The quantum harmonic oscillator

A mass on a spring is the archetypal classical oscillator: it can have any energy at all, set by how far it is pulled back before release, and it can sit perfectly still at the bottom of its potential well with exactly zero energy. Solve the Schrödinger equation for the same quadratic potential and both of those classical facts turn out to be wrong at the quantum scale.

The allowed energies of a quantum harmonic oscillator form a discrete ladder, En = (n + 1/2)ħω, where ω is the classical angular frequency of the oscillator and n = 0, 1, 2, ... labels the level. Two features of this formula are worth dwelling on. First, the levels are evenly spaced: moving from any level to the next one up always costs exactly the same amount of energy, ħω, no matter which level you start from — a strikingly different pattern from, say, the hydrogen atom, whose levels bunch closer together at higher energy. Second, and more surprising, even the lowest level n = 0 does not have zero energy: it has E0 = ħω/2, called the zero-point energy. A quantum oscillator can never be brought to perfect rest at the bottom of its potential well; even in its ground state it retains an irreducible minimum of both kinetic and potential energy, a direct consequence of the uncertainty principle — pinning the particle to an exact position with exact zero momentum would violate the trade-off between position and momentum uncertainty.

This zero-point energy is not a mathematical curiosity; it is measurable and physically consequential. It is part of why solids retain residual atomic vibration even cooled arbitrarily close to absolute zero, and the same evenly spaced-ladder structure appears throughout physics wherever a system is approximately harmonic near a stable equilibrium — vibrating diatomic molecules and normal modes of a crystal lattice are both, to good approximation, collections of quantum harmonic oscillators, inheriting both the even energy spacing and the unavoidable zero-point energy of the idealized model.