I want to find the expectation value [tex]\langle x^2 \rangle[/tex] in

[broegger]

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

$$\xi = \sqrt{\frac{m\omega}{\hslash}}x,$$​

and compute the expectation value $$\langle \xi^2 \rangle$$ like this:

$$\langle \xi^2 \rangle = \int\xi^2\vert\psi(\xi)\vert^2d\xi.$$​

Finally I change back to x:

$$\langle \xi^2 \rangle = \langle \frac{m\omega}{\hslash}x^2 \rangle = \frac{m\omega}{\hslash}\langle x^2 \rangle \Rightarrow \langle x^2 \rangle = \frac{\hslash}{m\omega}\langle \xi^2 \rangle.$$​

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]

$$\langle \xi^2 \rangle = \int\xi^2\vert\psi(\xi)\vert^2d\xi.$$​

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]

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 :rofl:

cheers,
Patrick.

 

[broegger]

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