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Time evolution from zero state

  1. Nov 5, 2008 #1
    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 [tex]|\psi(t)\rangle[/tex] at times t > 0.

    2. Relevant 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.
    Last edited: Nov 5, 2008
  2. jcsd
  3. Nov 5, 2008 #2
    What equation determines the time evolution of any state in quantum mechanics?

    Apply that equation!
  4. Nov 5, 2008 #3
    The SE.

    |\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] ?
  5. Nov 5, 2008 #4
    No...do the states |n> and |n+1> have the same energy?
  6. Nov 5, 2008 #5
    |\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)
  7. Nov 5, 2008 #6
    yeah...and what are the values of E_n and E_{n+1}?
  8. Nov 5, 2008 #7
    [tex]E_n = \hbar \omega(n + 0.5) [/tex]
    [tex]E_{n+1} = \hbar \omega(n + 1.5) [/tex]
  9. Nov 5, 2008 #8
    Right, substitute that in your last equation, take a common phase factor out, done.
  10. Nov 6, 2008 #9
    How does the common phase factor come out?
  11. Nov 6, 2008 #10
    That follows from [tex] \exp(i\phi) \exp(i\phi')=\exp(i(\phi+\phi')) [/tex]
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