quantum number n

The quantum number n is the integer that labels which energy level an oscillator occupies. It starts at zero for the ground state and counts upward: n equals zero, one, two, and so on. Each value of n picks out one rung of the energy ladder, one allowed wavefunction, and one definite energy, ℏω times the quantity n plus one half.

Quantum numbers like this are how quantisation shows up in practice. Instead of a continuous range, the oscillator's energies are tagged by a discrete counter, and physical quantities follow simple rules in terms of it. The number n also tells you how structured the wavefunction is: the state with quantum number n has exactly n nodes, the places where the wavefunction crosses zero, so higher n means a more wiggly, higher-energy shape.

In the particle interpretation, n is simply the number of quanta present — n photons, n phonons, n excitations of a field. This double meaning, as both an energy-level label and a particle count, is one of the deep unifying ideas that lets a single piece of mathematics describe vibrating springs and populations of particles in the same breath.