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Time Dependence of Expectation Values

  1. Oct 18, 2012 #1
    Hi,

    Please refer to this book (in google archive), and go to section 7.7 (page 85).

    http://books.google.com/books?id=MnY1jUP9nlIC&pg=PR11&lpg=PR11&dq=%22time+dependent+expectation+values%22+%22quantum%22+%22harmonic+oscillator%22+-abstract+-annihilation&source=bl&ots=cSOfuC8k9y&sig=Wdc327g7A5yA6n61L0ZLlKmu-Yk&hl=ar&sa=X&ei=dK-AUOagGueF4ASoxoGgCg&ved=0CFcQ6AEwCA#v=onepage&q&f=false

    I understand Ehrenfest theorem very well, but what the author does when he solved

    for the time-dependent expectation value of x, x^2, etc is strange.

    I cannot really understand what he is doing. If someone wants to help, you may consider x^2 case (the book solves all the cases so please refer to it).

    Thanks in advance!
     
  2. jcsd
  3. Oct 18, 2012 #2
    Is there a particular line of his work you are hung up on?
     
  4. Oct 19, 2012 #3
    yes. eqn 7.7.52 is not compatible with ehrenfest theorem.
     
  5. Oct 19, 2012 #4
    Are you sure about that? if you differentiate 7.7.52 wrt to time you get 7.7.51 which is the same thing you get by using 7.7.39.
     
  6. Oct 20, 2012 #5

    That's correct because in differentiating the constant vanish, but what about doing it the other way around...

    In particular, how originally do you get 7.7.52? specifically that constant term. I understand that the constant term is just the time-independent expectation value but what is the law? what is the relation used? that's the question.
     
  7. Oct 20, 2012 #6
    I think you may be over-thinking it. If df(t)/dt = C, then f(t) can be written as f(t) = f(0) + Ct, i.e. a first order Taylor expansion.

    That means that f(t) = <A>t can also be written as <A>t = <A>0 + d/dt(<A>t) * t.
     
  8. Oct 20, 2012 #7
    He just computes it using the definition of expectation for that operator at t=0 in 7.7.53. If you are asking why 52 has that form then the reason is the same that x(t)=x_o+Vt.
     
  9. Oct 21, 2012 #8
    Mathematically that's correct, but I want to see a law in Griffiths or Sakurai's text saying this.

    Or even better, I'd like to know what is the used "definition" for the time-dependent expectation value.
     
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