Engineering Physics 3(2+1)

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.