# Simple harmonic oscillator: zero point energy?

Today we had a lecture on the simple harmonic oscillator and its quantum mechanical treatment. My teacher derived the equation for it and finally concluded it has some zero point energy.

1. My question is what is the physical significance of this zero point energy?

2. And from where does this energy come from?

3. And yes one more thing. I'd like to know the physical examples where the concept of zero point energy is used.

• A good starting point is this. Sep 21, 2013 at 11:22
• A nice recent experiment on zero point energy in macroscopic bodies has been explained by PhD comic's Jorge Cham: youtu.be/pktWhH6m_DM Sep 21, 2013 at 13:55

Short answer: By the uncertainty principle, the harmonic oscillator can't be localized at the minimum value of potential energy, i.e., $x=0$, because, by the uncertainty principle, it's momentum would become large (strictly speaking, the expectation value of $p^2$, and thereby it's kinetic energy, becomes large). The lowest energy state of the harmonic oscillator is a compromise between minimizing potential energy (i.e., $x^2$) and kinetic energy (i.e., $p^2$), which cannot be done simultaneously, because $\langle x^2\rangle \langle p^2 \rangle \ge \frac{\hbar^2}{4}$.

As your probably know already, energy is quantized, that means it comes by steps, you can't have an arbitrary energy. That being said, zero point energy is just the lowest energy the system can have, there is no lower level there is no possible state wich energy is between zero and zero point energy.

It's not that energy comes from anywhere, is just that the wave function has discrete, very particular solutions and zero point energy is the one solution/state that has the lowest energy.

• Dear Sergio I edited you last sentence: please check the sense carefully Dec 21, 2013 at 1:03
• Whether or not energy is quantized depends on the spectrum of the Hamiltonian, and that's not necessarily discrete. In the case of the one-dimensional harmonical oscillator, every energy is a bound state, so the energy spectrum is discrete. But this discreteness should not be generalized to arbitrary situations without qualification. Dec 21, 2013 at 1:13

Below this energy, a particle can't go. That's what zero point energy means.