What quantum system has equally spaced energy levels?

The quantum harmonic oscillator is the quantum system with equally spaced energy levels. Its allowed energies are Eₙ = (n + 1/2)ℏω, where n = 0, 1, 2, …, so every adjacent level differs by exactly ℏω.

The model describes a particle subject to a restoring force proportional to its displacement, such as an idealized mass on a spring. It is important in quantum mechanics because many real systems behave approximately like harmonic oscillators near a stable equilibrium. Vibrating molecules, crystal lattices, and electromagnetic-field modes can all be treated this way.

The extra 1/2ℏω in the ground-state energy is called zero-point energy. This means the oscillator cannot have zero energy, even in its lowest state. The evenly spaced spectrum also distinguishes it from systems such as the hydrogen atom and particle in a box, whose level spacings change with the quantum number.

In quantum field theory, each quantized mode of vibration acts like an oscillator; the resulting units of vibrational energy are called phonons in solids and photons for electromagnetic modes.