Harmonic oscillator (quantum)

First four harmonic oscillator functions ψ_n. Potential V(x) is shown as reference. Function values are shifted upward by the corresponding energy values $(n+{\tfrac {1}{2}})\hbar \omega .$

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

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

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, and k is Hooke's spring constant. See the classical harmonic oscillator for further explanation of m and k.

The solutions of the Schrödinger equation are characterized by a vibration quantum number n = 0,1,2, .. and are of the form

$\psi _{n}(x)=\left({\frac {\beta ^{2}}{\pi }}\right)^{1/4}\;{\frac {1}{\sqrt {2^{n}\,n!}}}\;e^{-(\beta x)^{2}/2}\;H_{n}(\beta x)\quad {\hbox{with}}\quad E_{n}=(n+{\tfrac {1}{2}})\hbar \omega .$

Here

$\beta \equiv {\sqrt {\frac {m\omega }{\hbar }}}\quad {\hbox{and}}\quad \omega \equiv {\sqrt {\frac {k}{m}}}$

The functions H_n(x) are Hermite polynomials; the first few are:

$H_{0}(x)=1,\quad H_{1}(x)=2x,\quad H_{2}(x)=4x^{2}-2,\quad H_{3}(x)=8x^{3}-12x,\quad H_{4}(x)=16x^{4}-48x^{2}+12.$

The graphs of the first four eigenfunctions are shown in the figure. Note that the functions of even n are even, that is, $f_{2n}(-x)=f_{2n}(x)\,$, while those of odd n are antisymmetric $f_{2n+1}(-x)=-f_{2n+1}(x)\,.$