Schrodinger Wave Equation
Introduction
Schrödinger equation is one of the basic equation in quantum mechanics. This equation
can be applied for both macroscopic and microscopic particles. Schrödinger derived a
mathematical equation to describe the dual nature of matter waves.
The equation that describes the wave nature of a particle in mathematical form is known
as Schrödinger wave equation.
There are two forms of Schrodinger Wave Equations. They are
1. Schrödinger Time Independent wave equation
2. Schrödinger Time dependent wave equation
Schrödinger Time Independent wave equation
Consider a system of stationary waves associated with moving particle. Let x, y, z be the
coordinates of the particle and ψ the wave displacement for de-Broglie’s waves at any time t.
The classical differential equation of a wave motions is given by
 2  2  2 1  2



x 2 y 2 z 2 v 2 t 2
------------- (1)
Here, v is the wave velocity
The equation (1) is written as
 2 
1  2
v 2 t 2
-------------- (2)
Where  is the Laplacian operator.
The solution of equation (2) gives ψ as a periodic displacement in terms of time t,
 ( x, y, z, t )  0 ( x, y, z ) e it
-------------- (3)
Here, ψ0(x,y,z) is s function of x,y,z only and gives amplitude at the point considered.
Differentiating equation (3) with respect to t, we get

  i 0 e it
t
------------- (4)
After differentiating with respect to t
 2
 i 2 2 0 e it
2
t
------------ (5)
 2
   2
2
t
------------- (6)
Substituting equation (6) in equation (2) we have
 2 
2
v2
 0
------------- (7)
But, angular frequency = 2. Here, v is the frequency


v
2

------------- (8)
Squaring equation (8) on both sides, we get
2
v2

4 2
2
------------- (9)
Substituting equation (9) in equation (7) we have

2
4 2
2
 0
------------- (10)
We know that de-Broglie wavelength λ = h/mv on substituting the value of λ in equation (10) we
get
4 2 m 2 v 2

 0
h2
2
------------- (11)
If E is total energy of the particle, V is potential energy and 1/2mv2 is kinetic energy, then
Total energy (E) = Potential Energy (V) + Kinetic Energy 1/2mv2
E-V=1/2mv2
------------- (12)
Multiplying by m on both sides
m2v2 = 2m (E-V)
------------- (13)
Substituting equation (13) in equation (11), we get
 2 
8 2 m
(E  V )   0
h2
------------- (14)
This equation (14) is known as Schroedinger time Independent wave equation.
Let us now substitute h 
h
in equation (14),
2
Now, Schroedinger time-independent wave equation is written as
 2 
2m
h2
(E  V )   0
------------- (15)
Schrödinger Time Dependent wave equation
Schrödinger time dependent wave equation is derived form Schrödinger time independent wave
equation.
The solution of classical differential equation of wave motion is given by differentiating equation
(1) with respect to time t we get

  2iv 
t
We know E=hv and h 
Therefore,
------------- (3)
h
2

E
i 
t
h
------------ (4)
Multiplying i on both sides in equation (4)
i

E
  i 2  
t
h
Or E  i h

t
------------ (5)
Schrödinger time independent wave equation is given by
 2 
2m
h2
(E  V )   0
Substituting the value of Eψ from equation 5 in equation 6, we get
 2 
2m  

ih
V   0
2 

h  t
------------ (6)
Or
Or
  h2 2



  V   i h
 2m

t


H  E
------------ (7)
------------ (8)
Where, H is Hamiltonian operator and E is the Energy operator.This equation (8) is known as
Schrödinger time dependent wave equation.