Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The Hamiltonian of the particle is: $${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+{\frac {1}{2}}k{\hat {x}}^{2}={\frac {{\hat {p}}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}{\hat {x}}^{2}\,,}$$ where m is the particle's mass, k is the force constant, ${\textstyle \omega ={\sqrt {k/m}}}$ is the angular frequency of the oscillator, ${\displaystyle {\hat {x}}}$ is the position operator (given by x in the coordinate basis), and ${\displaystyle {\hat {p}}}$ is the momentum operator (given by ${\displaystyle {\hat {p}}=-i\hbar \,\partial /\partial x}$ in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.

The time-independent Schrödinger equation (TISE) is, $${\displaystyle {\hat {H}}\left|\psi \right\rangle =E\left|\psi \right\rangle ~,}$$ where ${\displaystyle E}$ denotes a real number (which needs to be determined) that will specify a time-independent energy level, or eigenvalue, and the solution ${\displaystyle |\psi \rangle }$ denotes that level's energy eigenstate.

Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function ${\displaystyle \langle x|\psi \rangle =\psi (x)}$, using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions, $${\displaystyle \psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}e^{-{\frac {m\omega x^{2}}{2\hbar }}}H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .}$$

The functions H_n are the physicists' Hermite polynomials, $${\displaystyle H_{n}(z)=(-1)^{n}~e^{z^{2}}{\frac {d^{n}}{dz^{n}}}\left(e^{-z^{2}}\right).}$$

The corresponding energy levels are $${\displaystyle E_{n}=\hbar \omega {\bigl (}n+{\tfrac {1}{2}}{\bigr )}.}$$The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be ${\textstyle \langle {\hat {x}}\rangle =0}$ and ${\textstyle \langle {\hat {p}}\rangle =0}$ owing to the symmetry of the problem, whereas:

${\displaystyle \langle {\hat {x}}^{2}\rangle =(2n+1){\frac {\hbar }{2m\omega }}=\sigma _{x}^{2}}$

${\displaystyle \langle {\hat {p}}^{2}\rangle =(2n+1){\frac {m\hbar \omega }{2}}=\sigma _{p}^{2}}$