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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.