How to Find the Time Evolution of a Polarized Cavity Mode in Quantum Mechanics?

[danjferg]

Homework Statement

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 Schrödinger picture. Evaluate the expectation of the electric field operator E_x and the uncertainty <ΔE_x²>, 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.

Homework Equations

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

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.


From the Schrödinger picture I'd start with

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

and set t=0. but how do the summations resolve (disappear from the equation)?

 

[member 553083]

What is the expectation of the electric field operator E_{x} and the uncertainty ΔE_{x}^{2}? I'm really confused on how to properly set up this problem.