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6.2 Quantum Mechanics
Quantum mechanics is based on the dual nature of matter. Experiments have verified that
particles such as electrons show typical wavelike properties, including interference. Particles of
small mass can be described by the wave function, , which represents the displacement of a
wave as it fluctuates with time and position,  = (q, t). The wave function can be expressed in
the form
 q

 = Aexp 2i   t 

 
(6.2-1)
where A is the amplitude of the wave,  is the wavelength, and  is the wave frequency. The
wave function can be expressed in terms of energy E using Einstein’s formula
E = h = mc2 =
p2
m
where h is Planck’s constant (h = 6.6310-34 Js), c is the velocity of light, and E is the particle or
wave energy. The relations between frequency , wavelength , energy, and momentum p can be
obtained from Einstein’s equation
=
E
h
=
c
ch
h
=
=
2

mc
p
h
h
=
agreed well with experimental data where
mv
p
v is the particle velocity. The wavelength of electron (m = = 9.10910-31 kg) travels at 0.05
speed of light is
The wavelength of particle predicted by  =
=
6.63  10 34
= 4.8510-11 m
9.109  10 31 (.05  3  108 )
Substituting the frequency and wavelength in terms of energy and momentum into Eq. (6.2-1)
yields
q
 2ipq 
 2iEt 


 = Aexp 2i  2it  = Aexp 
 exp  
 = AQ

h 


 h 

where
 2ipq 
Q = exp 
 , and
 h 
 2iEt 

h 

 = exp  
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Q and  are called the spatial and temporal parts of the wave function, respectively. The wave
function  can be used to determine the system mechanical properties such as momentum p and
energy E. These properties are obtained from differential operation on . From the definition of
the wave function
 2ipq 
 = AQ, Q = exp 
 , and
 h 
 2iEt 

h 

 = exp  
Operation on the wave function yields the product of the mechanical variables (p or E) and the
wave function as follows
  
 Q 
 2ip 
 2ipq   2ip 

 = A 
 = A 
 exp 
 =


 h 
 h   h 
 q  t
 q  t
 2iE 
 2iEt 
  
  
 exp  


 = AQ   = AQ  
h 
h 


 t  q
 t  q
h   
 2iE 
  
   E = 

 = 


2i  t  q
h 
 t  q 
When an operator operates on a function to produce a constant time the original function, the
  
constant is called an eigenvalue. For example, the operator   operates on the wave function
 q  t
 2ip 
 2ip 
 to produce a constant 
 time the wave function. The constant 
 is an eigenvalue.
 h 
 h 

 2iE 
Similarly  
 is an eigenvalue obtained from the operator   operating on the wave
h 

 t  q
function  . The momentum p̂ and energy Ê operators are then defined
h

2i
h 
Ê = 
2i t
p̂ =
The Hamiltonian operator Ĥ is formulated by replacing the momentum in the Hamiltonian
H(p,q) with the momentum operator.
H(p,q) =
p2
+ V(q)
2m
2
1  h 
h2
pˆ 2
Ĥ =
 + V = 
+V=
2 + V

2m  2i 
8 2 m
2m
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For conservative system the Hamiltonian H is the energy E of the system, therefore the
Hamiltonian operator Ĥ is just the energy operator Ê .
Ĥ = Ê = 
h 
2i t
When the Hamiltonian operator Ĥ is used on the wave function, the resultant equation is called
the Schrodinger equation that is the quantum mechanical equivalent of the equations of motion in
classical mechanics. Schrodinger equation can be used to explain many physical phenomena at
the atomic level.
h 
h2
=
2  + V 
2i t
8 2 m

Let h =
h
, and  = m, Schrodinger equation takes the following form
2
ih

h2 2
=
  + V
t
2
(6.2-1)
The wave function  only reflects the spatial probability of finding a particle. It does not reflect
an actual mechanical location of a selected particle. The probability of finding a particle in an
interval [a, b] is given by

b
a
 ( x)dx where (x) is called the probability density function. The
function (x) is related to the wave function  by the following expression
(x) =  (x)  (x) =  (x )
2
 (x) is the complex conjugate of  (x) as follow

 x
x


 (x) = Aexp 2i   t    (x) = Aexp  2i   t 



 

