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Change-of-variable

 
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Feb13-05, 04:20 AM   #1
 

Change-of-variable


I want to find the expectation value [tex]\langle x^2 \rangle[/tex] in some problem. To do this I make a change-of-variable,

[tex]\xi = \sqrt{\frac{m\omega}{\hslash}}x,[/tex]

and compute the expectation value [tex]\langle \xi^2 \rangle[/tex] like this:

[tex]\langle \xi^2 \rangle = \int\xi^2\vert\psi(\xi)\vert^2d\xi.[/tex]

Finally I change back to x:

[tex]\langle \xi^2 \rangle = \langle \frac{m\omega}{\hslash}x^2 \rangle = \frac{m\omega}{\hslash}\langle x^2 \rangle \Rightarrow \langle x^2 \rangle = \frac{\hslash}{m\omega}\langle \xi^2 \rangle.[/tex]

I really can't see what is wrong here, but something is! I've tried it 10 times and I keep getting the wrong result.
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Feb13-05, 04:22 AM   #2
 
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Quote by broegger
[tex]\langle \xi^2 \rangle = \int\xi^2\vert\psi(\xi)\vert^2d\xi.[/tex]
Be carefull that your psi is still correctly normalized when expressed in xi !

cheers,
Patrick.
Feb13-05, 06:02 AM   #3
 
Yep, that's it :D

Thank you very much! How do you spot these things right away?
Feb13-05, 06:16 AM   #4
 
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Change-of-variable


Quote by broegger
Yep, that's it :D

Thank you very much! How do you spot these things right away?
Because I'm an expert

An expert is someone who has made all possible and imaginable errors in a very small domain

cheers,
Patrick.
Feb13-05, 07:09 AM   #5
 
I see I'm on the right track, then...
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