Characteristics of Energy eigen value

Characteristics of Energy eigen value :

1. The minimum energy of oscillator is called ground state energy.

Zero point energy \((E_0\))=\( \frac{\hbar\omega}{2}\)

$$=\frac{h}{2\pi}\times\frac{2\pi\nu}{2}$$

$$=\frac{h\nu}{2}$$

Where, \(\nu\)= Frequency of oscillator.

2. This value of zero point energy or ground state energy is consequence of uncertainty principle [ Oscillator never be rest ]

$$E_1= \frac32 \hbar \omega=\frac32 h\nu $$

First exited state.

$$E_2=\frac52\hbar\omega=\frac52 h\nu$$

[ 2nd exited state] and so on.

3. the successive energy level of 1-dimensional simple harmonic oscillator are evenly spaced ( equally spaced) by an amount \(\hbar\omega\).

4. Energy of oscillator is quantised OR energy levels are discrete

5. Each energy states are non degenerate.

Graphical Representation

Using value of \(\lambda\) in equation (4) we get,

$$\frac{d^2H}{dy^2}-2y\frac{dH}{dy}+ (2n) H=0$$

$$\therefore\; \frac{d^2H}{dy^2}-2y\frac{dH}{dy}+ 2nH=0\dotsm(21)$$

Equation (21) is Hermite differential equation.

The solution is Hermite polynomial is,

$$H_n(y)= n^{th} order \;polynomial$$

Form of normalized equation of wave function for oscillator is,

$$\psi_n(y)= Ae^{-\frac{y^2}{2}} H_n(y)\dotsm(22)$$

We have to determine A.

The condition of normalization is

$$\int_{-\infty}^\infty \psi_n^*(y)\psi_n(y)dy=1$$

$$or,\;\; |A|^2\int_{-\infty}^\infty e^{-y^2}H_n(x)H_n(y)dy=1\dotsm(23)$$

We have from generating function of Hermite polynomial.

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

$$e^{2sx-s^2}= \sum_{m=0}^\infty \sum{H_m(x)\rho^m}{m!}$$

Multiplying above two equations we get,

$$e^{2tx-t^2}e^{2sx-s^2}= \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{H_n(x) H_m(x)}{n!}{m!} t^n s^m$$

Multiplying both sides by \(e^{-x^2}\) and integrating between \(-\infty \) to \(+\infty\).

$$\int_{-\infty}^\infty e^{-x^2-t^2-s^2+2tx+2sx-2st}\cdot e^{2st}dx=\int_{-\infty}^\infty \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{H_n(x)}{n!}\frac{H_m(x)}{m!}t^n s^m\cdot dx$$

Using standard integral of the form

$$\int_{-\infty}^\infty e^{-x^2}dx= \sqrt{\pi}$$

We get,

$$or,\;\; \sqrt{\pi}e^{2st}= \sum_{n=0}^\infty \sum_{m=0}^\infty \int_{-\infty}^\infty \frac{H_n(x)}{n!}\frac{H_m(x)}{m!}e^{-x^2}t^n s^m\cdot dx\dotsm(24)$$

$$or,\;\; \sqrt{\pi}\sum_{n=0}^\infty \frac{(2st)^n}{n!}= \sum_{n=0}^\infty \sum_{m=0}^\infty \int_{-\infty}^\infty \frac{H_n(x)}{n!}\frac{H_m(x)}{m!}e^{-x^2} t^n s^n \cdot dx$$

$$or,\sqrt{\pi}2^n\sum_{n=0}^\infty \frac{(st)^n}{n!}=\sum_{n=0}^\infty \sum_{m=0}^\infty \int_{-\infty }^\infty \frac{H_n(x)}{n!}\frac{H_m(x)}{m!}e^{-x^2}t^n s^m\cdot dx\dotsm(25)$$

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

Equating the coefficient of \(t^ns^m\) on both sides, we get

$$\int_{-\infty}^\infty \frac{H_n(x)}{n!}\frac{H_m(x)}{m!}e^{-x^2}dx=0$$

$$or,\;\int_{-\infty}^\infty H_n(x) H_m(x) e^{-x^2} dx=0\dotsm(26)$$

$$or,\;\; \int_{-\infty}^\infty [e^{\frac{-x^2}{2}}H_n(x)][ e^{\frac{-x^2}{2}} H_m(x)]dx=0$$

Hence different Harmonic oscillator wave function are orthogonal. i.e.

$$\int_{-\infty}^\infty e^{\frac{-\alpha^2 x^2}{2}} H_n(\alpha x)\cdot e^{-\frac{\alpha^2x^2}{2}}H_m(\alpha x)dx=0$$

Case (2) n= m :

Equating the coefficient of \(t^n s^n\) on both sides, we get,

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

$$\therefore\;\; \int_{-\infty}^\infty H_n(x) H_n(x) e^{-x^2} dx= \sqrt{\pi}2^n n!\dotsm(27)$$

Using equation (27) in (23), we get,

$$or,\;\; |A|^2\int_{-\infty}^\infty H_n(\alpha x)H_n(\alpha x)e^{-\alpha^2 x^2}d (\alpha x)=1$$

$$or,\;\; |A|^2\biggl[\frac{\sqrt{\pi}2^n n!}{\alpha}\biggr]=1$$

$$or,\;\; |A|^2=\frac{\alpha^2}{\sqrt{\pi}2^n n!}$$

$$\therefore\;\; A= \biggl[\frac{\alpha}{2^n \sqrt{\pi} n!}\biggr]^\frac{1}{2}\dotsm(28)$$

Equation (28) gives the normalization constant for harmonic oscillation.

The normalized wave function of one dimensional S.H.O is given by,

$$\psi_n(x)=\biggl[\frac{\alpha}{2^n\sqrt{\pi}n!}\biggr]^\frac12 H_n(\alpha x) e^{-\frac{\alpha^2 x^2}{2}}$$

Reference:

1. Mathews, P.M and K Venkatesan. A Text Book of Quantum Mechanics. New Delhi: Tata McGraw Hill Publishing Co. Ltd, 1997.
2. Merzbacher, E. Quantum Mechanics . New York: John Wiley, 1969.
3. Prakash, S and S Salauja. Quantum Mechanics. Kedar Nath Ram Nath Publishing Co, 2002.
4. Singh, S.P, M.K and K Singh. Quantum Mechanics. Chand & Company Ltd., 2002.