- #1
damarkk
- 2
- 1
- 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.
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.
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.