Wave velocity or phase velocity is the velocity with which a monochromatic wave propagates through a medium. It is represented by Vp and is equal to the ratio of the angular frequency (ω) to the wave number (k). The phase velocity describes the velocity at which the phase of the wave propagates in space, not the velocity of energy transport.

1. Wave Velocity or Phase Velocity
When a monochromatic wave travels through a medium,
its velocity of advancement in the medium is called the
wave velocity or phase velocity (Vp).
k
Vp


where is the angular frequency
and is the wave number.

 2



2

k

2. Wave velocity/phase velocity
 Wave motion is a form of disturbance which travels
through medium due to the repeated motion of the
particles of the medium about their mean positions, the
motion being handed over from one particle to the next.
 Every particle begins its vibrations a little later than its
predecessor and there is a progressive change of phase as
wave travel from one particle to the next.
 The phase relationship of these particles that we observe
as a wave and the velocity with which the plane of equal
phase travels through the medium, is known as phase
velocity.
 Hence the velocity with which monochromatic wave
propagates through medium is called the wave velocity.
2

3. The equation of motion for a
plane progressive wave is
given by
y=A cos (wt - kx) ……………(1)
Where y= displacement of the
wave depends upon space
coordinate ‘x’ and time ‘t’. A is
the amplitude and w is the
angular frequency which is
related to the frequency v by
w=2πv
k is the phase constant and given by k=2π/λ
3

4. The argument of the cosine function represents the phase of the
wave, ϕ, or the fraction of a complete cycle of the wave.
In-phase waves
Out-of-phase
waves
Line of equal phase = wavefront = contours of maximum field
4

5. 5

6. Phase velocity of de-Broglie waves
exceeds velocity of light

6

7. 
7

8. Group velocity: the concept of wave packet
The group velocity of a wave is the velocity with
which the overall shape of the waves' amplitudes
known as the modulation or envelope of the wave
propagates through space.
For example, imagine what happens if a stone is
thrown into the middle of a very still pond. When
the stone hits the surface of the water, a circular
pattern of waves appears. It soon turns into a
circular ring of waves with a quiescent center. The
ever expanding ring of waves is the wave group,
within which one can discern individual wavelets of
differing wavelengths traveling at different speeds.
The longer waves travel faster than the group as a
whole, but they die out as they approach the leading
edge. The shorter waves travel more slowly and
they die out as they emerge from the trailing
boundary of the group.
Solid line: A wave
packet. Dashed line:
The envelope of the
wave packet. The
envelope moves at the
group velocity.
8

9. The Group Velocity
This is the velocity at which the overall shape of the wave’s amplitudes, or the
wave ‘envelope’, propagates. (= signal velocity)
Here, phase velocity = group velocity (the medium is non-dispersive)
9

10. Dispersion: phase/group velocity depends on frequency
Black dot moves at phase velocity. Red dot moves at group velocity.
This is normal dispersion (refractive index decreases with increasing λ)
10

11. Expression for group velocity and
wave velocity
Let the wave packet is formed by the
superimposition of two waves of equal amplitude
‘A’ but of slightly different frequency w and w+dw
and propagation constants k and k+dk.
Displacement y1 and y2 of two waves
y1 = A cos(wt-kx)
y2 = A cos[(w+dw)t - (k+dk)x]
The resultant displacement y= y1+ y2
y= A {cos(wt-kx)+cos[(w+dw)t - (k+dk)x]}
11

12. 
12
Wave packet
produced by
superimposition
of two waves
with slightly
different
frequency

13. Group Velocity
So, the group velocity is the velocity with which the energy
in the group is transmitted (Vg).
dk
d
Vg


The individual waves travel “inside” the group with their
phase velocities.
In practice, we came across pulses rather than
monochromatic waves. A pulse consists of a number of
waves differing slightly from one another in frequency.
The observed velocity is, however, the velocity with which
the maximum amplitude of the group advances in a
medium.

