# Selection rules electrostatic field

• pstq
In summary, the conversation discusses using perturbation theory to find the selection rules for transitions from the ground state of a hydrogen atom in a weak time-dependent electric field. The Homework Equations section includes an equation for the transition probability and the Attempt at a Solution section mentions finding the values of m for which the sandwich vanishes. The poster also asks for help with computing the terms involving spherical harmonics.
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

Hint: Look up Y1,0.

## 1. What are selection rules in an electrostatic field?

Selection rules in an electrostatic field refer to the conditions that determine whether an electric dipole transition between two energy states is allowed or forbidden. They are based on conservation principles such as energy, momentum, and parity.

## 2. How do selection rules affect the emission or absorption of photons?

Selection rules determine which transitions are allowed in the emission or absorption of photons. If a transition is allowed, a photon can be emitted or absorbed, but if it is forbidden, no photon can be emitted or absorbed.

## 3. Can selection rules be violated?

Yes, selection rules can be violated in certain cases. This can occur through higher order processes or in systems with broken symmetry. However, these violations are usually very weak and occur at a much lower probability than allowed transitions.

## 4. What is the role of symmetry in selection rules?

Symmetry plays a crucial role in selection rules. In order for a transition to be allowed, the initial and final states must have the same symmetry. This is because the electric dipole operator, which governs the transition, must also have the same symmetry.

## 5. How are selection rules applied in spectroscopy?

Selection rules are used to interpret the observed emission or absorption lines in spectroscopy. By analyzing the allowed transitions, scientists can determine the energy levels and symmetries of the system being studied. This information is crucial in understanding the electronic and molecular structures of materials.

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