# Vacuum energy and harmonic oscillator

1. Sep 12, 2015

### astrolollo

Hi everyone
I was wondering, why is vacuum energy related to the zero point energy of an harmonic oscillator? The hamiltonian of an harmonic oscillator is $H= \frac{p^{2}}{2m} + \frac{1}{2} \omega x^{2}$. Where does the harmonic potential term come from in the vacuum?

2. Sep 12, 2015

### vanhees71

What do you mean by "vacuum".

The harmonic oscillator, described by the Hamiltonian,
$$\hat{H}=\frac{1}{2m} \hat{p}^2 + \frac{m \omega^2}{2} \hat{x}^2,$$
has a ground state with the energy eigenvalue $E_0=\hbar \omega/2$, which is the lowest possible energy the oscillator can have. That energy value is $>0$, while the lowest possible energy value of the classical harmonic oscillator is 0.

3. Sep 12, 2015

### astrolollo

4. Sep 12, 2015

### blue_leaf77

Because upon quantizing electromagnetic field in a cavity, the Hamiltonian turns out to be identical to that of a Harmonic oscillator. The eigenstates and eigenvalues of EM field's Hamiltonian can then be inferred to have the same properties as those corresponding to harmonic oscillator, where as vanhees pointed out has non-zero ground state energy. The ground state of a quantized EM field is what we termed as vacuum, this naming refers to the fact that the expectation value of number operator is zero for ground state.