14. Relation between phase velocity and
group velocity

14

15. 
15

16. Group Velocity of De-Broglie’s waves
The discrepancy is resolved by postulating that a moving
particle is associated with a “wave packet” or “wave
group”, rather than a single wave-train.
A wave group having wavelength λ is composed of a
number of component waves with slightly different
wavelengths in the neighborhood of λ.
Suppose a particle of rest mass mo moving with velocity v
then associated matter wave will have
h
mc2
2
  and
h
mv
k

2
 where 2
2
1 c
v
m
m o



17. and
On differentiating w.r.t. velocity, v
2
2
2
1
2
c
v
h
c
mo




2
2
1
2
c
v
h
v
m
k o



 2
3
2
2
1
2
c
v
h
v
m
dv
d o




(i)
 2
3
2
2
1
2
c
v
h
m
dv
dk o


 (ii)

18. Wave group associated with a moving particle also
moves with the velocity of the particle.
o
o
m
v
m
dk
dv
dv
d



2
2
. 
Dividing (i) by (ii)
g
V
v
dk
d



Moving particle wave packet or wave group


19. Heisenberg Uncertainty Principle
It states that only one of the “position” or “momentum” can be
measured accurately at a single moment within the instrumental
limit.
It is impossible to measure both the position and momentum
simultaneously with unlimited accuracy.
or
uncertainty in position
uncertainty in momentum

x

 x
p
then
2



 x
p
x

2
h


The product of & of an object is greater than or equal to
2

x
p

x


20. If is measured accurately i.e.
x
 0

x 


 x
p
Like, energy E and time t.
2



 t
E
2



 
L
The principle applies to all canonically conjugate pairs of quantities in
which measurement of one quantity affects the capacity to measure
the other.
and angular momentum L and angular position θ

21. Order of diameter of a nucleus ~ 5 x10-15 m
then
If electron exist in the nucleus then
Applications of Heisenberg Uncertainty Principle
(i) Non-existence of electron in nucleus
m
x 15
max 10
5
)
( 



2



 x
p
x
2
)
(
)
( min
max



 x
p
x
1
20
min .
.
10
1
.
1
2
)
( 





 s
m
kg
x
px

MeV
E 20
  relativistic

22. Thus the kinetic energy of an electron must be greater than 20
MeV to be a part of nucleus
Thus we can conclude that the electrons cannot be present
within nuclei.
Experiments show that the electrons emitted by certain unstable
nuclei don’t have energy greater than 3-4 MeV.

23. But
Concept of Bohr Orbit violates Uncertainty Principle
m
p
E
2
2

2
.



 p
x
m
p
p
E



m
p
mv
 p
t
x




p
x
t
E 



 .
.
2
.



 t
E

24. According to the concept of Bohr orbit, energy of an electron in a
orbit is constant i.e. ΔE = 0.
2
.



 t
E



 t
All energy states of the atom must have an infinite life-time.
But the excited states of the atom have life–time ~ 10-8 sec.
The finite life-time Δt gives a finite width (uncertainty) to the energy
levels.
Assignment:
Zero point energy of Harmonic Oscillator
Finite width of spectral lines
Radius of Bohr’s first orbit

25. Radius of Bohr’s first orbit
• Let are the uncertainty in position and
momentum of electron in the first orbit. We
know
p
xand

2



 x
p
x
2
2
2
16
4
mze
h
x
R






26. Wave function


 *
|
| 2

The quantity with which Quantum Mechanics is concerned is the
wave function of a body.
|Ψ|2 is proportional to the probability of finding a particle at a
particular point at a particular time. It is the probability density.
Wave function, ψ is a quantity associated with a moving particle. It
is a complex quantity.
Thus if iB
A

 iB
A

*

2
2
2
2
2
2
*
|
| B
A
B
i
A 




 


then
ψ is the probability amplitude.

27. Normalization

d
|Ψ|2 is the probability density.
The probability of finding the particle within an element of volume

 d
2
|
|
Since the particle is definitely be somewhere, so
1
|
| 2






 d
A wave function that obeys this equation is said to be normalized.
 Normalization

28. Properties of wave function
1. It must be finite everywhere.
If ψ is infinite for a particular point, it mean an infinite large
probability of finding the particles at that point. This would
violates the uncertainty principle.
2. It must be single valued.
If ψ has more than one value at any point, it mean more than
one value of probability of finding the particle at that point
which is obviously ridiculous.
3. It must be continuous and have a continuous first derivative
everywhere.
z
y
x 




 


,
, must be continuous
4. It must be normalizable.

