# Time Evolution of Polarized Cavity

1. Feb 24, 2012

### danjferg

1. The problem statement, all variables and given/known data
Suppose that a particular x-polarized cavity mode is described, at time t = 0, by the state

|ψ(0)> = (1/√2)(|n> + |n+1>)

Find |ψ(t)> for t > 0. This is best done in the Schrodinger picture. Evaluate the expectation of the electric field operator Ex and the uncertainty <ΔEx2>, both as a function of time. Plot your result for n = 1.​

I'm brand new to QM, and took this class after talking to the professor before the semester started. He assured me based on my past courses that I should be able to handle this class. I breezed through the first homework, but now we're doing advanced stuff using material that I have never seen before. Drop deadline passed after 1st homework so I'm stuck. I've got \$5000 on the line and I really need help learning this stuff! I really need some hand-holding for these first few problems so I can learn the math and notation. My understanding of QHO is shaky at best, so this new section is really killing me.

2. Relevant equations

$H = \hbar \sum_{j}\omega_{j}\left(\hat{a}_{j}^{\dagger}\hat{a}_{j}+\frac{1}{2}\right)$
$\left[\hat{a}_{i},\hat{a}_{j}^{\dagger}\right]=\delta_{ij}$
$\left[\hat{a}_{i},\hat{a}_{i}\right]=\left[\hat{a}_{i}^{\dagger},\hat{a}_{i}^{\dagger}\right]=0$

3. The attempt at a solution
I don't even know where to start. Usually I solve using the Heisenberg picture, and there I get

$\left|\psi,t\right\rangle=exp\left(\frac{-iHt}{\hbar}\right)\left|\psi\right\rangle$

but I'm unclear on how to merge the Hamiltonian into the state, or exactly what the state represents. I'm looking for guidance on how to approach these problems. What, notationally, do N and N+1 represent? How do I properly set it up? I can give more details if necessary, but that is all I know to be pertinent right now.

$\left|\psi\left(t\right)\right\rangle=\sum_{j}\sum_{n}c_{jn}\left|j,n\right\rangle e^{-i\left(n_{j}+1/2\right)\omega_{j}t}$