eigenfunction expansion

Eigenfunction expansion is the technique of writing an arbitrary quantum state as a weighted sum of energy eigenstates. Because the energy eigenstates of a Hamiltonian form a complete basis, any wavefunction whatsoever can be expressed as their combination, each with its own complex coefficient. Finding those coefficients is like decomposing a musical chord into its pure component tones.

Its power shows up in time evolution. A general state is hard to evolve directly, but each eigenstate evolves trivially — it just acquires a phase that turns at a rate set by its energy. So the recipe is simple: expand the state into eigenstates, let every component spin its own phase, then add them all back together. The hard differential equation becomes easy bookkeeping.

The coefficients also carry physical meaning by the Born rule. The squared magnitude of the coefficient on a given eigenstate is the probability that an energy measurement will return that eigenstate's energy. So the expansion does double duty: it solves the dynamics and, at the same time, tells you the odds of each possible energy outcome.