Quantum Mechanics (position uncertainty)

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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>=A2 ∞ = ∞

So, what have I done wrong here? :confused:

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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Answers and Replies

  • #2
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Check the evaluation of your upper and lower limits on your integrals again. ;)
 
  • #3
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Check the evaluation of your upper and lower limits on your integrals again. ;)
The upper limit is +∞ and the lowe limit is -∞. Here's what I did:

[itex]A^2 \left( \frac{e^{-2 \alpha \infty^2}}{4 \alpha} - \frac{e^{2 \alpha \infty^2}}{4 \alpha} \right)[/itex]

[itex]= A^2 \left( \frac{e^{-\infty^2}}{4 \alpha} - \frac{e^{\infty^2}}{4 \alpha} \right)[/itex]

[itex]= A^2 (0- 0) = 0[/itex]

Edit: this time I get 0. So <x>=0?
 
Last edited:
  • #4
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Yes! Not infinity as you were getting before. It makes sense to, a simple ground state harmonic oscillator spends most of the time close to origin.
 
  • #5
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Okay, but what about <x2>? The integral [itex]\int^{+\infty}_{-\infty} x^2 A^2 (e^{-\alpha x^2})^2 \ dx[/itex] turns out to be extremely messy. I'm just wondering if I am on the correct path? :confused:

If I integrate that I get this:

[itex]\left[ \frac{\sqrt{\pi/2} \ erf(\sqrt{2} \sqrt{\alpha} x)}{8 \alpha^{3/2}} - \frac{x e^{-2\alpha x^2}}{4 \alpha} \right]^{+\infty}_{-\infty}[/itex]

But I must get <x2>=ħ/2mω so that the position uncertainty would be the square root of that.
 
  • #6
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You are the right path, if you evaluate the integrals you should get the right value.
 

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