kinetic-energy operator

The kinetic-energy operator is the part of the Hamiltonian that represents the energy of motion. In classical physics kinetic energy is one-half mass times velocity squared, or equivalently momentum squared divided by twice the mass. Quantum mechanics keeps the same formula but promotes momentum to an operator, and momentum as an operator is a derivative in space.

Because momentum becomes a spatial derivative, squaring it makes the kinetic-energy operator a second derivative, written −(ℏ²/2m) times the curvature of the wavefunction. The physical reading is vivid: the more sharply the wavefunction bends and wiggles, the more kinetic energy it carries. A smooth, gently curving wave is calm and low in energy; a tightly oscillating one is fast and energetic.

This single term is responsible for some of quantum mechanics' most distinctive behaviour. It is why squeezing a particle into a small region costs energy — confinement forces sharper curvature — and so it underlies the zero-point energy that keeps a quantum particle restlessly jiggling even at absolute zero, never able to sit perfectly still at the bottom of a well.