Quantum Harmonic Oscillator Number & Ladder Operators

A quantum harmonic oscillator is similar in many ways to a classical one; it is simply a particle that undergoes repetitive motion, bound by a potential with an equilibrium point. The major difference between quantum and classical harmonic oscillators, however, is that in quantum systems, particles can occupy only certain discrete energy levels. Here, the number operator and ladder operators — operators concerned with energy levels — of quantum harmonic oscillators are derived.

Zero-Point Energy

The zero-point energy of a quantum mechanical system is the energy of the system in its ground state (when the system has the lowest possible energy). Take for example, a harmonically oscillating particle in a potential well. While classically, it may be possible to suppose that the particle could have zero energy if its kinetic energy and the energy potential at its position are both zero, quantum mechanically speaking, such a system is impossible. The Heisenberg uncertainty relation forbids such a scenario where both the particle's position and momentum are known. Here, the Heisenberg uncertainty relation and the Schrodinger equation are used to derive the zero-point energy of a quantum harmonic oscillator.