Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
QUANTUM MECHANICS
WAVE- PARTICLE DUALISM:
Wave: A wave is a spread-out disturbance specified by its amplitude, frequency, wavelength,
phase intensity etc…
Particle: A particle is a highly localized mass which is characterized by velocity, momentum
(p), Energy etc.
The phenomenon of interference, diffraction and polarization requires the presence of
two or more waves at the same time and at the same place. It is very clear that two or more
particles cannot simultaneously occupy the same position. So, one has to conclude that
radiation behaves like a wave.
It is well known that light exhibits the phenomenon like interference, diffraction,
Polarization, Photoelectric effect, Compton effect and discrete emission and absorption.
Interference, diffraction and polarization are explained by using wave theory, according to
which light has wave nature. Photoelectric effect, Compton effect and discrete emission and
absorption are explained on the basis of quantum theory, according to which light is propagated
in small packets of energy, each of E= h. These packets are called photons or quanta, which
behave like corpuscles (Particles). Thus, light possesses wave nature as well as particle nature.
It is referred to as wave particle dualism or dual nature of light.
In 1924, de-Broglie, made a daring suggestion that like light, matter also exhibits dual
nature. In other words, particles of matter like electrons, protons and neutrons also exhibit dual
nature i.e., wave nature as well as particle nature. The waves associated with the particles of
matter are known as matter or pilot waves or de-Broglie waves. According to de-Broglie
hypothesis, the wavelength  , associated with any moving particle of momentum P= mv is
given by  =
h
h
=
p mv
Where h is the Planck’s constant and  is called the de-Broglie wavelength.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
The expression for  can be derived considering photon to be a particle of mass m
moving with velocity C.
The energy associated with it is given by Einstein’s mass- energy relation i.e.
E = m c 2 ----- (1)
According to Planck’s hypothesis, energy of a photon of frequency  is given by
E = h  ------ (2)
From equation (1) and (2), we have h = m c 2  hc/ = m c 2   = h/ mc
De-Broglie wavelength of particles:
Suppose a particle at rest has been accelerated through a potential difference of V volts and
gains a velocity of ‘v’ m/sec. if m is the mass and e is the charge on the particle, then energy
gained by the particle = eV.
Also, kinetic energy of the particle =
1
mv 2
2
 Ve =
Velocity of an particle v =
1
mv 2
2
2eV
m
Using de-Brogile’s equation, wavelength of the particle  =
h
=
mv
h
2meV
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
If we consider the particle to be an electron then by using above equation, where
h = 6.625x10−34 J. sec ,
charge on electron
e = 1.6x10−19 C ,
mass of electron
m = 9.1x10−31 kg wavelength of electron is
=
12.27
V
Å
This shows that wavelength  is inversely proportional to the square root of accelerating
potential
CHARACTERISTICS OF MATTER WAVES
1. Matter waves are produced whenever the particles of the matter are in motion.
2. The de Broglie wavelength of a particle of mass m, moving with a velocity v is given
by  =
h
mv
3. Larger the mass and velocity, shorter will be the de Broglie wavelength.
4. de Broglie waves cannot be observed. It is a wave model to describe and study
matter.
5. When a particle is at rest, the wavelength associated with it becomes infinite. This
shows that only the moving particle produces the matter waves.
6. The wavelength of matter waves does not depend on the nature and change of the
particle.
7. The phase velocity of matter waves can be greater than that light, V phase =
c2
Vg
8. Matter waves propagate in the form of wave packet with group velocity V g .
9. Velocity Vphase of de Broglie wave is not the same as the velocity of the moving
particle.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
WAVE PACKET, PHASE VELOCITY AND GROUP VELOCITY:
A wave packet is formed by the superposition of a number of waves situated around the center
wavelength given by the de-Broglie formula. Such a wave packet will have resultant amplitude
which is appreciably different from zero only in a certain region of space having the
dimensions of the particle, this small region of space may be associated with the position of the
particle. The velocity of the component monochromatic waves making up of wave packet is
called phase velocity and the velocity of the wave group is itself is called group velocity.
Relation between group velocity and phase velocity:
For a wave group formed the superposition of number of waves the group velocity and phase
velocity are given by, v g =
d
dk
and
v ph =

k
 = v ph k
d
d
(v ph .k )
=
dk dk
dv ph
 v g = v ph + k .
dk
2 dv ph d
v g = v ph +
.
 d dk
2 dv ph − 2
v g = v ph +
.
2
 d
(
 dv ph
v g = v ph −  
 d
)



This is the relation between group velocity and phase velocity.
Relation between group velocity (Vg) and particle velocity (V or Vparticle):
Group velocity is given by v g =
d
dk
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Unit-4 Quantum mechanics
Hence,
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
v g = v particle
This is the relation between group velocity and particle velocity.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
RELATION BETWEEN VELOCITY OF LIGHT, GROUP VELOCITY AND PHASE
VELOCITY:
v phase .v g = c 2
HEISENBERG’S UNCERTAINTY PRINCIPLE:
Physical quantities like position, momentum, energy, time etc… can be measured
accurately in macroscopic systems. In the microscopic system measurements are not very
accurate. If the measurement of one is certain, then that of the other is uncertain. To account for
the uncertainty in the measurements, in 1927 Heisenberg proposed uncertainty principle.
In quantum mechanics, a moving particle can be the representative of the wave packet.
A wave packet cannot have a definite momentum. In such a case it is not possible to know the
exact location of the particle on the wave. It can be said that the particle should lie in a region
occupied by the wave and that its chance of being at a given point within this region is
proportional to the wave amplitude at that point. Thus, we can say that the particle is
somewhere in the wave packet moving with the group velocity.
It is impossible to determine simultaneously both the position and momentum of a
particle with accuracy.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
OR
In any simultaneous determination of the position and momentum of a particle, the
product of the corresponding uncertainties inherently present in the measurement is
equal to, or greater than h/4π.
where, Δx ----is uncertainty in position
Δp----- is uncertainty in momentum.
The problem of uncertainty is a logical consequence of dual nature of matter.
ILLUSTRATIONS OF HEISENBERG’S UNCERTAINTY PRINCIPLE:
Impossibility of existence of electrons in the atomic nucleus:
Heisenberg’s Uncertainty principle states that,
The diameter of the nucleus is of the order of 10-14m. If an electron is to exist inside the
nucleus, then the uncertainty in its position ∆x must not exceed the size of the nucleus
i.e,.
∆x≤10-14m.
Uncertainty in momentum is,
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
which is the uncertainty in the momentum of the electron.
Therefore, momentum of the electron must at least be equal to the uncertainty in the
momentum
There for,
From relativistic equation,
Neglecting the second term as it is smaller by more than the 3 orders
of the magnitude compared to first term.
Neglecting the second term as it is smaller by more than the 3 orders of the magnitude
compared to first term.
……………………(3)
Now by using the condition, we can say that in order that the electron may exist within the
nucleus, its energy E must be such that, Now from equation (3),
x 3 x 108
In order that an electron may exist inside the nucleus, its energy must be greater than or equal
to 9.7 MeV. But, the experimental investigations on beta decay shows that the kinetic energy of
the beta particles (electrons) is of the order of 3 to 4 MeV. This clearly indicates that electrons
cannot exist within the nucleus.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
WAVE FUNCTION:
Wave function: the quantity that characterizes the de –Broglie wave or matter wave is called

the wave function. It is usually denoted as  = (r , t ) or  = (x, y, z, t ). This gives complete
information about the state of a physical system at a particular time. It is also called the state
function and represents the probability amplitude. If Ψ is large the probability of finding he
particle is also large and if Ψ is small then the probability of finding the particle is small.
The wave function gives the likelihood of finding the particle at a given instant and at a
given position.
PHYSICAL SIGNIFICANCE OF WAVE FUNCTION:
The value of the wave function of a particle at a given point of space and time is related to the
likelihood of the particle’s being there at the time. By analogy with waves such as those of
sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an
expression for the amplitude of the particle wave (or de Broglie wave), although for such
waves amplitude has no physical significance. The square of the wave function, Ψ2, however,
does have physical significance: the probability of finding the particle described by a specific
wave function Ψ at a given point and time is proportional to the value of Ψ2.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
2
To be more specific, if  represents a single particle, then |  | is called the probability
density. It is the probability per unit volume that the particle will be found within a volume τ
containing the point x. The interpretation was first suggested by max born in 1928. If dτ is the
small volume element surrounding point x, then the probability of finding the particle in that
volume is dp =|  | 2 d
dp is the probability that the particle is within the volume element dτ. |  | 2 d is the
probability that the particle will be found in the volume element dτ surrounding the point x,
then the total probability of finding the particle somewhere in space τ must be equal to unity.
Hence  must satisfy the condition that ∫τ  d = 1
2
NORMALIZATION OF WAVE FUNCTION
The probability of finding the particle in volume d is given by  dx.dy.dz . For the total
2
probability of finding the particle somewhere is, of course, unit i.e., particle is certainly to be
found somewhere in space τ.
∫τ  d = 1 ……………(1)
2
This is called the normalization condition. So a wave function ψ(x,t) is said to be normalized if
it satisfies the condition (1)
EIGEN FUNCTIONS AND EIGEN VALUES:
In quantum mechanics to know the state of the system it is required to know about the wave
function Ψ. In order to find Ψ, the Schrodinger’s equation has to be solved. Since it is a second
order differential equation, there are several solutions. Here we have to select the wave
functions, which would correspond meaningfully to a physical system. Such a wave function
are said to be acceptable wave functions. These acceptable wave functions are called as Eigen
functions.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
The energy values that are evaluated from the Schrodinger’s wave equation by using the
Eigen functions are called as Eigen values.
PROPERTIES OR LIMITATIONS OF WAVE FUNCTIONS:
1.
Ψ must be finite for all values of x, y, z.
A function f(x) is not finite at P. At x=P, f(x) =∞. Thus, if f(x) were to be a wave function, it
signifies a large probability of finding the particle at a single location (x= R), which violates the
uncertainty principle. Hence the wave function becomes unacceptable.
2.
Ψ must be single valued. i.e., for each set of x, y and z.
3.
Ψ and its first derivatives with respect to its variables must be continuous
everywhere.
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Unit-4 Quantum mechanics
4.
The partial derivatives of Ψ i.e.,
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
  
must also be continuous.
,
,
x y z
SCHRODINGER’S WAVE EQUATION:
It is the wave equation capable of determining the wave function Ψ of the matter waves
in different physical situations.
Time independent Schrodinger’s wave equation:
According to de-Broglie theory, for a particle of mass ‘m’, moving with a velocity ‘v’,
associated with it is a wave of wavelength
=
h
p
The space and time dependent wave function for a de-Broglie wave can be written in complex
notation as (capital psi  ),
 = Ae i ( kx−t ) ..............(1)
The space or position dependent part of wave function can be taken as small psi ψ
where, A is a constant and ω is angular frequency.
Differentiate equation (1) with respect to ‘t’ twice,
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Unit-4 Quantum mechanics

Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT

d 2
= − 2 Ae i ( kx−t )
2
dt
d 2
= − 2 ...........(2)
2
dt
we have equation for the traveling wave as,
d2y
1 d2y
=
...........(3)
dx 2 v 2 dt 2
where, y is displacement and v is velocity of wave.
By analogy, we can write the wave equation for de-Broglie wave for the motion of a free
particle as,
d 2
1 d 2
=
..........(4)
dx 2
v 2 dt 2
this represents the de-Broglie wave propagating along x-direction with a velocity ‘v’ and Ψ is
displacement.
From equation (2) and (4),
d 2
1
= − 2  2
2
dx
v
d 2
1
(2 )2 2 
=−
2
2
dx
( )
here v = 
 = 2
 − frequency
d 2
4 2
=− 2 
dx 2

1
1 d 2
=
−
............(5)
2
4 2  dx 2
The kinetic energy of a moving particle of mass ‘m’ and velocity ‘v’ is given by
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
1 2 m2v 2
p2
mv =
=
2
2m
2m
But we have from equation (1), p = (h/λ)
K .E. =
h2 1
2m  2
Substitute for (1/ λ)2 from equation (5),
K .E =
− h2 1 d2
...............(6)
2m 4 2  dx 2
Let there be a field where the particle is present. Depending on its position in the field, the
particle will possess certain potential energy. Then
Total energy = Kinetic energy + Potential energy
E=
p2
+V
2m
p2
= E −V
2m
from equation (6), we can write
− h2 1 d 2
= (E − V )
8m 2  dx 2

d 2  8 2 m
+ 2 (E − V ) = 0
dx 2
h
This is the time independent Schrodinger’s wave equation in one-dimension.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
APPLICATIONS OF SCHRODINGER’S WAVE EQUATION:
Particle in 1-dimensional potential well of infinite height (Particle in a box):
Consider a particle of mass ‘m’ is freely moving in x- direction in the region from x=0
to x=a. Outside this region potential energy ‘V’ is infinity and within this region V=0.
Out side the box Schrodinger’s wave equation is
d 2  8 2 m
(E −  ) = 0............(1)
+
dx 2
h2
this equation holds good only if  =0 for all points outside the box i.e.,  = 0 , which means
2
that the particle cannot be found at all outside the box.
Inside the box, the Schrodinger’s equation is given by,
d 2  8 2 m
+ 2 E = 0
dx 2
h
2
d 
+ k 2  = 0..............(2)
2
dx
where, k 2 =
8m 2 E
............(2a)
h2
Discussion of the solution:
The solution of the above equation is given by
…………………….(3)
V=00
where C & D are constants depending on the boundary condition.
Now apply boundary conditions for this,
V=0
x=0
x
x=a
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
Condition:I
 = 0.
at x =0,
equation (3) becomes C = 0.
Condition:II
at x =a,  = 0

0= D sin(ka)

(Because the wave concept vanishes).
quantum numbers
Then equation (3) becomes,
………………….(4)
Substitute the value of ‘k’ in equation (2a).
We get,
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
when n=0, n = 0. which means to say that the electron is not present inside the box which is
not true. Hence the lowest value of ‘n’ is 1.
 The lowest energy corresponds to ‘n’ =1 is called the zero-point energy or ground state
E zero − po int =
energy.
h2
8ma 2
All the states for n 1 is called excited states.
To evaluate D in equation (3), one has to perform normalization of wave function.
Normalization:
Consider equation (4)
The integral of the wave function over the entire space in the box must be equal to unity
because there is only one particle within the box, the probability of finding the particle is 1.
a

2
dx = 1
0
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
Thus, the normalized wave function of a particle in a one-dimensional box is given by,
n =
2  n 
sin 
x
a  a 
where n=1,2,3……………
1 =
2  
sin   x
a a
2 =
2  2 
sin 
x
a  a 
Since the particle in a box is a quantum mechanical problem, we need to evaluate the most
probable location of the particle in a box and its energies at different permitted state.
Eigen function, Eigen energy values and probability density for a Particle in 1dimensional potential well of infinite height (Particle in a box):
Let us consider first three cases:
Case 1: n=1.
This is the ground state, and the particle is normally found in this state.
For n=1, the eigen function is
 
1 = B sin   x
a
In the above equation  =0 for both x=0 & x=a. but 1 has maximum value for x=a/2.
  a
1 = B sin   = B
a2
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Unit-4 Quantum mechanics
A plot of
1
2
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
the probability density versus ‘x’ is as shown. From the figure, it
indicates that the probability of finding the particle at different locations inside the box.
 1 =0
2
and 1
2
at x = 0 and x = a
is maximum at x = (a/2).
This means, in the ground state the particle cannot be found at the walls of the box and the
probability of finding the particle is maximum at the central region.
Energy in the ground state =
h2
= E0 .
8ma 2
Case 2: n =2
This is the first excited state. The eigen function for this state is given by
 2 
2 = B sin   x
 a 
now, 2 =0 for the values x = 0, a 2 , a
and 2 reaches maximum at x = a 4 , 3a 4
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
these facts are seen in the following plot.
From the figure it can be seen that 1
2
= 0 at x = 0, a 2 , a . It means that in the first excited
state the particle cannot be observed either at the walls or at the center.
The energy is E2 = 4E0 .
Thus, the energy in the first excited state is 4 times the zero point energy.
Case 3 : n =3
This is the second excited state and the Eigen function for this state is given by
 3
3 = B sin 
 a

x

20
Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
now, 3 =0 for the values x = 0, a , 2a , a
3
3
and 2 reaches maximum at x = a , a , 5a
6 2
6
The plot of 3
2
versus ‘x’ has maxima at x = a , a , 5a at which the particle is most likely
6 2
6
to be found.
The energy corresponds to second excited state is given by E3 = 9E0 .
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
Questions from Unit 4
1. State de Broglie’s hypothesis and important characteristics of matter waves.
2. Define group velocity, phase velocity, and obtain an expression relating group
velocity and phase velocity.
3. Based on matter wave concept obtain relation between group velocity and particle
velocity.
4. Using uncertainty principle, prove that free electron does not exist inside the
nucleus.
OR
State uncertainty principle and give its application.
5. Give the characteristics and physical significances of a wave function.
6. What are Eigen values and Eigen functions? Discuss their nature.
7. Set up the one-dimensional time independent Schrodinger wave equation.
8. Assuming the time independent Schrodinger wave equation discuss the solutions
for energy of a particle in one dimensional infinite potential well.
9. Find the wavefunction, energy values and probability densities in the first three
energy levels for a particle in one dimensional infinite potential well.
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Unit-4 Quantum mechanics
Dr. Hitha D Shetty
Associate Professor
Dept. of Physics, NMIT
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