# 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.