naggy
Oct24-10, 05:55 PM
1. The problem statement, all variables and given/known data
Harmonic oscillator is in the first excited state |1> when a constant electric field E is switched on. Find the time evolution of the wave function
2. Relevant equations
Schrodinger equation
H\Psi = E_n\Psi
H = \frac{P^2}{2m} + \frac{m\omega^2x^2}{2}-qEx
3. The attempt at a solution
The electric field will give us a shifted Harmonic oscillator. The time evolution is
\Psi (x,t) = \sum_{n=0}^{\infty} C_n \phi_{n}(x-\frac{qE}{m\omega})e^{-iE_n t/ \hbar}
now, I'm stuck at finding the coefficients C_ns. That is, I donīt know how to solve
C_n = \langle \phi_n(x-\frac{qE}{m\omega} | \phi_1(x) \rangle
where the \phi_n(x) are the stationary state of the undisturbed harmonic oscillator
We were given a hint: The ground state of the harmonic oscillator is a coherent state. I'm not sure how to use that.
Harmonic oscillator is in the first excited state |1> when a constant electric field E is switched on. Find the time evolution of the wave function
2. Relevant equations
Schrodinger equation
H\Psi = E_n\Psi
H = \frac{P^2}{2m} + \frac{m\omega^2x^2}{2}-qEx
3. The attempt at a solution
The electric field will give us a shifted Harmonic oscillator. The time evolution is
\Psi (x,t) = \sum_{n=0}^{\infty} C_n \phi_{n}(x-\frac{qE}{m\omega})e^{-iE_n t/ \hbar}
now, I'm stuck at finding the coefficients C_ns. That is, I donīt know how to solve
C_n = \langle \phi_n(x-\frac{qE}{m\omega} | \phi_1(x) \rangle
where the \phi_n(x) are the stationary state of the undisturbed harmonic oscillator
We were given a hint: The ground state of the harmonic oscillator is a coherent state. I'm not sure how to use that.