Homework Help: Time evolution of Expectation

1. Sep 28, 2010

OGrowli

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 28, 2010

planck42

Are you asking about how quantum mechanical expectation values evolve with time? If so, then it evolves according to the differential equation

$$\frac{d}{dt}<{\psi}|O|{\psi}> = \frac{i}{\hbar}<{\psi}|[H,O]|{\psi}> + <{\psi}|\frac{{\partial}O}{{\partial}t}|{\psi}>$$

With O being a Hermitian operator.

Last edited: Sep 28, 2010
3. Sep 28, 2010

OGrowli

i don't know what happened to my original post, but I am having an issue with the following problem:

Show that:

\frac{d}{dt}<x^2> =\frac{1}{m}(< xp_x> +<p_xx>)

for a three dimensional wave packet.

relevant equations:

Ehrenfest Theorem:(1)
$$i\hbar\frac{d}{dt}<O>=<[O,H]>+i\hbar<\frac{\partial }{\partial t}O>$$

where O is an operator
(2)
$$\frac{d}{dt}\int_{V}d^3r\psi ^*O\psi$$

I tried using both ways illustrated above and I arrived at the same answer:

$$\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\bigtriangledown ^2(x^2\psi)-x^2\bigtriangledown ^2\psi]$$

$$=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\frac{\partial }{\partial x} (x^2\psi'+2x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi]$$

$$=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[ 2\psi+2x\frac{\partial }{\partial x}\psi+2x\frac{\partial }{\partial x}\psi+(x^2-x^2)\frac{\partial^2 }{\partial x^2}\psi]$$

$$=\frac{1}{m}\int_{V}d^3r[\psi ^*(-i\hbar)\psi+\psi^*x(-i\hbar\frac{\partial }{\partial x})\psi+\psi^*(-i\hbar\frac{\partial }{\partial x})x\psi]$$

$$=\frac{1}{m}(<xp_x>+<p_xx>)-\frac{i\hbar}{m}$$

Am I doing anything wrong? Where does the extra term come from, and does it mean anything?

4. Sep 28, 2010

OGrowli

nvm, I got it:

$$=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\frac{\partial }{\partial x} (x^2\psi'+2x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi]$$

$$=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[x^2\frac{\partial^2 }{\partial x^2}\psi+2x\frac{\partial }{\partial x}\psi+2\frac{\partial }{\partial x}(x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi]$$

$$=\frac{1}{m}\int_{V}d^3r[\psi^*x(-i\hbar\frac{\partial }{\partial x})\psi+\psi^*(-i\hbar\frac{\partial }{\partial x})x\psi]$$

$$=\frac{1}{m}(<xp_x>+<p_xx>)$$

I went wrong thinking I could just rearrange the derivatives.