB	70	IB 2015 Quantum Mechanics	Paper 3, Section II, D	2015

Define the angular momentum operators $\hat{L}{i}$ for a particle in three dimensions in terms of the position and momentum operators $\hat{x}{i}$ and $\hat{p}{i}=-i \hbar \frac{\partial}{\partial x{i}}$. Write down an expression for $\left[\hat{L}{i}, \hat{L}{j}\right]$ and use this to show that $\left[\hat{L}^{2}, \hat{L}{i}\right]=0$ where $\hat{L}^{2}=\hat{L}{x}^{2}+\hat{L}{y}^{2}+\hat{L}{z}^{2}$. What is the significance of these two commutation relations?

Let $\psi(x, y, z)$ be both an eigenstate of $\hat{L}{z}$ with eigenvalue zero and an eigenstate of $\hat{L}^{2}$ with eigenvalue $\hbar^{2} l(l+1)$. Show that $\left(\hat{L}{x}+i \hat{L}{y}\right) \psi$ is also an eigenstate of both $\hat{L}{z}$ and $\hat{L}^{2}$ and determine the corresponding eigenvalues.

Find real constants $A$ and $B$ such that

$$\phi(x, y, z)=\left(A z^{2}+B y^{2}-r^{2}\right) e^{-r}, \quad r^{2}=x^{2}+y^{2}+z^{2},$$

is an eigenfunction of $\hat{L}{z}$ with eigenvalue zero and an eigenfunction of $\hat{L}^{2}$ with an eigenvalue which you should determine. [Hint: You might like to show that $\left.\hat{L}{i} f(r)=0 .\right]$