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Selection rules electrostatic field 
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#1
Mar812, 01:25 PM

P: 6

Hi all , I need some help with this problem,
1. The problem statement, all variables and given/known data A hydrogen atom, which is in its ground state 1 0 0 > , is put into a weak timedependent external electric field, which points into the z direction: [tex]\boldsymbol{E}(t,\boldsymbol{r}) = \frac{C\hat{\text{e}}_{z}}{t^{2}+\tau ^{2}}[/tex], where C and [tex]\tau > 0[/tex] and e are constants. This gives rise to a perturbation potential [tex]V(t) = C\frac{e\hat{z}}{t^{2}+\tau^{2}}[/tex]. Using lowestorder timedependent 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. 2. Relevant equations [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 First, I have to see the values of m for which the sandwich vanish i.e. [tex]< n l m \hat{z}  1 0 0 > =0 [/tex] [tex]< n l m \hat{z}  1 0 0 > = I (radial ) χ \int Y^*_{l,m} Cos [\theta]d \Omega Y_{0,0} [/tex] The radial part is always non zero, Therefore , i have to compute [tex]\int d \Omega Y^*_{l,m} \cos (\theta) Y_{0,0} = [/tex] But I don't know what I can use, to compute the terms \cos (\theta) X spherical harmonics thanks for your help! 


#2
Mar912, 02:21 AM

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Thanks
PF Gold
P: 11,731

Hint: Look up Y_{1,0}.



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