# Question on the Hydrogen atom

## Homework Statement

Use
$$[H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}$$
for the Hydrogen atom (where the j's denote the jth components in Cartesian coordinates) to prove that
$$<n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}>=-i\mu\omega<n_{f},l_{f},m_{l,f}|r_{j}|n_{i},l_{i},m_{l,i}>$$

## Homework Equations

$$[H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}$$

## The Attempt at a Solution

I'm really at a loss on how to begin here. I don't see how I can use the commutator to prove this.

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## Homework Statement

Use
$$[H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}$$
for the Hydrogen atom (where the j's denote the jth components in Cartesian coordinates) to prove that
$$<n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}>=-i\mu\omega<n_{f},l_{f},m_{l,f}|r_{j}|n_{i},l_{i},m_{l,i}>$$
Just substitute for $p_{j}$ the communtator $[H_{0},r_{j}]$
$$<n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}> = \frac{\mu}{i\hbar}<n_{f},l_{f},m_{l,f}|[H, r_j]|n_{i},l_{i},m_{l,i}>$$
and take it from there. Remember that $H$ is Hermitian and what its eigenstates and eigenvalues are.

Thanks a lot mathfeel. It took me a while when you mentioned the hermicity of the Hamiltonian, but after staring it down for a straight 5 minutes I felt dumb since it's so obvious, hahaha.