15 Hermite Polynomials
15.1 Definition of Hermite Polynomials
The Hermite polynomials \(H_{n}\left(x\right)\) are defined over the Hilbert space \(\mathcal{L}^{2}(-\infty,\infty)\) with weight function \(w\left(x\right)=\exp\left(-x^{2}\right)\).
- Hermite’s differential equation
\[y''+\left(2n+1-x^{2}\right)y=0\]
is solved by the Hermite functions \[\psi_{n}\left(x\right)=\left(2^{n}n!\sqrt{\pi}\right)^{-1/2}\exp\left(-x^{2}/2\right)H_{n}\left(x\right)\]
where \(H_{n}\left(x\right)\) are the Hermite Polynomials.
First few terms: \[\begin{aligned} H_{0}\left(x\right) & =1\\ H_{1}\left(x\right) & =2x\\ H_{2}\left(x\right) & =4x^{2}-2 \end{aligned}\]
Rodrigues’ formula
\[H_{n}\left(x\right)=\left(2x-\text{d}_{x}\right)^{n}\cdot1\]
by which it is meant that the operator in parentheses acts on 1, e.g.
\[\begin{aligned} H_{2}\left(x\right) & =\left(2x-\text{d}_{x}\right)^{2}\cdot1\\ & =\left(2x-\text{d}_{x}\right)\left(2x-\text{d}_{x}\right)\cdot1\\ & =\left(4x^{2}-2x\text{d}_{x}-\text{d}_{x}\left(2x\right)-\text{d}^{2}_{x}\right)\cdot1\\ & \downarrow\text{(chain rule on 3rd term)}\nonumber \\ & =\left(4x^{2}-2x\text{d}_{x}-2-2x\text{d}_{x}-\text{d}^{2}_{x}\right)\cdot1\\ & =4x^{2}-2. \end{aligned}\]
- Recursion formula: given \(H_{0}\left(x\right)\),
\[H_{n+1}\left(x\right)=2xH_{n}\left(x\right)-H_{n}'\left(x\right)\]
- Generating function
\[\exp\left(2xt-t^{2}\right)=\sum^{\infty}_{n=0}\frac{1}{n!}H_{n}\left(x\right)t^{n}\]
meaning
\[H_{n}\left(x\right)=\left.\partial^{n}_{t}\exp\left(2xt-t^{2}\right)\right|_{t=0}.\]
- Contour integral expression
\[H_{n}\left(z\right)=\frac{n!}{2\pi i}\oint_{C}\exp\left(-t^{2}+2tz\right)t^{-n-1}\text{d}t\]
where \(C\) encircles the origin anticlockwise.
15.2 Orthogonality, Normalisation and Completeness
The Hermite functions are orthogonal and normalised: \[\int^{\infty}_{-\infty}\psi_{n}\left(x\right)\psi_{m}\left(x\right)\textrm{d}x=\delta_{nm}.\]
They are also complete:
\[\sum^{\infty}_{n=0}\psi_{n}\left(x\right)\psi_{n}\left(y\right)=\delta\left(x-y\right).\]
Hence, they form a complete orthonormal basis for the Hilbert space \(\mathcal{L}^{2}\left(\mathbb{R}\right)\).
15.3 QM example: Quantum Harmonic Oscillator
A very important problem in quantum mechanics is the quantum harmonic oscillator:
\[\hat{V}=\frac{1}{2}m\omega^{2}\hat{x}^{2}.\]
In the position basis, the time independent Schroedinger equation reads
\[-\frac{\hbar^{2}}{2m}\frac{\text{d}^{2}\phi_{n}\left(x\right)}{\text{d}x^{2}}+\frac{1}{2}m\omega^{2}x^{2}\phi_{n}\left(x\right)=E_{n}\phi_{n}\left(x\right).\label{eq:SHO TISE}\]
It is convenient to rescale using \(x=\alpha y\):
\[-\frac{\hbar^{2}}{2m\alpha^{2}}\frac{\text{d}^{2}\phi_{n}\left(y\right)}{\text{d}y^{2}}+\frac{1}{2}m\omega^{2}\alpha^{2}y^{2}\phi_{n}\left(y\right)=E_{n}\phi_{n}\left(y\right)\]
and selecting
\[\alpha^{2}=\frac{\hbar}{m\omega}\]
gives
\[\frac{1}{2}\left(-\frac{\text{d}^{2}\phi_{n}\left(y\right)}{\text{d}y^{2}}+y^{2}\phi_{n}\left(y\right)\right)=\epsilon_{n}\phi_{n}\left(y\right) \tag{15.1}\]
where
\[\epsilon_{n}\triangleq\frac{E_{n}}{\hbar\omega}.\]
Equation 15.1 is a second order ODE. Substitute
\[\phi_{n}\left(y\right)=H_{n}\left(y\right)\exp\left(-\frac{y^{2}}{2}\right)\]
to reduce Equation 15.1 to
\[H_{n}\left(y\right)''-2yH_{n}\left(y\right)'+\left(2\epsilon-1\right)H_{n}\left(y\right)=0.\]
This is Hermite’s equation. It can be solved with Frobenius series to yield \(H_{n}\left(y\right)\), the Hermite polynomials for \(n\ge1\):
\[H_{n}\left(y\right)=\left(-1\right)^{n}\exp\left(y^{2}\right)\frac{\text{d}^{n}}{\text{d}y^{n}}\exp\left(-y^{2}\right)\]
(where \(H_{0}=1\)) with energy eigenvalues
\[\epsilon_{n}=n+1/2\]
for integer \(n\ge0\). Returning to the original scaling we have
\[E_{n}=\hbar\omega\left(n+\frac{1}{2}\right).\label{eq:SHO eigenvalues}\]