Time evolution from zero state

[Confundo]

Homework Statement

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

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

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

Homework Equations

I'm a little confused of what to do with this one, I know that [tex]|n\rangle = \frac{(\hat a^{\dagger})^n}{\sqrt{n!}}|0\rangle[/tex] 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
[tex]
|\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)
[/tex] ?

 

[borgwal]

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

 

[Confundo]

[tex]
|\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)
[/tex]

 

[borgwal]

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

 

[Confundo]

[tex]E_n = \hbar \omega(n + 0.5) [/tex]
[tex]E_{n+1} = \hbar \omega(n + 1.5) [/tex]

 

[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 [tex] \exp(i\phi) \exp(i\phi')=\exp(i(\phi+\phi')) [/tex]