energy eigenvalue

An energy eigenvalue is one of the specific energies that a quantum system is allowed to have. When you solve the time-independent Schrödinger equation Ĥψ = Eψ, the equation only has well-behaved solutions for certain values of the number E. Those permitted values are the energy eigenvalues, and each comes paired with its own energy eigenstate.

For a bound system — an electron trapped near a nucleus, a particle confined to a box — these allowed energies are discrete, separated by gaps. This is the precise mathematical meaning of 'quantised energy levels': not a vague philosophy, but the simple fact that the equation refuses to be solved for the in-between values. The forbidden energies are the ones whose wavefunctions would blow up or fail the boundary conditions.

Energy eigenvalues are the link between the equation and the laboratory. The light an atom emits and absorbs comes in sharp colours whose energies are exactly the gaps between eigenvalues, which is why atomic spectra are the fingerprints we read across the cosmos. Solving for the eigenvalues is, in many problems, the whole point of doing quantum mechanics at all.