QM: expectation value of a harmonic oscillator

[ktravelet]

first post! but for bad reasons lol

Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2

i solved for x: x=sqrt(h/2mw)((a+)+(a-))
so <x> integral of si x ((a+)+(a-)) x si.
therefor the integral of si(n+1) x si + si(n-1) x si.
si(n+1) x si as far as I know is always 0 so this would mean <x>=0?
this same convention would be used for <p>.

sorry for the sloppy work but I'm pretty sure <x>, and <p> shouldn't = 0

please help, and thanks for the posts!

 

[nrqed]

first post! but for bad reasons lol

Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2

i solved for x: x=sqrt(h/2mw)((a+)+(a-))
so <x> integral of si x ((a+)+(a-)) x si.
therefor the integral of si(n+1) x si + si(n-1) x si.
si(n+1) x si as far as I know is always 0 so this would mean <x>=0?
this same convention would be used for <p>.

sorry for the sloppy work but I'm pretty sure <x>, and <p> shouldn't = 0

please help, and thanks for the posts!

welcome!

Your result is correct! They are both zero. You can see it directly if you think in terms of raising and lowering operator since, for any eigenstate |n>, we have

$$\langle n| a | n\rangle = \langle n | a^\dagger | n \rangle = 0$$

 

[ktravelet]

Alright thanks a lot! I have absolutely no idea how to find <T>. If you can give me a pointer in the right direction that would be greatly appreciated. Thank you very much for all the help.