Homework Help: Time evolution from zero state

1. Nov 5, 2008

Confundo

1. The problem statement, all variables and given/known data

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.

2. Relevant 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.

Last edited: Nov 5, 2008
2. Nov 5, 2008

borgwal

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

Apply that equation!

3. Nov 5, 2008

Confundo

The SE.

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

4. Nov 5, 2008

borgwal

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

5. Nov 5, 2008

Confundo

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

6. Nov 5, 2008

borgwal

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

7. Nov 5, 2008

Confundo

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

8. Nov 5, 2008

borgwal

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

9. Nov 6, 2008

Confundo

How does the common phase factor come out?

10. Nov 6, 2008

borgwal

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