Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 8
The Schrödinger Equation
And
Some Simple Applications
- Let us start with the classical one-dimensional wave
equation
 2u
1  2u

x 2 v2 t 2
(1)
where u (x, t) is the amplitude of the string
v=the speed of the moves along the string
- We can write u (x, t) as
u (x , t )   (x ) cos t
(2)
where is the spatial amplitude of the wave
and ω =2πν
- If we substitute Eq.2 into Eq.1
d 2  2

 (x )  0
(3)
dx 2 v 2
- But total energy = kinetic energy + potential energy
- For linear momentum
p2
E 
 U (x )
2m

 p  2m  E U (x )
1
1
2
(4)
(5)
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 8
h
- From de Broglie formula   p

h
h

p 2m  E U (x )

1/2
(6)
v
  2 and   
Putting all this together, we have
4 2 2 4 2 2m  E U (x )

 2 
2
v2
v2

2
Substituting this into Eq. 3 we find
d 2 2m
 2 [E U (x )] (x )  0
2
dx
(7)
This equation can be rewritten in the form
d 2
E (x )  
 U (x ) (x )
2m dx 2
2
(8)
The above equation can represents what is called
Schrödinger equation
2
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 8
The Schrödinger Equation can be Formulated as an
Eigen value problem
An Operator: is a symbol that tells you to do something
to whatever follows the symbol.
For example, we can consider dy/dx to be the d/dx
operator operating on the function y(x). √,and

y
all
this make as operators
We shall usually denote an operator by capital letter
g(x)= Â f(x)
to indicate that the operator  operates on f(x) to give
a new function g(x).
- Let us go back to equation 8 we can write it
2


d2
E (x )   

U
(
x
)
 (x ) (9)

2
 2m dx

- If we denote the operator in brackets by Ĥ, then the
equation (9) is
Ĥ ψ(x) = E ψ(x)
(10)
This formula the Schrödinger equation as an eigen
value problem.
Operator Ĥ is called Hamiltonian operator.
2


d2



U
(
x
)


2
Ĥ
(11)
 2m dx

The energy is an eigenvalue of the Hamiltonian
operator.
3
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 8
Some examples on operator
Example1:
Perform the following operations.
d2
A 2
dx
(a)
Â(2x)
,
(b)
Â(x2)
,
A
d2
d
2
3
2
dx
dx
(c)
Â(xy3)
,
A

y
Solution:
(a)
d2
Â(2x)= 2 (2x )  0
dx
2
d2 2
)= 2 x  2 d x 2  3x 2  2  4x  3x 2
dx
dx
(b)
Â(x
(c)
Â(xy3)=
4

xy 3  3xy 2
y