energy levels

Energy levels are the discrete, allowed values of energy that a confined quantum system may have. A trapped particle cannot hold any energy it likes; it must sit on one of a fixed set of rungs, with the gaps between them strictly off-limits. This is the defining signature of the quantum world for bound systems, in sharp contrast to a classical object like a pendulum, whose energy can be tuned to any value at all.

The reason for this discreteness is the wave nature of matter combined with confinement. A bound particle's wavefunction must fit cleanly into its trap and satisfy conditions at the edges, just as a vibrating string can only sound certain notes set by its length. Only particular energies allow a self-consistent wavefunction, and those become the levels. The lowest is the ground state; the higher ones are excited states.

Energy levels are not abstract bookkeeping — they are visible. When an atom or molecule jumps from a higher level to a lower one, it emits light whose colour is fixed by the gap between them, and the reverse jump absorbs light of the same colour. The crisp lines in the spectrum of a glowing gas are a direct photograph of its energy levels, and reading those lines is how we decode the make-up of stars and distant galaxies.