29. Schrodinger’s time independent wave equation



x
A
2
sin

One dimensional wave equation for the waves associated with a
moving particle is
From (i)




 x
A
x
2
sin
4
2
2
2
2




ψ is the wave amplitude for a given x.
where
A is the maximum amplitude.
λ is the wavelength
(i)




2
2
2
2
4




x
(ii)

30. v
m
h
o


2
2
2
2
1
h
v
mo


 2
2
2
1
2
h
v
m
m o
o 






2
2
2
1
h
K
mo


where K is the K.E. for the non-relativistic case
(iii)
Suppose E is the total energy of the particle
and V is the potential energy of the particle
)
(
2
1
2
2
V
E
h
mo




31. This is the time independent (steady state) Schrodinger’s wave
equation for a particle of mass mo, total energy E, potential
energy V, moving along the x-axis.
If the particle is moving in 3-dimensional space then
Equation (ii) now becomes



)
(
2
4
2
2
2
2
V
E
m
h
x
o 




0
)
(
2
2
2
2







V
E
m
x
o

0
)
(
2
2
2
2
2
2
2
2















V
E
m
z
y
x
o


32. For a free particle V = 0, so the Schrodinger equation for a
free particle
0
2
2
2


 
 E
mo

0
)
(
2
2
2



 
 V
E
mo

This is the time independent (steady state) Schrodinger’s wave
equation for a particle in 3-dimensional space.

33. Schrodinger’s time dependent wave equation
)
( px
Et
i
Ae


 

Wave equation for a free particle moving in +x direction is
(iii)


2
2
2
2

p
x




where E is the total energy and p is the momentum of the particle
Differentiating (i) twice w.r.t. x
(i)
2
2
2
2
x
p






  (ii)
Differentiating (i) w.r.t. t



iE
t




t
i
E





 

34. For non-relativistic case
Using (ii) and (iii) in (iv)
(iv)



V
x
m
t
i 






2
2
2
2


E = K.E. + Potential Energy
t
x
V
m
p
E ,
2
2




 V
m
p
E 


2
2
This is the time dependent Schrodinger’s wave equation for a
particle in one dimension.

35. Linearity and Superposition
2
2
1
1 

 a
a 

If ψ1 and ψ2 are two solutions of any Schrodinger equation of a
system, then linear combination of ψ1 and ψ2 will also be a solution
of the equation..
Here are constants
Above equation suggests:
2
1 &a
a
is also a solution
(i) The linear property of Schrodinger equation
(ii) ψ1 and ψ2 follow the superposition principle

36. 2
1 

 

Then
Total probability will be
2
2
1
2
|
|
|
| 

 


P
due to superposition principle
)
(
)
( 2
1
*
2
1 


 


)
)(
( 2
1
*
2
*
1 


 


1
*
2
2
*
1
2
*
2
1
*
1 






 



1
*
2
2
*
1
2
1 


 


 P
P
P
2
1 P
P
P 

Probability density can’t be added linearly
If P1 is the probability density corresponding to ψ1 and P2 is the
probability density corresponding to ψ2

37. Expectation values
dx
x
f




 2
|
|
)
( 
Expectation value of any quantity which is a function of ‘x’ ,say f(x)
is given by
for normalized ψ
Thus expectation value for position ‘x’ is

 )
(x
f
dx
x




 2
|
|

 x
Expectation value is the value of ‘x’ we would obtain if we
measured the positions of a large number of particles described by
the same function at some instant ‘t’ and then averaged the
results.

38. dx
x


1
0
2
|
|
Solution

 x
1
0
4
2
4







x
a
Q. Find the expectation value of position of a particle having wave
function ψ = ax between x = 0 & 1, ψ = 0 elsewhere.
dx
x
a 

1
0
3
2

 x
4
2
a


39. Operators


p
i
x 



(Another way of finding the expectation value)
For a free particle
An operator is a rule by means of which, from a given function
we can find another function.
)
( px
Et
i
Ae


 

Then
Here
x
i
p




^
is called the momentum operator
(i)

40. 

E
i
t 




Similarly
Here
t
i
E


 
^
is called the Total Energy operator
(ii)
Equation (i) and (ii) are general results and their validity is the
same as that of the Schrodinger equation.

41. U
x
i
m
t
i 











2
2
1 

If a particle is not free then
This is the time dependent Schrodinger equation
^
^
^
.
. U
E
K
E 

^
^
2
^
2
U
m
p
E
o



U
U 

^
U
x
m
t
i 






2
2
2
2





U
x
m
t
i 






2
2
2
2



42. If Operator is Hamiltonian
Then time dependent Schrodinger equation can be written as
U
x
m
H 



 2
2
2
^
2


 E
H 
^
This is time dependent Schrodinger equation in Hamiltonian
form.

