Change-of-variable

broegger

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.

vanesch

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.

broegger

Yep, that's it :D

Thank you very much! How do you spot these things right away?

vanesch

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.

broegger

I see I'm on the right track, then...

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broegger	Yep, that's it :D Thank you very much! How do you spot these things right away?