Step of Normalization

Step of Normalization:

If \(\psi(r,t) \) does not satisfy the condition of normalization i.e.

$$\iiint \psi^* (r,t) \psi(r,t) dxdydz\ne1$$

then \(\psi(r,t) can not acts as the amplitude for probability distribution. It can not represent a particle.

We should normalize \(\psi(r,t)\) to make probability distribution function.

Step 1:

If \(\psi(r,t)\) is not normalize then let, \(\psi_N= N\psi(r,t) \dotsm(1)\) be normalized wave function. Where, N is Normalization constant to be determined.

Step 2:

\(\psi_N\) satisfy normalization condition. i.e.

$$\iiint\psi_N^* \psi_N dxdydz=1\dotsm(1)$$

Put value of \(\psi_N\) from (1) in equation (2)

$$\iiint[N\psi(r,t)]^*[N\psi(r,t)]dxdydz=1$$

$$|N|^2\int_{-\infty}{\infty}\int_{-\infty}{\infty}\int_{-\infty}{\infty} \psi^*(r,t) \psi(r,t)dxdydz=1$$

$$\therefore\;\; N= \frac{1}{\sqrt{\iiint\psi^*(r,t)\psi(r,t)dxdydz}}\dotsm(3)$$

The equation (3) gives the value of normalization constant.

Step 3:

Substituting equation (3) in (1) we get,

$$\psi_N(r,t)= \frac{\psi(r,t)}{\sqrt{\iiint \psi^*(r,t)\psi(r,t)dxdydz}}\dotsm(4)$$

Equation (4) gives the normalized wave function.

Question:

(a) Normalize the wave function

$$\psi(x)=Ae^{-\alpha^2x^2/2}$$

over \(-\infty\) to \(+\infty\).

$$OR$$

If \(\psi(x)= Ae^{\frac{-\alpha^2x^2}{2}}\) is normalized over \(-\infty\) to \(+\infty\) then find the value of A

(b) Find the probability density.

(c) Plot both \(\psi(x)\) and \(\rho(x)\) for both A and \(\alpha\) real.

(d) fin the region where probability of finding particle is maximum.

(e) Also find J(x) [ Probability current density ]

\(\Rightarrow \) Given, \(\psi(x)= Ae^{\frac{-\alpha^2 x^2}{2}}\) over \(-\infty\) to \(+\infty\).

(a) The condition of normalization.

$$\int_{-\infty}^\infty \psi^*(x) \psi(x) dx=1$$

$$or,\;\; \int_{-\infty}^{+\infty} \biggl(Ae^{\frac{-\alpha^2x^2}{2}}\biggr)^*\biggl(Ae^{\frac{-\alpha^2 x^2}{2}}\biggr)dx=1$$

$$or,\;\; |A|^2\int_{-\infty}^{+\infty} e^{-2\cdot \alpha^2x^2/2}dx=1$$

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

Let,

$$I= \int_{-\infty}^{+\infty} e^{-\alpha^2}x^2 dx$$

$$=2\int_0^{\infty} e^{-alpha^2x^2}dx$$

Put, \(\alpha^2x^2=y\Rightarrow x=\frac{\sqrt{y}}{alpha}\)

$$dx=\frac{1}{2\sqrt{y}}\frac{dy}{\alpha}$$

When x=0, y=0, x=\(\infty\), y= \(\infty\)

$$I= 2\int_0^\infty e^{-y} \frac{1}{2\sqrt{y}}\frac{dy}{\alpha}$$

$$=\frac{1}{\alpha}\int_0^\infty e^{-y} y^{\frac{-1}{2}} dy$$

$$=\frac{1}{\alpha}\int_0^\infty e^{-y}y^{\frac12-1}dy$$

$$=\frac{1\sqrt{\frac12}}{\alpha}$$

$$=\frac{\sqrt{\pi}}{\alpha}$$

From (1), \(|A|^2 \frac{\sqrt{\pi}}{\alpha}=1\)

$$\therefore\; A= \biggl(\frac{\alpha}{\sqrt{\pi}}\biggr)^\frac12= normalization\;constant$$

The normalized wwave function is,

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

(b) Probability density:

$$\rho(x)= \psi_N^*(x)\psi_N(x)$$

$$\therefore\;\; \rho(x)=\frac{\alpha}{\sqrt{\pi}}e^{-\alpha^2x^2}$$

(c) Plot \(\psi_N(x)\) and \(\rho(x)\):

Fig

(d) the region where the probability of finding of particle is maximum, must have maximum value of probability density i.e.

\(\frac{d\rho(x)}{dx}=0\) [ maxima and minima]

$$\frac{d}{dx}\biggl[\frac{\alpha}{\sqrt{\pi}}e^{-\alpha^2x^2}\biggr]=0$$

$$or,\;\; \frac{\alpha}{\sqrt{\pi}}\biggl[e^{-\alpha^2 x^2 }\times (-2\alpha^2 x)\biggr]=0$$

Either, x=0 or, \(e^{-\alpha^2 x^2}=0\)

For finite value of x \( e^{-\alpha^2 x^2}\ne0 \) \(\therefore\;; x=0\)

So, the region where, probability of finding particle is maximum is around x=0.

(e) Value of J(x)

$$J(x)=\frac{\hbar}{2im}\biggl[\psi^*\frac{d\psi}{dx}-\psi\frac{d\psi^*}{dx}\biggr]$$

Here, \(\psi^*(x)=\psi(x)\) because A and \(\alpha\0 are real.

$$\therefore\;\; J(x)=0$$

So, the probability current density is zero. i.e. there is localized around x=0, it is not moving.

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.