Eigen Value Equation

Eigen Value Equation:

An equation of the form \(\hat A\psi= \lambda \psi\dotsm(1) \) is known as eigen value equation of operator \(\hat A\) over wave function \(\psi\). Here \(\lambda\) is known as eigen value or measured value of a physical quantity represented by operator \(\hat A\)in the state \(\psi\).

It is the method of measurement in which the process of measurement avoids the interaction between the system and measuring device. All possible values of \(\lambda\) is known as spectrum of eigen value. It may be discrete, continuous or both.

1. Linear momentum Operator (\(\hat P_x\)):-

Let us take a particle of mass 'm' moving along x-axis is represented by wave packet of the form.

$$\psi(x,t)= Ae^{i(kx-\omega t)}\dotsm(1)$$

Where,

$$k=\frac{2\pi}{\lambda}$$

$$=\frac{2\pi}{\frac hp}$$

$$=\frac{P_x}{\hbar}$$

and

$$\omega= 2n\nu$$

$$=2\pi\frac Eh$$

$$=\frac {E}{\hbar}$$

$$\therefore\; \psi (x,t)= Ae^{i(\frac{P_x\cdot x}{\hbar}-\frac{Et}{\hbar})}\dotsm(2)$$

Differentiating equation (2) with respect to

$$\frac{\partial \psi(x,t)}{\partial x}=\biggl(\frac{iP_x}{\hbar}\biggr)\biggl[Ae^{i(\frac{P_x\cdot x}{\hbar}-\frac{Et}{h})}\biggr]$$

$$or,\;\; \frac{\hbar}{i}\frac{\partial \psi(x,t)}{\partial x}= P_x\psi(x,t)$$

$$or,\;\; \frac{\hbar}{i}\cdot \frac{\partial}{\partial x}[\psi(x,t)]= P_x [ \psi(x,t)]\dotsm(3)$$

Comparing equation (3) with Eigen value equation.

$$\hat A\psi(x,t)= \lambda\psi(x,t)\dotsm(4)$$

Where, \(\hat A=\frac{\hbar}{i}\frac{\partial}{\partial x}\) and [\(lambda= P_x\)]

Here, \(\hat A= \frac{\hbar}{i}\frac{\partial}{\partial x} \) is called as linear momentum operator.

\(\therefore\; \hat P_x= \) x- component of linear momentum operator.

$$=\frac{i\hbar}{i^2}\frac{\partial}{\partial x}$$

$$=\frac{-i\hbar\partial}{\partial x}$$

\(\therefore\; \hat P_y= -i\hbar\frac{\partial}{\partial y}\) and \(\hat P_z= -i\hbar \frac{\partial}{\partial z}\)

In 3-dimension,

$$\hat P= \hat i \hat P_x+ \hat j\hat P_y+ \hat k \hat P_z$$

$$=i\hat h\biggl[ \hat i\frac{\partial}{\partial x}+\hat j\frac{\partial}{\partial y}+\hat k\frac{\partial}{\partial z}\biggr]$$

$$\therefore \hat P= -i\hbar \nabla$$

Where, \(\nabla= \hat i \frac{\partial}{\partial x}+\hat j\frac{\partial}{\partial y}+\hat k\frac{\partial }{\partial z}\)

$$\therefore \hat P= \frac{\hbar}{i}\nabla$$

2. Square of linear momentum operator:

We have

$$\hat P^2= \hat P\cdot\hat P$$

$$= (-i\hbar\nabla)\cdot(-i\hbar\nabla)$$

$$=i^2\hbar^2 \nabla^2$$

$$\therefore\;\; \hat P^2= -\hbar^2 \nabla^2$$

In one dimension,

$$\hat P_x^2= \hat P_x \cdot \hat P_x$$

$$=\biggl(-\hbar\frac{\partial}{\partial x}\biggr)\cdot\biggl(-i\hbar\frac{\partial}{\partial x}\biggr)$$

$$=-\hbar^2\frac{\partial^2}{\partial x^2}$$

Similarly,

$$\hat P_y^2= -\hbar^2\frac{\partial}{\partial y^2}$$

$$\hat P_z^2= -\hbar^2 \frac{\partial^2}{\partial z^2}$$

3. Kinetic energy operator (\(\hat T\)):-

Differentiating equation (2) with respect to x we get,

$$\frac{\partial\psi(x,t)}{\partial x}= \frac{iP_x}{\hbar}\psi(x,t)$$

$$or,\;\; \frac{\partial^2\psi(x,t)}{\partial x^2}= \biggl(\frac{iP_x}{\hbar}\biggr)^2\psi(x,t)$$

$$or,\;\; -\hbar^2\frac{\partial ^2 \psi(x,t)}{\partial x^2}= P_x^2 \psi(x,t)$$

$$or,\;\; \frac{-\hbar^2}{2m}\frac{\partial^2\psi(x,t)}{\partial x^2}= \frac{P_x^2}{2m}\psi(x,t)$$

$$or,\;\; \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}[\psi(x,t)]=\frac{P_x^2}{2m}[\psi(x,t)]\dotsm(5)$$

Comparing equation (5) with eigen value equation [ eqn (4)] we get,

$$\hat A= \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}=\hat T_x= K.E \; operator$$

$$T_x= \frac{P_x^2}{2m}= Eigne \;value \;of\; K.E$$

In 3-dimension:

$$\hat T= \frac{-\hbar^2}{2m}\biggl[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\biggr]$$

$$\hat T= \frac{-\hbar^2}{2m}\nabla^2=\frac{\hat P^2}{2m}$$

4. Time dependent Total energy Operator:

Differentiating equation (2) with respect to 't' we get

$$\frac{\partial \psi(x,t)}{\partial t}=\biggl(\frac{-iE}{\hbar}\biggr)Ae^{i(\frac{p_x x}{\hbar}-\frac{Et}{\hbar})}$$

$$or,\;\; \frac{\partial \psi(x,t)}{\partial t}=\frac{-iE}{\hbar}\psi(x,t)$$

$$or,\;\; \frac{-\hbar}{i}\frac{\partial}{\partial t}[\psi(x,t)]=E[\psi(x,t)]\dotsm(6)$$

Comparing (6) with (4) we get,

$$\hat A= \frac{-\hbar}{9}\frac{\partial}{\partial t}= energy\;operator$$

$$\lambda= E= energy\;eigen\;value$$

$$E_{op}=i\hbar\frac{\partial}{\partial t}$$

5. Time independent energy operator [ Hamiltonian operator ]

We have from the time independent schrodinger equation

$$\frac{-\hbar^2}{2m}\nabla^2\psi(r)+V(r)\psi(r)=E\psi(r)$$

$$or\;\; \biggl[\frac{-\hbar^2}{2m}\nabla^2+ \hat V(r)\biggr]\psi(r)=E\psi(r)$$

$$or,\;\; [\hat H\psi(r)= E\psi(r)\dotsm(1)$$

Where, \(\hat H=\frac{-\hbar^2}{2m}\nabla^2\psi(r)+V(r)\) is called Hamiltonian operator.

Equation (1) is eigen value equation of Hamiltonian operator.

Here, \(\hat V(r)\psi(r)= V(r)\psi(r)\dotsm(2)\)

Where, \(\hat V(r)\) = Potential energy operator.

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.