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

AI Thread Summary
The discussion focuses on proving a relationship involving the momentum operator and position operator for the hydrogen atom using the commutation relation of the Hamiltonian. Participants suggest substituting the momentum operator with the commutator of the Hamiltonian and position operator, leading to the equation involving eigenstates and eigenvalues. The importance of the Hermitian nature of the Hamiltonian is emphasized, as it plays a crucial role in the proof. One user expresses initial confusion but gains clarity after considering the Hermicity. The conversation highlights the significance of understanding quantum mechanics principles in solving such problems.
arenaninja
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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
&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.
 
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arenaninja said:

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.
 
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.
 
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