Harmonic oscillator (quantum)

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}k x^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, &nabla;&sup2; is the Laplace operator (del squared), and k is Hooke's spring constant.