Orthogonality property of Hermite polynomial

Orthogonality property of Hermite Polynomial:

We have Hermite differential equations as:

$$\frac{d^2y}{dx^2}-2x\cdot\frac{dy}{dx}+2ny=0\dotsm(1)$$

Since \(H_n(x)\) is a solution of equation (1), then we can write

$$\frac{d^2H_n(x)}{dx^2}- 2x\cdot \frac{dH_n(x)}{dx}+2nH_n(x)=0\dotsm(2)$$

Hints:

\(\frac{d^2H_n(x)}{dx^2}-2x\cdot \frac{dH_n(x)}{d(x)}+2nH_n(x)=0\)

Put\( \frac{dH_n(x)}{dx}=\phi\)

\(or,\;\; \frac{d\phi}{dx}-2x\phi=Q\)

\(\frac{dy}{dx}+Py=Q\), I.F =\(e^{\int pdx}\)

\(I.F= e ^{\int 2xdx}=e^{-2}\cdot \frac{x^2}{2}=e^{-x^2}\)

Multiplying equation (2) on both side by \(e^{-x^2}\) we get,

$$ e^{-x^2}\frac{d^2H_n(x)}{dx^2}- e^{-x^2}\cdot 2x\frac{dH_n(x)}{dx}+2ne^{-x^2}H_n(x)=0$$

$$or,\;\; \frac{d}{dx}\biggl[e^{-x^2} \frac{dH_n(x)}{dx}\biggr] + 2ne^{-x^2} H_n(x)=0\dotsm(*)$$

For m order solution i.e. \(y= H_m(x)\) we can write

$$\frac{d}{dx}\biggl[e^{-x^2}\frac{dH_m(x)}{dx}\biggr] + 2me^{-x^2}H_m(x)=0\dotsm(3)$$

Multiplying equation (*) with \(H_m(x) \) and (3) with \(H_n(x)\) and subtract (*) we get,

$$or\;\; H_m(x)\cdot \frac{d}{dx}\biggl[\frac{e^{-x^2} dH_n(x)}{dx}\biggr]- H_n(x) \frac{d}{dx}\biggl[e^{-x^2}\frac{dH_m(x)}{dx}\biggr]+ 2(n-m) e^{-x^2} H_n(x) H_m(x)=0$$

$$or,\;\; \frac{d}{dx}\biggl[H_m(x)e^{-x^2} \frac{dH_n(x)}{dx}-H_n(x) e^{-x^2}\frac{dH_m(x)}{dx}\biggr]+2(n-m) e^{-x^2} H_n(x)H_m(x)=0$$

Because: \( H_m(x) \frac{d}{dx}\biggl[e^{-x^2}\frac{dH_n(x)}{d(x)}\biggr]+ e^{-x^2} \frac{dH_n(x)}{dx}\cdot \frac{dH_m(x)}{dx}-H_n(x)\frac{d}{dx}\biggl[e^{-x^2}\frac{dH_m(x)}{dx}\biggr]-\frac{e^{-x^2} d_m(x)}{dx}\cdot\frac{dH_n(x)}{dx}\)

Integrating above equation from \(-\infty\;to\;+\infty\) we get,

$$\int_{-\infty}^{+\infty} \frac{d}{dx}\biggl[H_m(x) e^{-x^2}\frac{dH_n(x)}{dx}-H_n(x)e^{-x^2}\frac{dH_m(x)}{dx}\biggr]dx+2(n-m) \int_{-\infty}^{+\infty}e^{-x^2} H_n(x)H_m(x) dx=0$$

$$or,\;\; H_m(x) e^{-x^2}\frac{dH_n(x)}{dx}- H_n(x) e^{-x^2} \frac{dH_m(x)}{dx}\biggl]_{-\infty}^{+\infty}+ 2(n-m) \int_{-\infty}^{+\infty}e^{-x^2} H_n(x) H_m(x)dx=0$$

$$or,\;\; 2(n-m) \int_{-\infty}^{+\infty} e^{-x^2}H_n(x)H_m(x) dx=0\dotsm(4)$$

Case I: When \(n\ne m\)

$$\int_{-\infty}^{+\infty} e^{-x^2}H_n(x) H_m(x) dx=0\dotsm(5)$$

Case II: when n=m

$$\int_{-\infty}^{+\infty} e^{-x^2} H_n(x) H_n(x) dx=?\dotsm(6)$$

We have generating function of Hermite polynomial

$$\sum_{n=0}^\infty \frac{H_n(x) t^n}{n!}= e^{2xt- t^2}$$

Squaring both side we get,

$$,\;\;\sum_{n=0}^\infty \frac{t^{2n} H_n(x) H_n(x)}{(n!)^2}= e^{4xt-2t^2}$$

Multiplying both side by \(e^{-x^2}\) we get,

$$or,\;\; \sum_{n=0}^\infty \frac{t^{2n}}{(n!)^2} e^{-x^2} H_n(x) H_n(x)= e^{-x^2}\cdot e^{4xt-2t^2}$$

On integrating both sides from \(-\infty \) to \(\infty\) we get,

$$or,\;\; \sum_{n=0}^\infty\frac{t^2n}{(n!)^2}\int_{-\infty}^{+\infty} e^{-x^2} H_n(x) H_n(x) dx=\int_{-\infty}^{+\infty} e^{-x^2+4xt-2t^2}dx$$

$$=\int_{-\infty}^{+\infty} e^{-(x-2t)^2+2t^2}dx$$

$$=e^{2t^2}\int_{-\infty}^{+\infty} e^{-(x-2t)^2}dx= e^{2t^2}\sqrt{\pi}$$

Put \(y= x-2t\)

$$dy= dx$$

$$or,\;\; \int_{-\infty}^{+\infty} e^{-x^2}H_n(x) H_n(x) dx= e^{2t^2}\sqrt{\pi}$$

$$= \sqrt{\pi} \biggl[1+\frac{2t^2}{1!}+\frac{(2t^2)^2}{2!}+\dotsm+\frac{(2t^2)^n}{n!}+\dotsm\biggr]$$

Now collecting the coefficient of \(t^2n\) on both side

$$\frac{1}{(n!)^2}\int_{-\infty}^{+\infty} e^{-x^2} H_n(x) H_n(x) dx=\sqrt{\pi} \dotsm\frac{2^n}{n!}$$

$$or,\;\; \int_{-\infty}^{+\infty} e^{-x^2} H_n(x)H_n(x)dx= \sqrt{\pi}2^n n!\dotsm(7)$$

From (6) and (7)

$$or,\;\; \int_{-\infty}^{+\infty} e^{-x^2} H_n(x) H_m(x) dx= \sqrt{\pi}2^n n! \delta_{mn}$$

Where,

\(\delta_{mn}\)=1 for m=n

\(\delta_{mn}=0\) for m\(\ne\)n

Combining equation (5) and (6) we get,

$$\int_{-\infty}^{+\infty} e^{-x^2}H_n(x) H_m(x) dx= \sqrt{\pi}2^n n! \delta_{mn}$$

This is orthogonality relation of Hermite polynomial.

Reference:

1. Dass, H. K.Mathematical Physics. New Delhi: S. Chand, 2005.
2. Vaughn, Michael T.Introduction to Mathematical Physics. Weinheim: Wiley-VCH, 2007
3. Butkov, Eugene.Mathematical Physics. Reading, MA: Addison-Wesley Pub., 1968.
4. Carroll, Robert W.Mathematical Physics. Amsterdam: North-Holland, 1988.
5. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.