Site pages
Current course
Participants
General
MODULE 1. Magnetism
MODULE 2. Particle Physics
MODULE 3. Modern Physics
MODULE 4. Semicoductor Physics
MODULE 5. Superconductivty
MODULE 6. Optics
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.