# Potential and kinetic energies in Quantum Oscillator

1. Jun 5, 2010

### alchemistoff

1. The problem statement, all variables and given/known data

Problem 9. Evaluate the matrix elements $$\langle n + \nu|x^2|n\rangle$$ and $$\langle n + \nu|p^2|n\rangle$$ in
the harmonic oscillator basis, for $$\nu = 1, 2, 3, 4$$ :
1. Using the closure property and the matrix elements.
2. Applying the operators $$x^22$$ and $$p^2$$ , expressed in terms of the $$a+, a$$ on the eigenstates.
3. Find the ratio $$\langle n + \nu|K|n\rangle/\langle n + \nu|V|n\rangle$$ $$(\nu = 0, \pm2)$$ between the kinetic
and the potential energy matrix elements. Justify the differences in sign
on quantum mechanical grounds.

2. Relevant equations

$$H=\frac{p^2}{2m}+\frac{m\omega ^2}{2}x^2$$

3. The attempt at a solution

$$\frac{\langle n|K|n \rangle}{\langle n|V|n \rangle}=-\frac{\langle n\pm 2|K|n \rangle}{\langle n\pm 2|V|n \rangle}=1$$
...but I cannot justify the difference in sign on quantum mechanical grounds!!!