(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A hydrogen atom, which is in its ground state 1s (i.e. [tex]\|1,0,0\rangle[/tex]), is put into a weak time-dependent external electric field, which points into thezdirection:

[tex]\boldsymbol{E}(t,\boldsymbol{r}) = \frac{C\hat{\text{\boldsymbol{e}}}_{z}}{t^{2}+\tau^{2}}[/tex], where C and [tex]\tau > 0[/tex] are constants. This gives rise to a perturbation potential [tex]V(t) = C\frac{e\hat{z}}{t^{2}+\tau^{2}}[/tex], where e denotes the electron charge.

(a) Using lowest-order time-dependent perturbation theory, find the selection rules for transitions from the ground state, i.e. find out which final state values for the quantum numbersn,landmare possible in transitions from the ground state.

(b) Calculate the probability of transition from the ground state 1s to to the state 2p during an infinitely long period of time, setting [tex]t_{0}\rightarrow -\infty[/tex] and [tex]t\rightarrow\infty[/tex].

2. Relevant equations

This is useful, isn't it?

[tex]P_{fi}(t,t_{0})\equiv |\langle\phi_{f}|\psi(t)\rangle|^{2}\approx \frac{1}{\hbar^{2}}\left|\int_{t_{0}}^{t}\text{d}t_{1}\langle\phi_{f}|V_{S}(t_{1})|\phi_{i}\rangle \text{e}^{\text{i}(E_{f}-E_{i})t_{1}/\hbar}\right|^{2}.[/tex]

3. The attempt at a solution

(b) I probably should use the equation given in the section "2. Relevant equations" and "sandwich" perturbation potential between the states [tex]\langle 2,1,m|[/tex] and [tex]|1,0,0\rangle[/tex], shouldn't I? What m-value should I use?

Any hints to (a)?

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# Homework Help: A hydrogen atom in a weak time-dependent perturbation

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