Development of time dependent and independent Schrodinger equation

Development of time dependent Schrodinger equation:

Consider a particle of mass 'm' is travelling along X-axis with momentum \(P_x= mv_x=mv_g\), Where, \(V_g\) is group velocity associated with particle. The moving particle is represented by wave packet of the form,

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

Where,

A= Amplitude

K= \(\frac{2\pi}{\lambda}=\frac{P_x}{\hbar}=\) Wave number

\(\omega=\frac{E}{\hbar}\)= Angular frequency

i = \(\sqrt{-1}\)

E= Total energy of particle.

Substituting above we get,

$$\psi(x,t)= A\cdot e^{i(\frac{p_x x}{\hbar}-\frac{Et}{\hbar})}\dotsm(2)$$

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

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

Again differentiating above equation with respect to x, we get

$$\frac{\partial^2\psi(x,t)}{\partial x^2}=\biggl(\frac{iP_x}{\hbar}\biggr)^2\biggl[Ae^{i\biggl(\frac{P_x\cdot x}{\hbar}-\frac{Et}{\hbar}\biggr)}\biggr]$$

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

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

Dividing both side by 2m, we get

$$\frac{-\hbar^2}{2m}\frac{\partial^2 \psi(x,t)}{\partial x^2}=\frac{P_x^2}{2m}\psi(x,t)\dotsm(3)$$

Again, differentiating equation (2) with respect to 't' we get,

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

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

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

$$or,\;\; i\hbar \frac{\partial \psi(x,t)}{\partial t}=E\partial (x,t)\dotsm(4)$$

Since the non-relativistic T.E is the sum of K.E and P.E, We can write

$$E= K.E+ P.E$$

$$E=\frac{P_x^2}{2m}+V(x)\dotsm(5)$$

Multiplying both sides by \(\psi(x,t)\)

$$E\psi(x,t)= \frac{P_x^2}{2m}\psi(x,t)+V(x) \psi(x,t)$$

Using equation (3) and (4) in above equation.

$$\biggl[ i\hbar \frac{\partial(x,t)}{\partial t}= \frac{-\hbar^2}{2m}\frac{\partial^2\psi(x,t)}{\partial x^2}+V(x)\partial (x,t)\biggr]\dotsm(6)$$

Equation (6) is the required one-dimensional time dependent Schrodinger equation.

In time-dimension, equation (6) can be written as,

$$i\hbar \frac{\partial \psi(\vec r, t)}{\partial t}=\frac{-\hbar^2}{2m}\biggl[\frac{\partial^2}{\partial x^2}+{\partial^2}{\partial y^2}+\frac{partial^2}{\partial z^2}\biggr]\psi(\vec r, t)+ V(\vec r)\psi(\vec r, t)$$

$$\therefore i\hbar \frac{\partial \psi(\vec r,t)}{\partial t}= \frac{-\hbar^2}{2m}\nabla^2\psi(\vec r , t)+ V(\vec r)\psi(\vec r, t)\dotsm(7)$$

Which is the required time dependent Schrodinger equation in three dimension.

It is 2^nd order, linear, homogenous, partial differential equation.

Time independent Schrodinger equation :

Let us suppose the form of wave packet be size,

$$\psi(r,t)=\Phi(t)\cdot\psi(r)\dotsm(8)$$

Substituting equation (8) in (7), we get,

$$\frac{-\hbar^2}{2m}\nabla^2 [ \phi(t)\cdot\psi(r)]+V(r) [\Phi(t)\cdot\psi(r)]= \frac{i\hbar\partial}{\partial t} [\Phi(t)\cdot\psi(r)]$$

$$or,\;\; \Phi(t)\biggl[ \biggl( \frac{-\hbar^2}{2m}\nabla^2 \psi(r)\biggr)+V(r)[psi(r)]\biggr]=\psi(r)i\hbar\frac{\partial \Phi(t)}{partial t}$$

Dividing both sides by \(\Phi(t)\cdot\psi(r)\) we get,

$$\frac{\biggl[\frac{-\hbar^2}{2m}\nabla^2\psi(r)+V(r)[\psi(r)]\biggr]}{\psi(r)}=\frac{i\hbar\partial \Phi(t)}{\Phi(t)}=E\dotsm(9)$$

Where, E is separation constant which is independent of both r and t.

This separation constant is indentified as total energy of particle.

Taking 1^st and last term of equation (9) we get,

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

$$\therefore \frac{-\hbar^2}{2m}\nabla^2\psi(r)+V(r) \psi(r)= E\psi(r)\dotsm(10)$$

Equation (10) is the required time independent Schrodinger equation. It is 2^nd order. Linear , homogeneous , partial differential equation.

Taking 2^nd and last term, we get

$$i\hbar\frac{\partial \Phi(t)}{\partial t}= E\Phi(t)\dotsm(11)$$

Equation (11) is eigen value equation of energy and \(i\hbar\frac{\partial }{\partial t}\) is identified as 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.