Proving <n_f,l_f,m_l,f|p_j|n_i,l_i,m_l,i> for Hydrogen Atom - Homework Help

[arenaninja]

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
&lt;n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}&gt;=-i\mu\omega&lt;n_{f},l_{f},m_{l,f}|r_{j}|n_{i},l_{i},m_{l,i}&gt;

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.

 

[mathfeel]

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
&lt;n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}&gt;=-i\mu\omega&lt;n_{f},l_{f},m_{l,f}|r_{j}|n_{i},l_{i},m_{l,i}&gt;

Just substitute for p_{j} the communtator [H_{0},r_{j}]
&lt;n_{f},l_{f},m_{l,f}|p_{j}|n_{i},l_{i},m_{l,i}&gt; = \frac{\mu}{i\hbar}&lt;n_{f},l_{f},m_{l,f}|[H, r_j]|n_{i},l_{i},m_{l,i}&gt;
and take it from there. Remember that H is Hermitian and what its eigenstates and eigenvalues are.

 

[arenaninja]

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.