- #1

renec112

- 35

- 4

## Homework Statement

A free particle in one dimension is described by:

## H = \frac{p^2}{2m} = \frac{\hbar}{2m}\frac{\partial^2}{\partial x^2}##

at ##t = 0##

The wavefunction is described by:

## \Psi(x,0) = N(a^2-x^2) e^{i k x}## for ##|x| \leq a##

outside ##a##, ## \Psi = 0##. Use Ehrenfest to find the expectation value for all later times ##<x(t))>## of the particles position for all time ## t \geq 0 ##.

## Homework Equations

Ehrensfest:

##

\frac{d<Q>}{dt} = \frac{i}{\hbar}<[H,Q]> + <\frac{\partial Q}{\partial t}>

##

Where ##Q## is an operator.

## The Attempt at a Solution

We need to find it for all later times, Ehrensfest will show how an operator evolves in time. So set ##Q = x## and use Ehrenfest. Then we know the poisiton for all later time.

##

\frac{d<x>}{dt} = \frac{i}{\hbar}<[H,x]> + <\frac{\partial x}{\partial t}>

##

Since the operator does not change in time we have:

##

\frac{d<x>}{dt} = \frac{i}{\hbar}<[H,x]>

##

Here is where i am stuck. I am trying to do the commutator:

##

\frac{d<x>}{dt} = \frac{i}{\hbar}<(Hx -xH>

##

However, these do commute and hence everything should be zero.

What do you think?