What is the minimum possible energy of a quantum harmonic oscillator called?

The minimum possible energy of a quantum harmonic oscillator is called zero-point energy.

Unlike a classical oscillator, a quantum oscillator cannot be perfectly motionless at zero energy. Its lowest allowed energy is one-half of the oscillator frequency multiplied by the reduced Planck constant. This residual energy is called zero-point energy.

The result follows from the structure of quantum states and the position–momentum uncertainty relation. If both position and momentum were exactly zero, the oscillator would have a precisely defined state that quantum mechanics does not permit. The ground state therefore retains unavoidable fluctuations.

Zero-point energy appears in many systems, including molecular vibrations, electromagnetic modes, and quantized fields. It is related to vacuum fluctuations, but the terms are not identical: zero-point energy describes the lowest energy of a mode, while vacuum fluctuations refer to fluctuations of quantum fields in their vacuum state. Its interpretation in cosmology remains an active subject.