## LESSON 10. Derivation for Schrodinger Wave Equation

Equation of motion of matter waves

(1) Time Independent Schrodinger Wave Equation

According to be Broglie theory, a particle of mass m is always associated with a wave whose wavelength is given by

$\lambda$ = $\frac{h}{{mv}}$  . If the particle has wave properties, it is expected that there should be some sort of wave equation which describes the behavior of the particle.

Suppose, a system of stationary waves associated with a particle.

x, y, z be the coordinates of the particle and $\Psi$  the wave displacement for the de Broglie waves at any time .

The classical differential equation of wave motion is given by

$\frac{{{\partial ^2}\Psi }}{{\partial {t^2}}}$ = ${v^2}\left( {\frac{{{\partial ^2}\Psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\Psi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\Psi }}{{\partial {z^2}}}} \right)$ = ${v^2}{\nabla ^2}\Psi$  …….. (1)

Where, ${\nabla ^2}$  = $\frac{{{\partial ^2}}}{{\partial x}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}}$

${\nabla ^2}$ Is being Lasplacian operator and  is wave velocity.

The solution of equation (1) gives $\Psi$ as a periodic displacement in term of time i.e.

$\Psi \left( {x,y,z,t} \right) = {\Psi _o}\left( {x,y,z} \right){e^{ - i\omega t}}$ ……..(2)

Where ${\Psi _o}\left( {x,y,z} \right)$ is a function of  x, y, z and gives the amplitude at the point considered.

Equation (2) can also be expressed as

$\overrightarrow {\Psi \left( {r,t} \right)}$ = $\overrightarrow {{\Psi _o}\left( r \right)} {e^{ - i\omega t}}$  ……….(3)

Differentiating equation (3) twice we get

$\frac{{{\partial ^2}\Psi }}{{\partial {t^2}}}$ = $- {\omega ^2}{\Psi _o}\left( r \right){e^{ - i\omega t}}$

$\frac{{{\partial ^2}\Psi }}{{\partial {t^2}}}$ =  $- {\omega ^2}\Psi$

Substituting the value of  $\frac{{{\partial ^2}\Psi }}{{\partial {t^2}}}$  in equation (1) we have

$\frac{{{\partial ^2}\Psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\Psi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\Psi }}{{\partial {z^2}}}$ +
$\frac{{{\omega ^2}}}{{{v^2}}}\Psi$ = 0 …….(4)

But,   $\omega= 2\pi v = 2\pi \frac{v}{\lambda }$

Because  $\frac{\omega }{v}=\frac{{2\pi }}{\lambda }$

Substituting the value of  $\frac{\omega }{v}$ in equation (4) we have

$\frac{{{\partial ^2}\Psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\Psi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\Psi }}{{\partial {z^2}}}$ +
$\frac{{4{\pi ^2}}}{{{\lambda ^2}}}\Psi$ = 0

${\nabla ^2}\Psi$ + $\frac{{4{\pi ^2}}}{{{\lambda ^2}}}\Psi$ = 0 ……. (5)

Now from de Broglie relation  $\lambda$ = $\frac{h}{{mv}}$

${\nabla ^2}\Psi$ + $\frac{{4{\pi ^2}}}{{{h^2}}}{m^2}{v^2}\Psi$ = 0 ……. (6)

If  E and V be the total and potential energies of the particle respectively, then its kinetic energy  $\frac{{m{v^2}}}{2}$ is given by

$\frac{{m{v^2}}}{2}$ = $E - V$

$m{v^2}$ = $2m\left( {E - V{\text{}}} \right)$  ……... (7)

From equation (6) and (7) we have

${\nabla ^2}\Psi$ + $\frac{{4{\pi ^2}}}{{{h^2}}}2m\left( {E - V{\text{}}} \right)\Psi$ = 0

Or

${\nabla ^2}\Psi$ + $\frac{{8{\pi ^2}m}}{{{h^2}}}\left( {E - V{\text{}}} \right)\Psi$ = 0 ……… (8)

It is called Schrodinger time independent wave equation

Substituting $\hbar$   = $\frac{h}{{2\pi }}$ in equation (8), the Schrodinger wave equation can be written as

${\nabla ^2}\Psi$ + $\frac{{2m}}{{{\hbar ^2}}}\left( {E - V{\text{}}} \right)\Psi$  = 0……… (9)

For free particle  V = 0  , hence the Schrodinger wave equation for a particle can be expressed as

${\nabla ^2}\Psi$ + $\frac{{2mE}}{{{\hbar ^2}}}\Psi$ = 0………. (10)

(2) Time Dependent Schrodinger Wave Equation

The Schrodinger time depending wave equation may be obtained from Schrodinger time independent wave equation by eliminating E .

Differentiating equation (3) with respect to  t , we get

$\frac{{\partial \Psi }}{{\partial t}}$ =  $- i\omega {\Psi _o}\overrightarrow {\left( r \right)} {e^{ - i\omega t}}$

$\frac{{\partial \Psi }}{{\partial t}}$ =  $- i2\pi v{\Psi _o}\overrightarrow {\left( r \right)} {e^{ - i\omega t}}$

$\frac{{\partial \Psi }}{{\partial t}}$ = $- 2\pi i$ $\frac{E}{h}$ $\Psi$ =    ………….(11)

Substituting the value of  E  and $\Psi$ in Schrodinger time independent wave equation, we get

${\nabla ^2}\Psi$ + $\frac{{2m}}{{{\hbar ^2}}}\left( {i\hbar \frac{{\partial \Psi }}{{\partial t}} - V{\text{}}\Psi } \right)$ = 0

${\nabla ^2}\Psi$ = $- \frac{{2m}}{{{\hbar ^2}}}\left( {i\hbar \frac{{\partial \Psi }}{{\partial t}} - V{\text{}}\Psi } \right)$

$- \frac{{{\hbar ^2}}}{{2m}}$ ${\nabla ^2}\Psi$  + $V{\text{}}\Psi$  = $i\hbar \frac{{\partial \Psi }}{{\partial t}}$  ……………. (12)

This equation is known as Schrodinger time independent wave equation

It can be written as

$\left( { - \frac{{{\hbar ^2}}}{{2m}}{\text{}}{\nabla ^2}{\text{}} + {\text{}}V{\text{}}} \right)$ $\psi$  = $i\hbar$ $\frac{\partial }{{\partial t}}\Psi$

Or

$H\Psi$ = $E\Psi$ ………….. (13)

Where $H$  = $- \frac{{{\hbar ^2}}}{{2m}}{\text{}}{\nabla ^2}{\text{}} + {\text{}}V$  and $E$ = $i\hbar$ $\frac{\partial }{{\partial t}}$

The equation (13) describes the motion of a non relativistic material particle.