# Homework Help: Time dependent quantum state probability calculation

1. May 27, 2014

### kdlsw

For part a I have (H0-ω$\hbar$m)|nlm>, which I think the (H0-ω$\hbar$m) part is the eigenvalue of the Hamiltonian, also is the energies?

And mainly, I am not sure how to approach part b, the time variable is not in any of the states. I saw this in our lecture notes: ψ(r,t)=∑Cnψn(r) e-iEnt/$\hbar$. Do I simply add the e-iEnt/$\hbar$ term after each ψ state? And then what? Please help me with this, thanks

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2. May 28, 2014

### Staff: Mentor

$H_0$ is the Hamiltonian for the hydrogen atom in the absence of the external magnetic field. You should be able to find its eigenvalues.

Basically, yes. The time evolution of a state $\psi$ (with a time-independent Hamiltonian $H$) is given by
$$\psi(t) = e^{-i H t / \hbar} \psi(0)$$
For $\phi_n$ and eigenstate of $H$ with eigenvalue $E_n$,
$$e^{-i H t / \hbar} \phi_n = e^{-i E_n t / \hbar} \phi_n$$
Therefore, if $\psi$ is a superposition of eigenstates $\phi_n$,
$$\psi(0) = \sum_n c_n \phi_n$$
then
$$\psi(t) = \sum_n c_n e^{-i E_n t / \hbar} \phi_n$$

When you've done a correctly, you'll have $E_{nlm}$ for each state $| n l m \rangle$, and therefore can get $|\Psi(t)\rangle$ for any time $t$. You will then have to calculate projections like $\langle \psi_1 | \Psi(t) \rangle$, from which you can calculate the probabilities.