- #1

- 1,266

- 11

## Homework Statement

I need some help with the following problem:

http://img827.imageshack.us/img827/4061/prob1y.jpg [Broken]

## Homework Equations

For some particle in state Ψ the expectation value of x is given by:

[itex]\left\langle x \right\rangle = \int^{+\infty}_{-\infty} x |\Psi(x, t)|^2 \ dx[/itex]

The wave function in ground state:

[itex]\psi_0 (x)= A e^{\frac{-m \omega}{\hbar}x^2}[/itex]

## The Attempt at a Solution

Let α = mω/2ħ, so

[itex]\left\langle x \right\rangle = \int^{+\infty}_{-\infty} x A^2 (e^{-\alpha x^2})^2 \ dx[/itex]

[itex]= A^2 \int^{+\infty}_{-\infty} x (e^{-2 \alpha x^2}) \ dx[/itex]

[itex]=A^2 \left[ \frac{-e^{-2 \alpha x^2}}{4 \alpha} \right]^{+\infty}_{-\infty} \ dx[/itex]

But since e

^{∞}=∞ and e

^{-∞}=0 (i.e. the limits as x approaches ±∞) we get:

<x>=A

^{2}∞ = ∞

So, what have I done wrong here?

Also for <x

^{2}> we have the following

[itex]\left\langle x^2 \right\rangle = \int^{+\infty}_{-\infty} x^2 A^2 (e^{-\alpha x^2})^2 \ dx[/itex]

But similarly here I will encounter the same problem. So how can I evaluate the <x> and <x

^{2}> without getting infinity?

Clearly they can't equal to infinity since we must obtain:

[itex]\Delta x = \sqrt{\left\langle x^2 \right\rangle -\left\langle x \right\rangle^2} = \sqrt{\frac{\hbar}{2m \omega}}[/itex]

Any help is greatly appreciated.

Last edited by a moderator: