# Uncertainty of energy in a quantum harmonic oscillator

1. Apr 4, 2013

### bobred

1. The problem statement, all variables and given/known data
Find the uncertainty of the kinetic energy of a quantum harmonic oscillator in the ground state, using

$\left\langle p^2_x \right\rangle = \displaystyle\frac{\hbar^2}{2a^2}$ and
$\left\langle p^4_x \right\rangle = \displaystyle\frac{3\hbar^2}{4a^2}$

2. Relevant equations
$\Delta E_{kin}=\sqrt{\left\langle E^2_{kin} \right\rangle - \left\langle E_{kin} \right\rangle^2}$

$\left\langle E_{kin} \right\rangle = \displaystyle\frac{\left\langle p^2_x \right\rangle}{2m}$

3. The attempt at a solution
With $\left\langle E_{kin} \right\rangle^2$ I have no problem with but am I valid in saying

$\left\langle E^2_{kin} \right\rangle = \displaystyle\frac{\left\langle p^4_x \right\rangle}{4m^2}$?

2. Apr 4, 2013

Yes.