Quantum harmonic oscillator

Example

Let $m > 0$ be a mass, $\omega > 0$ an angular frequency and $\hbar > 0$ the reduced Planck constant. The quantum harmonic oscillator is the Schrödinger operator $$H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2} m \omega^2 x^2$$ on $L^2(\R)$. Its spectrum is purely discrete, and its eigenvalues are $$E_n = \hbar \omega \left( n + \tfrac12 \right), \qquad n = 0, 1, 2, \ldots,$$ each of multiplicity one. With $a = m\omega/\hbar$, an eigenfunction for $E_n$ is $\psi_n(x) = H_n(\sqrt{a}\, x)\, e^{-a x^2/2}$, where $H_n$ is the $n$-th Hermite polynomial. Up to normalisation these are the Hermite functions, and they form an orthonormal basis of $L^2(\R)$ after normalisation.

Check for $n = 0$: with $\psi_0(x) = e^{-a x^2/2}$ one has $\psi_0'' = (a^2 x^2 - a)\psi_0$. So $H\psi_0 = \left( -\frac{\hbar^2 a^2}{2m} x^2 + \frac{\hbar^2 a}{2m} + \frac{m\omega^2}{2} x^2 \right) \psi_0$. Since $\hbar^2 a^2 / (2m) = m\omega^2/2$, the terms in $x^2$ cancel, and $H\psi_0 = \frac{\hbar \omega}{2} \psi_0$.