Time Dependence of Expectation Values

1. Oct 18, 2012

phys_student1

Hi,

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

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).

2. Oct 18, 2012

Jorriss

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

3. Oct 19, 2012

phys_student1

yes. eqn 7.7.52 is not compatible with ehrenfest theorem.

4. Oct 19, 2012

bp_psy

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.

5. Oct 20, 2012

phys_student1

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.

6. Oct 20, 2012

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.

7. Oct 20, 2012

bp_psy

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.

8. Oct 21, 2012

phys_student1

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.