Wave Packets and
Phase and Group Velocity
Prof. B.N. Jagatap
Department of Physics
IIT Bombay
Recapitulate
Everything (matter and radiation) has both wave
and particle properties.
de Broglie Wavelength
For a photon, momentum p h / c h /
So for a particle of momentum p, the wavelength is
dB h / p h / mv h / m0 v
dB = de Broglie wavelength
Interference is a result of wave property
When the object reaches the screen, it is detected
as a particle.
Wave Particle Duality
Wave Particle Duality
Particle: Localized, Definite position,
momentum, confined in space
Wave: Delocalized, spread out in space
and time
How to find a description of a particle which
Fits the wave description Wave
And localized in space Packet
Wave Packet
If several waves of different wavelengths and phases are
superimposed together, what we get is a localized wave packet.
Example: Beat formation in superposition of two sinusoidal waves
Spatial beats by superposition of sinusoidal waves of
nearby wavelengths
2
A sin x A sin kx
[sin(5x) + sin(6x)]/2 [sin(5x) + sin(5.5x) + sin(6x)]/3
[sin(5x) + sin (5.25x) + sin(5.5x) + sin (5.75x) + sin(6x)]/5
[sin(5x) + sin (5.125x) + sin (5.25x) + sin(5.375x) + sin(5.5x) + sin (5.625x)
+ sin (5.75x) + sin(5.875x) + sin(6x)]/9
[sin(5x) + sin(5.0625x) + sin (5.125x) + sin(5.1875x) + sin (5.25x) +
sin(5.3125x) + sin(5.375x) + sin(5.4375x) + sin(5.5x) + sin(5.5625x) + sin
(5.625x) + sin(5.6875x) + sin (5.75x) + sin(5.8125x) + sin(5.875x) +
sin(5.9375x) + sin(6x)]/17
Particle
Wave packet
A wave packet is a group of waves with slightly
different wavelengths interfering with one another in
a way that the amplitude of the group (envelope) is
non-zero in the neighbourhood of the particle.
A wave packet is localized; it is a good representation
of a particle
Phase Velocity and Group Velocity
Consider an ideal wave A sin(kx t )
k 2 / 2
A k measured in
0 wavenumber
x
A
Take a point at t = 0 for which = 0. Let time increase to t. What
would be x to maintain = 0.
x Phase
kx t 0 vp
t k Velocity
Phase velocity is the velocity of a point of constant phase on the wave.
Now consider superposition of two waves
1 A sin( kx t )
1 A sin[(k k ) x ( )t ]
1 2 A sin(kx t ) A sin[(k k ) x ( )t ]
(2k k ) x (2 )t kx t
2 A sin cos
2 2 2 2
Since k and are infinitesimally small quantities
2 k k 2 k , 2 2
kx t
2 A sin(kx t ) cos
2 2
kx t
2 A sin(kx t ) cos
2 2
Slowly varying envelope of frequency
kx t and propagation constant k
cos
2 2
Group velocity is the velocity with
which the envelope of the wave
packet moves.
d
vg as k
k dk
sin( kx t )
vg is the velocity with which the wave packet moves.
Particle v
v g d / dk
Wave packet vp / k
Phase velocity
Relation between p and k
2
vp p h / 2h/2 k
k 2 /
Wavelength
h / p h / mv
Frequency
m0 c 2
p 2 c 2 m02 c 4
E / h mc 2 / h
h 1 v2 / c2 h
Particle v
vp / k v g d / dk
Wave packet
Suppose the velocity of the de Broglie wave associated with the
moving particle is vp
2 h / p h / mv
vp
k 2 /
E / h mc / h
2
2
c
vp vp c since vc
v
The de Broglie wave associated with the particle would leave the
particle behind. This is against the wave concept of the particle.
v vp
Is v = vg?
E h mc 2
mc 2 / h h / p h / mv
2 2mc 2 / h k 2 / 2mv / h
2m0 c 2
2m0 v
k
h 1 v2 / c2 h 1 v2 / c2
d d / dv
vg
dk dk / dv
2m0 v 2m0
d / dv dk / dv
h(1 v 2 / c 2 ) 3 / 2 h(1 v 2 / c 2 ) 3 / 2
vg v
de Broglie wave group associated with a moving body
travels with the same velocity as the body!
de Broglie wave group
associated with a moving
body travels with the same
velocity as the body!
In general, many waves having a continuous distribution of
wavelengths must be added to form a packet that is finite
over a limited range and really zero everywhere else. In this
case,
d d
vg vg
dk dk k0
where the derivative is to be evaluated at the central k0 .
Relationship between vg and vp
vp
k d dv p
vg (kv p ) v p k
d dk dk k
vg
0
dk
Since k 2 /
dv p dv p dv p
k vg vp
dk d d 0
Dispersion Relations
Relation between and k is known as dispersion relation.
Plot of vs k is called the dispersion curve.
If dv p / dk 0
dv p dv p
vg v p k v p
dk k0 d 0
vg v p
Since v p / k , dv p / dk 0 kv g kv
Non-dispersive medium:
dv p / dk 0 vg vp
v p c / nr nr = Refractive index
All component waves have the same speed!
Dispersion Relations
dv p dv p Non-dispersive medium:
vg v p k
dk v p
k0 d
0 dv p / dk 0 vg v p
Dispersive medium: dv p / dk 0 vg vp
Dispersive occurs when phase velocity depends on k (or ): v p c / nr ( )
Normal dispersion Anomalous dispersion
dv p / d 0 dv p / d 0
nr (red ) nr (blue), dnr / d 0,
vg vp
vg vp
Dispersion Relation for de Broglie Wave
Phase velocity v p Group velocity
h/ p p k
vg vp k
dv p
dk k0
p 2 c 2 m02 c 4
E/h
h mc 2
1 / 2
c2
2 v g c 1
p c m c
2 2 2 4
m0 c k 0 vp
vp c 1
0
k0
p k
Since v p c / v
2
dv p / dk 0
vg v
v is the particle velocity
All media are dispersive for de
Broglie wave [This derivation is identical to the
derivation on slide no. 15]
De Broglie Wave: Dispersion relation
Vs k relation
p 2 2k 2
E
2m 2m
2
k 2
h 2
k
2m 2m
E h
