- #1
Urvabara
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Homework Statement
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 the z direction:
[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 numbers n, l and m are 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].
Homework 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]
The Attempt at a Solution
(b) I probably should use the equation given in the section "2. Homework 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)?