43. Eigen values and Eigen function
Schrodinger equation can be solved for some specific values of
energy i.e. Energy Quantization.


 a

^
Suppose a wave function (ψ) is operated by an operator ‘α’ such
that the result is the product of a constant say ‘a’ and the wave
function itself i.e.
The energy values for which Schrodinger equation can be solved
are called ‘Eigen values’ and the corresponding wave function are
called ‘Eigen function’.
then
ψ is the eigen function of
a is the eigen value of
^

^


44. Q. Suppose is eigen function of operator then find
the eigen value.
The eigen value is 4.
Solution.
x
e2

 2
2
dx
d
2
2
^
dx
d
G 
2
2
^
dx
d
G

  )
( 2
2
2
x
e
dx
d

x
e
G 2
^
4



 4
^

G

45. Particle in a Box
Consider a particle of rest mass mo enclosed in a one-dimensional
box (infinite potential well).
Thus for a particle inside the box Schrodinger equation is
Boundary conditions for Potential
V(x)=
0 for 0 < x < L

{ for 0 > x > L
Boundary conditions for ψ
Ψ =
0 for x = 0
{0 for x = L
0
2
2
2
2






E
m
x
o

x = 0 x = L


V 

V
particle
0

V
0

V inside
(i)

46. Equation (i) becomes
p
h


k

2


p
k 


E
mo
2

2
2 2

E
m
k o


(k is the propagation constant)
(ii)
0
2
2
2






k
x
(iii)
General solution of equation (iii) is
kx
B
kx
A
x cos
sin
)
( 

 (iv)

47. Equation (iv) reduces to
Boundary condition says ψ = 0 when x = 0
(v)
0
.
cos
0
.
sin
)
0
( k
B
k
A 


1
.
0
0 B

 0

 B
kx
A
x sin
)
( 

Boundary condition says ψ = 0 when x = L
L
k
A
L .
sin
)
( 

L
k
A .
sin
0 
0

A 0
.
sin 
 L
k

n
L
k sin
.
sin 


48. Put this in Equation (v)
(vi)
L
n
k


L
x
n
A
x

 sin
)
( 
Where n # 0 i.e. n = 1, 2, 3…., because n=0 gives ψ = 0
everywhere.

n
kL
Put value of k from (vi) in (ii)
2
2 2

E
m
k o

2
2
2

E
m
L
n o






 

49. Where n = 1, 2, 3….
Equation (vii) concludes
o
m
k
E
2
2
2


 2
2
2
8 L
m
h
n
o
 (vii)
1. Energy of the particle inside the box can’t be equal to zero.
The minimum energy of the particle is obtained for n = 1
2
2
1
8 L
m
h
E
o
 (Zero Point Energy)
If momentum i.e.
0
1 
E 0
 0

p



 x
But since the particle is confined in the box of
dimension L.
L
x 
 max

51. Thus zero value of zero point energy violates the
Heisenberg’s uncertainty principle and hence zero value is
not acceptable.
2. All the energy values are not possible for a particle in
potential well.
Energy is Quantized
3. En are the eigen values and ‘n’ is the quantum number.
4. Energy levels (En) are not equally spaced.
n = 1
n = 3
n = 2
3
E
1
E
2
E

52. Using Normalization condition
L
x
n
A
x
n

 sin
)
( 
1
sin
0
2
2

 dx
L
x
n
A
L

1
|
)
(
| 2





dx
x
n

1
2
2






 L
A
L
A
2


The normalized eigen function of the particle are
L
nx
L
x
n

 sin
2
)
( 

53. Probability density figure suggest that:
1. There are some positions (nodes) in the box that will never be
occupied by the particle.
2. For different energy levels the points of maximum probability
are found at different positions in the box.
|ψ1|2 is maximum at L/2 (middle of the box)
|ψ2|2 is zero L/2.

54. Eigen function
z
y
x 


 
L
z
n
L
y
n
L
x
n
A
A
A z
y
x
z
y
x



 sin
sin
sin

L
z
n
L
y
n
L
x
n
L
z
y
x 


 sin
sin
sin
2
3









2
2
2
2
2
8
)
(
mL
h
n
n
n
E z
y
x 


Particle in a Three Dimensional Box
z
y
x E
E
E
E 


Eigen energy