Selection rules electrostatic field

[pstq]

Hi all , I need some help with this problem,

Homework Statement

A hydrogen atom, which is in its ground state |1 0 0 > , is put into a weak time-dependent external electric field, which points into the z direction:
$$\boldsymbol{E}(t,\boldsymbol{r}) = \frac{C\hat{\text{e}}_{z}}{t^{2}+\tau ^{2}}$$, where C and $$\tau > 0$$ and e are constants. This gives rise to a perturbation potential $$V(t) = C\frac{e\hat{z}}{t^{2}+\tau^{2}}$$.
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.

Homework Equations

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

The Attempt at a Solution

First, I have to see the values of m for which the sandwich vanish i.e. $$< n l m| \hat{z} | 1 0 0 > =0$$

$$< n l m| \hat{z} | 1 0 0 > = I (radial ) χ \int Y^*_{l,m} Cos [\theta]d \Omega Y_{0,0}$$

The radial part is always non zero,

Therefore , i have to compute
$$\int d \Omega Y^*_{l,m} \cos (\theta) Y_{0,0} =$$ But I don't know what I can use, to compute the terms \cos (\theta) X spherical harmonics
thanks for your help!

 

[vela]

Hint: Look up Y_1,0.