course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Quantum Mechanics |
IB |
64 |
|
Paper 4, Section II, C |
2021 |
(a) Consider the angular momentum operators $\hat{L}{x}, \hat{L}{y}, \hat{L}{z}$ and $\hat{\mathbf{L}}^{2}=\hat{L}{x}^{2}+\hat{L}{y}^{2}+\hat{L}{z}^{2}$ where
$$\hat{L}{z}=\hat{x} \hat{p}{y}-\hat{y} \hat{p}{x}, \quad \hat{L}{x}=\hat{y} \hat{p}{z}-\hat{z} \hat{p}{y} \text { and } \hat{L}{y}=\hat{z} \hat{p}{x}-\hat{x} \hat{p}_{z} .$$
Use the standard commutation relations for these operators to show that
$$\hat{L}{\pm}=\hat{L}{x} \pm i \hat{L}{y} \quad \text { obeys } \quad\left[\hat{L}{z}, \hat{L}{\pm}\right]=\pm \hbar \hat{L}{\pm} \quad \text { and } \quad\left[\hat{\mathbf{L}}^{2}, \hat{L}_{\pm}\right]=0$$
Deduce that if
(b) A harmonic oscillator of mass
$$\hat{H}=\frac{1}{2 M}\left(\hat{p}{x}^{2}+\hat{p}{y}^{2}+\hat{p}_{z}^{2}\right)+\frac{1}{2} M \omega^{2}\left(\hat{x}^{2}+\hat{y}^{2}+\hat{z}^{2}\right) .$$
Find eigenstates of
Verify that the ground state for
Why should you expect to find joint eigenstates of
[ The first two eigenstates for an oscillator in one dimension are