1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Expectation value of raising/lowering operators

  1. Dec 15, 2009 #1
    1. The problem statement, all variables and given/known data

    This has been driving me CRAZY:

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


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

    2. Relevant 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)

    3. 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...
  2. jcsd
  3. Dec 15, 2009 #2


    User Avatar
    Homework Helper

    First of all, thankyouthankyouthankyou for making a decent attempt at the problem o:)

    Here's the catch: when you did that calculation, you implicitly assumed that [itex]|\psi(0)\rangle[/itex] was an energy eigenstate with a particular eigenvalue [itex]E = \hbar \omega[/itex]. It's not, in general, an eigenstate, and so it doesn't necessarily evolve according to a single exponential factor [itex]e^{-i\omega t}[/itex]. You can't write
    [tex]|\psi(t)\rangle = |\psi(0) e^{-i\omega t}\rangle[/tex]
    unless you know that the state is an energy eigenstate.

    Before you try to generalize that calculation, take another look at this:
    Note that you can calculate [itex]\langle a(t)\rangle[/itex] the same way, just make the coefficients [itex]c_k[/itex] functions of time. Specifically,
    [tex]|\psi(t)\rangle = \sum_k c_k(t) |n_k\rangle = \sum_k c_k e^{-i\omega_k t} |n_k\rangle[/tex]
    Since this is a harmonic oscillator, you know what [itex]\omega_k[/itex] is in terms of [itex]k[/itex].
    Be careful there: you started out with a delta function that requires [itex]k = l-1[/itex]. So you shouldn't be winding up with [itex]|c_k|^2[/itex], since that results from the term where [itex]k = l[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook