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Unit –III QUANTUM MECHANICS Engineering Physics
Dr. P.Sreenivasula Reddy M.Sc, Ph.D.
Website: www.engineeringphysics.weebly.com Page 1
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Generally a wave is nothing but rather a spread out disturbance. A wave is specified by
its frequency, wave length, phase or wave velocity, amplitude and intensity.
A particle (matter) has mass and it is located at some definite point. It can move from
one place to another and it gives energy slowed down or stopped. The particle is specified by
its mass, momentum, velocity and energy.
We cannot explain the Photoelectric effect, Black body radiation and Compton Effect by
using the wave nature of radiation. Similarly we cannot explain the interference, diffraction
and polarization phenomena by using the particle nature of radiation
In 1900, Max Planck, successfully explained the black body radiation phenomenon by
particle nature of radiation. Similarly we can explain the Einstein’s photoelectric effect and
Compton effects by the particle nature of radiation. We can explain the interference, diffraction
and polarization phenomena by the wave nature of light.
Thus the radiation sometimes exhibits particle nature and sometimes exhibits wave
nature; hence we can say that the radiation has dual nature. This is also called as wave
particle dualism.
. de Broglie hypothesis
In 1924, de Broglie extended the dual nature of light to material particles or micro
particles like electrons, protons, neutrons etc. According to de Broglie hypothesis, a moving
particle is associated with a wave which is known as de Broglie wave or matter wave (the
waves associated with the material particle is called matter wave of de Broglie wave).
According to de Broglie the wave length of the matter wave is given by
p
h
mv
h
Where 'm' is the mss of the material particle, ‘ v ’ is its velocity and ‘ p ’ is its momentum.
Expression for de Broglie wave length
According to Planck’s quantum theory,
E = h υ
Where ‘ h ’ is a Planck’s constant
According to Einstein’s mass energy relation
2
E = mc ……………. 2
Where ‘
m ’ is the mass of the photon and ‘
c ’ is the velocity of the photon.
From equations (1) and (2)
h υ
2
= mc
c
h
2
= mc
c
Q
mc
h
mc
hc
2
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Prepared by: Dr. P.Sreenivasula Reddy M.Sc, (PhD)
Website: www.palleti.webnode.com Page 2
mc
h
Since, mc = p
momentum of a photon
p
h
In case of material particles
Momentum p = mv
So the de Broglie wave length of a material particle is
mv
h
d
d e
e B
B
r
r o
o g
g l
l i
i e
e w
w a
a v
v e
e l
l e
e n
n g
g t
t h
h i
i n
n t
t e
e r
r m
m s
s o
o f
f k
k i
i n
n e
e t
t i
i c
c e
e n
n e
e r
r g
g y
y
If ‘ E ’ is the kinetic energy of the material particle then
m
p
m
m v
E mv
2 2 2
2
∴ de Broglie wave length
mE
h
d
d e
e B
B
r
r o
o g
g l
l i
i e
e w
w a
a v
v e
e l
l e
e n
n g
g t
t h
h a
a s
s s
s o
o c
c i
i a
a t
t e
e d
d w
w i
i t
t h
h a
a p
p a
a r
r t
t i
i c
c l
l e
e a
a c
c c
c e
e l
l e
e r
r a
a t
t e
e d
d b
b y
y a
a p
p o
o t
t e
e n
n t
t i
i a
a l
l V
V
W
W
h
h e
e n
n a
a c
c h
h a
a r
r g
g e
e d
d p
p a
a r
r t
t i
i c
c l
l e
e c
c a
a r
r r
r y
y i
i n
n g
g a
a c
c h
h a
a r
r g
g e
e q ,
i
i s
s a
a c
c c
c e
e l
l e
e r
r a
a t
t e
e d
d t
t h
h r
r o
o u
u g
g h
h a
a p
p o
o t
t e
e n
n t
t i
i a
a l
l
d
d i
i f
f f
f e
e r
r e
e n
n c
c e
e o
o f
f V
V
v
v o
o l
l t
t s
s ,
t
t h
h e
e n
n k
k i
i n
n e
e t
t i
i c
c e
e n
n e
e r
r g
g y
y
E = qV
d
d e
e B
B
r
r o
o g
g l
l i
i e
e w
w a
a v
v e
e l
l e
e n
n g
g t
t h
h
mqV
h
λ =
d dee BBrroogglliiee wwaavvee lleennggtthh aassssoocciiaatteedd wwiitthh eelleeccttrroonnss
Let us consider the an electron of rest mass
0
m and charge e being accelerated by a
potential V volts. If v is the velocity attained by the electron due to acceleration then
m v = eV
2
0
0
2
m
eV
v =
0
m
eV
v =
According to de Broglie concept
mv
h
0
eV
m
m
h
mv
h
0
0 0
m eV
h
0
V
31 19
34
− −
−
× × × ×
×
0
A
V
The above equation shows the wave length associated with electron accelerated to V volts.
If V = 100 volts.
0
λ = 12. 26 100 A
0
= 1. 226 A
p = 2 mE
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Prepared by: Dr. P.Sreenivasula Reddy M.Sc, (PhD)
Website: www.palleti.webnode.com Page 4
O
O
n
n e
e d
d i
i m
m e
e n
n s
s i
i o
o n
n a
a l
l Schrödinger time independent wave equation
Schrödinger derived a second order differential equation to describe wave the motion of
de Broglie waves associated with a particle.
Let us consider a particle of mass ‘m’ moving with velocity ‘v’ along x direction. The
displacement of the wave function ′ is
〶䙦〸け⡹ぐぇ䙧
Where A is amplitude of the wave.
Differencing the equation (1) w.r.t x , twice
〶䙦〸け⡹ぐぇ䙧
⡰
⡰
⡰
⡰
〶䙦〸け⡹ぐぇ䙧
⡰
⡰
⡰
Since ᡣ 㐄
⡰ゕ
ゐ
〱
ㄘ
〱け
ㄘ
⡲ゕ
ゐ
ㄘ
⡰
According to de Broglie hypothesis the wave length of matter wave associated with a
particle of mass m moving with a velocity v is
p
h
mv
h
In terms of momentum, the kinetic energy of the particle can be written as
m
p
m
m v
KE mv
2 2 2
2
Substituting equation (3) in equation (4)
2
2
2 m λ
h
KE =
h
Q p =
2 2
h
m KE
Substituting equation (5) into equation (2)
( )
=− Ψ
∂
∂Ψ
2
2
2
2
2
h
m KE
x
π
2
2
2
2
h
m KE
x
Total energy of systemE can be written as,
E = KE + PE
E = KE + V
Q PE = V
KE = E − V (7)
Substituting equation (7) into equation (6)
2
2
2
2
h
m E V
x
( ) 0
2
2
2
2
E V
h
m
x
π
( ) 0
2 2
2
E V
m
x h
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Unit –III QUANTUM MECHANICS Engineering Physics
Dr. P.Sreenivasula Reddy M.Sc, Ph.D.
Website: www.engineeringphysics.weebly.com Page 5
The above equation shows the one dimensional Schrodinger time independent wave equation
moving along x axis.
The three dimensional Schrodinger wave equation can be written as
( ) 0
2
2
∇ Ψ+ E − V Ψ=
m
h
Where
2
2
2
2
2
2
2
x y ∂ z
O
O
n
n e
e d
d i
i m
m e
e n
n s
s i
i o
o n
n a
a l
l Schrödinger time dependent wave equation
By eliminating the total energy in the Schrodinger time independent wave equation, we
get Schrodinger time dependent wave equation. The wave function is given by
〶䙦〸け⡹ぐぇ䙧
Differencing the equation (1) w.r.t ' t ', we get
〶䙦〸け⡹ぐぇ䙧
〶䙦〸け⡹ぐぇ䙧
Since ᠱ 㐄 ᡠ , 㐄
〆
〵
〱
〱ぇ
Substituting the value of ᠱ′ in Schrodinger time dependent wave equation,
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
⡰
Three dimensional Schrodinger time dependent wave equation can be written as
⡰
⡰
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Unit –III QUANTUM MECHANICS Engineering Physics
Dr. P.Sreenivasula Reddy M.Sc, Ph.D.
Website: www.engineeringphysics.weebly.com Page 7
From equation (4)
0 = A Cos 0 + BSin 0
∴ A = 0 6
From equation (5)
0 = A CosKa + BSin Ka
0 = 0 Cos Ka + BSin Ka
0 = 0 + BSin Ka
BSin Ka = 0
From the above equation Sin Ka = 0
Q B ≠ 0
Ka = n π
a
n
K
= 7
Where n = 0, 1, 2, 3… represents the quantum numbers.
Put equation (6) and (7) in equation (3) we get
a
n x
x BSin
π
ψ =
Eigen functions:- 8
By applying normalization we will find out the value of B
The normalization condition is
0
x dx
a
ψ 9
Put equation (8) in equation (9)
0
2 2
dx
a
n x
B Sin
a π
0
2
dx
a
n x
B Cos
a
π
[ ( 0 ) ( 0 0 )] 1
2
a − − − =
B
2
B a
a
B
Substituting equation (10) in equation (8) we get
2
a
a
n
a
nx
Sin
x
B
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Prepared by: Dr. P.Sreenivasula Reddy M.Sc, (PhD)
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( )
a
n x
Sin
a
x
π
ψ
Eigen values
From equations (2) and (7)
( E )
h
m
a
n
2
2
2
2 2
π 8 π
2
2 2
8 ma
n h
E = 12
From equations (11) and (12) we will explain the motion of electrons in one dimensional box of
potential zero.
Case (i) for n = 1
( )
a
x
Sin
a
x
1
and
2
2
1
8 ma
h
E =
Hence ( x )
1
ψ is a maximum at exactly middle of the box as shown in figure.
Case (ii) for n = 2
a
x
Sin
a
x
2
1 2
2
2
E
ma
h
E = =
Hence
( x )
2
is a maximum at quarter distance from either sides of the box as shown in the
figure.
The values of E
n
are known as Eigen values and the corresponding wave functions
n
ψ as
Eigen functions of the particle.
Probability of the location of particle:
The probability of finding a particle over a small distance ᡖᡶ is given by
ぁ
⡰
⡰
Thus the probability density for one dimensional motion is
⡰
The probability density is maximum when
For ᡦ 㐄 1, the most probable position of the particle is at ᡶ 㐄 ᡓ/
For ᡦ 㐄 2, the most probable positions of the particle is at ᡶ 㐄
〨
⡲
and
⡱〨
⡲
( x )
1
ψ
( x )
2
ψ
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