- #1
roam
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Homework Statement
I need some help with the following problem:
http://img827.imageshack.us/img827/4061/prob1y.jpg
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>=A2 ∞ = ∞
So, what have I done wrong here?
Also for <x2> 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 <x2> 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.
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