Harmonic oscillator (quantum)

First four harmonic oscillator functions. Potential is shown as reference. Zero of function n=0,1,2,3 is shifted upward by the energy value (n+½) h ν

In quantum mechanics, the one-dimensional harmonic oscillator is one of the few systems that can be treated exactly. Its time-independent Schrödinger equation has the form

$\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {1}{2}}kx^{2}\right]\psi =E\psi$

The two terms between square brackets are the Hamiltonian (energy operator) of the system: the first term is the kinetic energy operator and the second the potential energy operator. The quantity $\hbar$ is Planck's reduced constant, m is the mass of the oscillator, ∇² is the Laplace operator (del squared), and k is Hooke's spring constant.