Zero-Point Energy

The quantum harmonic oscillator is one of the most important exactly solvable systems in quantum mechanics. Its energy spectrum is equally spaced:

Here $\omega$ is the classical angular frequency of the oscillator and $n$ is the quantum number. Each allowed energy is separated from the next by a fixed gap $\hbar\omega$.

The ground state and zero-point energy

The ground state is the lowest-energy state, reached by setting $n = 0$:

This non-zero minimum energy is called the zero-point energy. It is a direct consequence of the Heisenberg uncertainty principle: a particle confined near the bottom of the well cannot simultaneously have zero kinetic energy and a precisely known position. The uncertainty in momentum $\Delta p$ required by $\Delta x \, \Delta p \geq \hbar/2$ contributes kinetic energy that can never be removed, even at absolute zero temperature.

Why the spectrum is equally spaced

The energy levels $E_0, E_1, E_2, \ldots$ are equally spaced with gap $\hbar\omega$. Going from level $n$ to level $n+1$ requires absorbing exactly one quantum of energy:

This is consistent with treating each energy quantum as a single phonon (for a lattice vibration) or photon (for an electromagnetic mode). The harmonic oscillator model therefore underlies the quantum theory of radiation, the Debye model of specific heats, and the ladder operators of quantum field theory.

A numerical example

Suppose $\hbar\omega = 0.1\,\text{eV}$ (a typical vibrational energy scale for a diatomic molecule). The first few energy levels are:

The spacing between adjacent levels is always $\Delta E = \hbar\omega = 0.1\,\text{eV}$.

Try it

This is a numerical exercise — return a number. An oscillator has $\hbar\omega = 0.1\,\text{eV}$. What is the zero-point energy $E_0$ in eV?

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