Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 9
Classical-Mechanical Quantities Are Represented by linear
Operators in Quantum Mechanics
- From Hamiltonian operator which equal
2


d2
 U (x ) 
Ĥ  
2
 2m dx

But we know that total energy = kinetic energy +potential
energy.
So, Kinetic energy operator will equal
2
d2
K  
2m dx 2
(12)
- Furthermore, classically, k=p2/2m, and so we conclude
that
p  
2
2
d2
dx 2
(13)
- Operators can be imaginary or complex quantities. We
shall see that the x component of the momentum can be
represented in quantum mechanics by an operator as a
form
p x  i
[1]

x
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 9
The operator p̂2 can be factored as
p 
2
2
d2
d
d
 (i
)(i
) (14)
2
dx
dx
dx
- For example(2):
Show that eikx is an eigenfunction of the momentum operator,
p x  i

x
. What is the eigenvalue?
Solution:
We apply p̂x to eikx and find

eikx= ħk eikx
x
and so we see that eikx is an eigenfunction and ħk is an
eigenvalue of the momentum operator.
p̂x eikx = i
Free particle in a one –dimensional box:
- In this section we shall study the case of a free particle
of mass m constrained to lie along the x axis between
x=0 and x=a. This is called the problem of a particle in
a one- dimensional box.
- The free particle means that the particle experiences
no potential energy U(x)=0
- So the Schrodinger equation (7) for a free particle in a
one- dimensional box is
d 2 2mE
 2  (x )  0
2
dx
0 x a
- The particle is restricted to the region
cannot be found outside this region.
[2]
(15)
0  x  a and
so
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 9
The Energy of a Particle in a Box Is Quantized
The general solution of Eq.15
k 
(2mE )
1
2
(16)
Suppose ka=n π , n=1,2,…..
Compare Eqs. 15,16
h 2n 2
E n 
8ma 2
n=1,2,……
The energy of the particle is quantized and n is called a
quantum number.
These wave function are plotted in Figure 1.
[3]
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 9
Example(3)
The model of a particle in a one dimensional box has
been applied to the π electrons in linear conjugated
hydrocarbons such as butadiene move along a straight
line whose length can be estimated as 5.78Å .what is
the energy required to make a transition from the n=2
state to the n=3 state
Solution:
h 2n 2
E n 
8ma 2
h2
2
2
E n 
(3

2
)
2
8ma
(6.6x 1034 J .S )2  5
E n 
 9.02 1019 J
31
10
2
8(9.1110 kg )(5.87 10 m )
And so we see that this very simple model, called the freeelectron model ,is somewhat successful.
[4]