- #1

quasar_4

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## Homework Statement

This has been driving me CRAZY:

Show that [tex] \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle [/tex]

and

[tex] \langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle [/tex]

## Homework Equations

Raising/lowering eigenvalue equations:

[tex] a |n \rangle = \sqrt{n} |n-1 \rangle [/tex]

[tex] a^{\dagger} |n \rangle = \sqrt{n+1} |n+1 \rangle [/tex]

Time development of stationary states: psi(x)*exp(-i*En*t/hbar)=psi(x)*exp(-i*w_n*t)

## The Attempt at a Solution

Suppose we've got the system in some state [tex] \psi(0) [/tex].

Then expanding into the [tex] | n(0)\rangle [/tex] basis (looking just at a, not a dagger here) we have

[tex] \langle a(0) \rangle = \langle \psi(0) | a | \psi(0) \rangle = \langle \sum_k c_k n_k | a | \sum_k c_k n_k \rangle = \sum_k \sum_l c_k^* c_l \langle n_k | a | n_l \rangle = \sum_k \sum_l c_k^* c_l \sqrt{n_l} \langle n_k | a | n_{l-1}\rangle = \sum_k \sum_l c_k^* c_l \sqrt{n_l} \delta_{k, l-1} [/tex]

so for non-trivial <a(0)>, [tex] \langle a(0) \rangle = \sum_k |c_k|^2 \sqrt{n_k} [/tex]

But now

[tex] \langle a(t) \rangle = \langle \psi(0) e^{-i \omega t} | a | \psi(0) e^{-i \omega t} \rangle = \langle \psi(0)| e^{i \omega t} a e^{-i \omega t} | \psi(0) \rangle = \langle \psi(0) | a | \psi(0) \rangle \neq e^{-i \omega t} \langle a(0) \rangle [/tex]

because as far as I know, a does not act on exp(i*w*t) and the two exponential terms cancel out! I get the same sort of problem with a dagger. So... what's the deal?? This is really bugging me, I'd love to know how to do this problem...