What is the difference between probabilistic and physicists Hermite polynomials?

Question

In the equation they differ only by a fraction of 2 and in the results one is without a constant in x^n and one has those constants. Why are they defined differently and what are their separate uses?

Answer 1

In Wikipedia's notation, the probabilistic functions are $\operatorname{He}_n(x):=(-1)^ne^{x^2/2}\tfrac{d^n}{dx^n}e^{-x^2/2}$; the physical ones are $H_n(x):=(-1)^ne^{x^2}\tfrac{d^n}{dx^n}e^{-x^2}=2^{n/2}\operatorname{He}_n(x\sqrt{2})$. I'll give one motivation for each:

• In terms of the $N(0,\,1)$ PDF $\phi(x):=\tfrac{1}{\sqrt{2\pi}}\exp\tfrac{-x^2}{2}$, $\int_{\Bbb R}\operatorname{He}_m(x)\operatorname{He}_n(x)\phi(x)dx=n!\delta_{mn}$, i.e. the functions $\tfrac{1}{\sqrt{n!}}\operatorname{He}_n(x)$ are orthogonal with respect to weight $\phi$.
• The quantum simple harmonic oscillator has $n$th-excitation $x$-space wavefunction $\psi_n(\sqrt{m\omega/\hbar}x)$ with $\psi_n(x):=(2^nn!\sqrt{\pi})^{-1/2}e^{-x^2/2}H_n(x)$.