Calculating Averages in a Unidimensional Quantum System

[dirac68]

Homework Statement

Hi, i would to resolve this problem of quantum mechanics.

I have hamiltonian operator of a unidimensional system:

$\hat{H}={\hat{p}^2 \over 2 m}-F\hat{x}$

where m and F are costant; the state is described by the function wave at t=0

$\psi (x, t=0)=A e ^{-x^2-x}$

where A is a costant.

How can I calculate the the avarage of x and p at time t after t=0 ( so $<x>_t$ and $<p>_t$ )?

what is the fast procedure to solve it?

Homework Equations

$\hat{H}={\hat{p}\over 2 m}-F\hat{x}$

$\psi (x, t=0)=A e ^{-x^2-x}$

The Attempt at a Solution

I found a solution but it seems very long and boring...

 

[Simon Bridge]

$$\psi(x,t)=\psi(x,0)e^{-iEt/\hbar}$$... where E is given by: $$\hat{H}\psi=E\psi$$

note: shouldn't the momentum operator appear squared in that hamiltonian?

 

[dirac68]

$$\psi(x,t)=\psi(x,0)e^{-iEt/\hbar}$$... where E is given by: $$\hat{H}\psi=E\psi$$

note: shouldn't the momentum operator appear squared in that hamiltonian?

oh yes it's p²/2m... but find eigenvalue E is too hard!

 

[vela]

Use the Ehrenfest theorem.

 

[Simon Bridge]

avarage of x and p

Ahhh yes - that's easier.

You don't have discrete E eigenvalues because you don't have a lower bound - but you don't need them. Sorry, my bad.