Question

Expectation valu of (momentum)^4 in one dim linear harmonic oscilator

Answer 1

Answer

n-th state of one dim linear harmonic oscilator is given by

$|n>=(\frac{\alpha}{2^n n! \sqrt{\pi}}) ^{1/2}e^{-\alpha^2 x^2/2} H_n(\alpha x)$

Now

Expectation value of momentum^4

$<p^4>=\frac{<n|p^4n>}{<n|n>}$

Putting

$P=\frac{-i\hbar}{2m}$ And state n

So expectation value is

$<p^4>=(\frac{\hbar m\omega}{2}) ^2(6n^2+6n+3)$