energy spectrum

The energy spectrum of a quantum system is the complete set of energies it is allowed to have — the full list of all the eigenvalues of its Hamiltonian. It is the system's energetic fingerprint, and reading it off is one of the central goals of quantum mechanics, because it predicts the light a system emits, absorbs, and how it responds to being probed.

Spectra come in two flavours, and which one you get depends on the potential. A particle bound in a well has a discrete spectrum: a ladder of separated, quantised levels, like the rungs that give an atom its sharp spectral lines. A particle that is free or can escape to infinity has a continuous spectrum, with energies allowed across an unbroken range. Many real systems show both at once, discrete below an escape threshold and continuous above it.

The spacing and pattern of the spectrum reveal the physics inside. Equally spaced levels betray a harmonic oscillator; levels that crowd together toward a limit betray the attractive pull of an atom; gaps and bands betray a crystal. By measuring an energy spectrum in the laboratory and comparing it with what the Schrödinger equation predicts, physicists reverse-engineer the forces and structure of matter.