Electric sinusoidal field on a hydrogen atom - Quantum Mechanics

[damarkk]

Homework Statement

some doubts to clarify

Relevant Equations

Hamiltonian, spherical harmonic functions of hydrogen atom

Hello to everyone. I have some doubts about one problem of quantum mechanics.

Consider an hydrogen atom under the action of a electric field $E(t)= E_0 sin(\omega t)$ along $z$ axis. We can put $\frac{|E_2-E_1|}{\hbar} = \omega_0$, where$E_1$, $E_2$ are respectively the ground state and first excited energy states.

If the system is in a ground state for $t=0$, then using dependent time perturbation theory on the first order, find the state of a system at generic $t>0$.
Consider only transition for n=1 to $n=2$ states.

My attempt.

I need to calculate the coefficient $W_{ij}=<\psi_i | H' |\psi_j>$ where $H' = -eE(t)z$ is a perturbation term in the hamiltonian and $|\psi_i> = |\psi_{nlm}>$. We have four states and sixteen terms to calculate, respectively for the states $\psi_{100}, \psi_{200}, \psi_{210}, \psi_{211}, \psi_{21-1}$.

After this work, because the dependance of time, I can assume that the coefficients of the linear combination of the states are functions of time and for compute these term $c_ni = -\frac{i}{\hbar}\int W_{ij}exp(-i\omega_{ni}t)dt$.


Then, the state is $|\psi>= c_{100}(t)exp(-iE_{100}t/\hbar)|\psi_{100}>+c_{200}(t)exp(-iE_{200}t/\hbar)|\psi_{200}>+c_{210}(t)exp(-iE_{210}t/\hbar)|\psi_{210}>+c_{211}(t)exp(-iE_{211}t/\hbar)|\psi_{211}>+c_{21-1}(t)exp(-iE_{21-1}t/\hbar)|\psi_{21-1}>$


Is this correct?
I'm sorry for my poor english.

 

[damarkk]

Some suggestions? Is my attempt correct?
Thanks in advance.

 

[TSny]

Overall, I think you have the correct approach.

I can assume that the coefficients of the linear combination of the states are functions of time and for compute these term $c_ni = -\frac{i}{\hbar}\int W_{ij}exp(-i\omega_{ni}t)dt$.

On the left side, I think you meant to type $c_{ni}$ instead of $c_ni$.

Am I right that the subscript $i$ refers to the initial state $|\psi_{100}\rangle$ and the subscript $n$ refers to one of the states $|\psi_{100}\rangle, |\psi_{200}\rangle, |\psi_{211}\rangle, |\psi_{210}\rangle,|\psi_{21-1}\rangle$?

On the right side, you have $W_{ij}$. Do you have the correct subscripts here?

Also, you didn't tell us what the notation $\omega_{ni}$ represents. Check the sign of the argument of the exponential function.

What are the upper and lower limits for the integral in the expression for $c_{ni}$ ?

Your formula for $c_{ni}$ appears to be the formula for obtaining the first-order contribution to the expansion coefficient. So, I would write the left side as $c_{ni}^{(1)}$, where the superscript $(1)$ denotes the first-order contribution.

Up through first order, the expansion coefficients will be $c_{ni} = c_{ni}^{(0)}+ c_{ni}^{(1)}$, where $c_{ni}^{(0)}$ is the the zeroth-order approximation. From the setup of the problem, you should be able to deduce the values of the zeroth-order terms $c_{ni}^{(0)}$ for the various states $|\psi_n \rangle$.

Then, the state is $|\psi>= c_{100}(t)exp(-iE_{100}t/\hbar)|\psi_{100}>+c_{200}(t)exp(-iE_{200}t/\hbar)|\psi_{200}>+c_{210}(t)exp(-iE_{210}t/\hbar)|\psi_{210}>+c_{211}(t)exp(-iE_{211}t/\hbar)|\psi_{211}>+c_{21-1}(t)exp(-iE_{21-1}t/\hbar)|\psi_{21-1}>$

This looks right.

 

[DrClaude]

Note that you have to calculate the actual value of the $c(t)$.