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
[tex][H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}[/tex]
for the Hydrogen atom (where the j's denote the jth components in Cartesian coordinates) to prove that
[tex]<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}>[/tex]

Homework Equations

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

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
[tex][H_{0},r_{j}]=\frac{i\hbar}{\mu}p_{j}[/tex]
for the Hydrogen atom (where the j's denote the jth components in Cartesian coordinates) to prove that
[tex]<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}>[/tex]

Just substitute for [itex]p_{j}[/itex] the communtator [itex][H_{0},r_{j}][/itex]
[tex]<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}>[/tex]
and take it from there. Remember that [itex]H[/itex] 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.