Time Dependence of Expectation Values

[phys_student1]

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!

 

[Jorriss]

Is there a particular line of his work you are hung up on?

 

[phys_student1]

Is there a particular line of his work you are hung up on?

yes. eqn 7.7.52 is not compatible with ehrenfest theorem.

 

[bp_psy]

yes. eqn 7.7.52 is not compatible with ehrenfest theorem.

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.

 

[phys_student1]

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.

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.

 

[Jasso]

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.

 

[bp_psy]

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.

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.

 

[phys_student1]

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.

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.