The probability of finding the particle somewhere in the entire x coordinate must be equal to 1.
Hence



 ( x)dx =



( x ) dx = 1
2
The Schrodinger equation (6.2-1) can be solved if the potential V and the initial condition
 (x, y, z, 0) = f(x, y, z)
are given. We assume that  (x, y, z, t) can be separated into u(x, y, z), a function of spatial
coordinates only, and T(t), a function of t alone.
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 (x, y, z, 0) = u(x, y, z) T(t)
In terms of the new functions, Eq. (6.2-1) becomes
ihu
dT
h2
=
T2u + V uT
dt
2
Dividing the equation by uT yields
ih
1 dT
h2 2
=
 u+V=E
T dt
2u
where E is the separation constant. The differential equation on u is called the time-independent
Schrodinger equation.
h2 2

 u + Vu = Eu
2
Several simple solution of the Schrodinger equation can be used to explain the energy storage in
atoms and molecules.
Example 6.2-1
Solve the Schrodinger equation for a small particle confined to move freely inside an interval [0,
L] with potential
 0 if 0  x  L
V(x) = 
 otherwise
The initial wave function is known:  (x, 0) = f(x)
Solution
The one-dimensional Schrodinger equation with V = 0 is
1 dT
h 2 d 2u
ih
=
=E
T dt
2u dx 2
The time dependent function is
ih
1 dT
 iEt 
= E  T =Aexp  

T dt
 h 
The time independent function is
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
2 E
d 2u
h 2 d 2u
=
Eu

=  2u, where 2 = 2
2
2
h
dx
2  dx
For E > 0, the time independent wave function has the form
u(x) = acos(x) + bsin(x)
The constant a and b are determined from the boundary conditions that the probability of finding
the particle must approach zero at x = 0 and x = L since a particle with finite energy cannot exist
where V .
u(0) = 0 = a
u(L) = 0 = bsin(L)  L = n  2 =
2 E
n 2 2
= 2
2
h
L
The separation constants En are called the eigenvalues according to the definition
En =
n 2 2 h 2
2 L2
The solutions un(x) are the eigenfunctions
 n 
x
un(x) = bn sin 
 L 
The one-dimensional wave function is then
 (x, t) =


n 1
 iE t 
 n 
x  exp   n 
bn sin 
 L 
 h 
The constants bn can be obtained from the initial condition
 (x, 0) = f(x) =


n 1
bn =
 n 
x
bn sin 
 L 
2 L
n
f ( x ) sin(
x )dx

L 0
L
Let consider the particular solutions of the one-dimensional time independent wave equation
 n 
x
u(x) = b sin 
 L 
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The constant b in this case can be determined from the requirement that the probability of finding
the particle within the interval [0, L] must be equal to 1.
L
 n 
x dx = 1
= 1  b2  sin 2 
0
0
 L 
1  cos 2 x
Using the identity sin2x =
, the above equation becomes
2
L
 uu dx
b2 
L
0
b2
2
1
2nx 
1  cos
 dx = 1
2
L 
L
L

b2
 2 n x  
x

sin
L=1b=

 =

2 n
 L  0 2

2
L
The full wave function solutions are therefore
2
 n
sin 
L
 L
u(x) =

x ,

n = 1, 2, 3, ...
For the 3-dimensional case, the time-independent equation becomes
  2u  2u  2u 
 2  2  2  = Eu
y
z 
 x
h2

2
Using separation of variable and the requirement that
Volume

0
uu dxdydz = 1
The 3-D solutions are
2
u(x, y, z) =  
 L
3/ 2
 n y 
n  
n  
sin  x x  sin 
y  sin  z z 
 L 
 L 
 L 
The eigenvalues for the solutions are
En =
 2h 2
( n x2 + n 2y + n z2 )
2
2 L
For a given En there are a number of different combinations of nx, ny, and nz. The wave function
for each combination will be different but the total energy of the system will be the same. Hence
the system may occupy a given energy level En in a number of different ways.
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