What is the Time-Evolved State of a Single-Mode Cavity Field?

[Confundo]

Homework Statement

Suppose the state of a single-mode cavity field is given at time t=0 by

$$|\Psi(0) \rangle = \frac{1}{\sqrt{2}}(|n \rangle + e^{i\phi}|n+1 \rangle)$$

where phi is some phase. Find the state $$|\psi(t)\rangle$$ at times t > 0.

Homework Equations

I'm a little confused of what to do with this one, I know that $$|n\rangle = \frac{(\hat a^{\dagger})^n}{\sqrt{n!}}|0\rangle$$ and think I make have to substitue for the n eigenvalue using that somehow of that somehow and integrate.

 

[borgwal]

What equation determines the time evolution of any state in quantum mechanics?

Apply that equation!

 

[Confundo]

The SE.

Just
$$|\Psi(t) \rangle = e^\frac{-iEt}{\hbar}|\Psi(0) \rangle = e^\frac{-iEt}{\hbar}\frac{1}{\sqrt{2}}(|n \rangle + e^{i\phi}|n+1 \rangle)$$ ?

 

[borgwal]

No...do the states |n> and |n+1> have the same energy?

 

[Confundo]

$$|\Psi(t) \rangle = \frac{1}{\sqrt{2}}(e^\frac{-iE_{n}t}{\hbar}|n \rangle + e^\frac{-iE_{n+1}t}{\hbar}e^{i\phi}|n+1 \rangle)$$

 

[borgwal]

yeah...and what are the values of E_n and E_{n+1}?

 

[Confundo]

$$E_n = \hbar \omega(n + 0.5)$$
$$E_{n+1} = \hbar \omega(n + 1.5)$$

 

[borgwal]

Right, substitute that in your last equation, take a common phase factor out, done.

 

[Confundo]

How does the common phase factor come out?

 

[borgwal]

That follows from $$\exp(i\phi) \exp(i\phi')=\exp(i(\phi+\phi'))$$