- #1
natugnaro
- 64
- 1
Homework Statement
Is there any way to find [tex]<\varphi_{n}(x)|x|\varphi_{m}(x)|>[/tex] (where phi_n(x) , phi_m(x) are eigenfunction of harmonic oscillator) without doing integral ?
Homework Equations
perhaps orthonormality of hermite polynomials ?
[tex]\int^{+\infty}_{-\infty}H_{n}(x)H_{m}(x)e^{-x^{2}}dx=\delta_{nm}(Pi)^{1/2}2^{n}n![/tex]
The Attempt at a Solution
Actually i need to find <x> from known [tex]\psi(x,t)[/tex].
[tex]<x>=<\psi(x,t)|x|\psi(x,t)|>[/tex]
This gives me a lot of [tex]<\varphi_{n}(x)|x|\varphi_{m}(x)|>[/tex] terms, some of them cancel out (for m=n) ,
but at the end I'm left with three terms thath I have to calculate ([tex]<\varphi_{1}(x)|x|\varphi_{2}(x)>,<\varphi_{2}(x)|x|\varphi_{3}(x)>, <\varphi_{3}(x)|x|\varphi_{1}(x)>[/tex]
and that also looks like a lot of work.
final result is <x>=0
(This is problem from Schaums QM (supplementary prob.))