quantum mechanics:wave packet, phase velocity and group
TRANSCRIPT
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of the group (envelope) is non-zero only in the neighbourhoodof the particle
A wave packet is localized a good representation for aparticle!
The spread of wavepacket in wavelength depends on the requireddegree of localization in space the central wavelength is given by
What is the velocity of the wave packet?
If several waves of different wavelengths (frequencies) and phasesare superposed together, one would get a resultant which is a
localized wave packet A wave packet is a group of waves
withslightly different wavelengths interferingwith one another in a way that the amplitude
p
h
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The velocities of the individual waves which superpose toproduce the wave packet representing the particle are
different - the wave packet as a whole has a differentvelocity from the waves that comprise it
Phase velocity: The rate at which the phase of the wave
propagates in space
Group velocity: The rate at which the envelope of the wavepacket propagates
Here c is the velocity of light and v is the velocity of the particle .
v
cvp
2
vvg
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Suppose velocity of the De-Broglie wave associated with the moving particlebe vp then,
Since the velocity of the particle is always less than c , therefore vp should
always be greater than cwhich shows that De-Broglie wave associated withparticle would leave the particle far behind. This is against the wave concept
of the particle.
The above difficulty was overcome by considering that the moving particle is
associated withAWAVEPACKETrather than a single wave train.
The velocity with which this wave packet moves forward in the medium is
called GROUP VELOCITY. The average velocity of the advancement of
individual monochromatic wave in the medium with which a wave packet is
constructed is called WAVE VELOCITY OR PHASE VELOCITY.
p
2
p
v
cv
v
2mc: frequency
hh h
: wavelength =p mv
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1
2
1 2
Group velocity
y Acos(kx t)
y Acos[(k dk)x ( d )t]
The resultant displacement 'y' at any time t and at any position x isy=y y
y=Acos(kx t)
Acos[(k dk)x ( d )t]
(2k dk)x (2 d )t dkx d ty=2A[cos cos( )]
2 2 2Since d and dk are infinitesimally small quantities therefore,
2 d 2
2k+dk 2k
dky=2A[cos(kx t)cos(
x d t )]2 2
Hence second term is the modified amplitude of the wavepacket which is modulatedin the space and time by a very slowly varying envelop of frequency d/2 and thepropagation constant dk/2 and has maximum value 2A. The effect of the modulation
is to produce successive groups
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(x,t) =kx-t
=0, x=0, t=0New position of =0 at t
k x - t = 0
1y Acos(kx t)
x tk
Phase velocity =The velocity with which the constant phasemoves
p
xv
t k
dk dy=2A[cos(kx t)cos( x t)]
2 2
Group velocity=The velocity with which the wave packet moves
g
dv
k dk
as k 0
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Superposition of Two Waves: Formulas
Add two waves of equal amplitude and nearly equal .
1 1 2 2, cos cos
, c2
so cos2
2
o o
o
y x t y k x
k
t y k x t
y x y x ttkx t
Wave #1 Wave #2
Wave Envelope
(2) waves
1 2 1 2
2 1 2 1
where
and
,2 2
,
k kk
k k k
Wave Envelope
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Imp2
g p
cShow that v v and v
v
Now, the angular frequency and the propagation constant of De-Broglie
waves associated with a body of mass mo and moving with velocity v are
2
2
2
2
o
2
2
2
E h ; E=mc
mch
2 mc
h
2 m c= (1)
vh 1
c
o
2
2
2k
h h=p mv
2 mvk=
h
2 m vk= (2)v
h 1c
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g
2
o
2
2
o
2 32
2
dd dvv (3)
dkdkdv
From eq (1)
2 m cd d
dv dv vh 1
c
2 m v
= (4)vh(1 )
c
o
2
2
o
23
22
and from eq(2)
2 m vdk d
dv dvvh 1c
2 m= (5)
vh(1 )
c
g
d dkSubstituting and in eq (3)
dv dv
v v
Thus de-Broglie wave group associated with a movingbody travels with the same velocity as the body. Thewave velocity vp of the de-Broglie waves evidently has nosimple physical significance. Hence a moving particle is
equivalent to a wave packet or a group of waves.
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Relation between Phase and Group Velocities:Wave Packet
For multiple waves, must define two velocities: Phase velocity vp
:
Group velocity vg :
and
only in non-dispersive media, i.e.
0
p g
pg p p
p
g p
dv v
k dk
dvdv kv v k dk dk
dvv v
dk
d
dv
vvp
pg
vg< vp Normal Dispersion
vg> vp Anomalous Dispersion
Show that
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, 2 cos2 2
cos
oy x t ykx
kx tt
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vp>vg
t = 0
Normal Dispersion
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vp>vg Normal Dispersion
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vp>vg Normal Dispersion
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vp>vg Normal Dispersion
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t = 4
vp>vg Normal Dispersion
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t = 5
vp>vg Normal Dispersion
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t = 6
Normal Dispersionvp>vg
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t = 7
vp>vg Normal Dispersion
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t = 8
vp>vg Normal Dispersion
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t = 9
vp>vg Normal Dispersion
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t = 10
Normal Dispersionvp>vg
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vp < vg
t = 0
Anomalous Dispersion
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vp < vg
t = 1
Anomalous Dispersion
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vp < vg
t = 2
Anomalous Dispersion
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vp< vg
t = 3
Anomalous Dispersion
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vp < vg
t = 4
Anomalous Dispersion
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vp < vg
t = 5
Anomalous Dispersion
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vp < vg
t = 6
Anomalous Dispersion
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vp < vg
t = 7
Anomalous Dispersion
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vp< vg
t = 8
Anomalous Dispersion
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vp < vg
t = 9
Anomalous Dispersion
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vp
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vp= vg
t = 0
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 1
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 2
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 3
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 4
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 5
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 6
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 7
only in non-dispersive media, i.e. 0 pg pdv
v vdk
d
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vp= vg
t = 8
only in non-dispersive media, i.e. 0 pg pdv
v vdk
dv
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vp= vg
t = 9
only in non-dispersive media, i.e. 0 pg pdv
v vdk
dv
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vp= vg
t = 10
only in non-dispersive media, i.e. 0 pg pdv
v vdk
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scheme
ff d d
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Differences in speed cause spreading or
dispersion of wave packets
The group velocity is the speed of the wavepacket
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The group velocity is the speed of the wavepacket
The phase velocity is the speed of the individual waves
Phase velocity = Group Velocity
The entire waveformthecomponent waves andtheir
envelopemoves as one. non-
dispersive wave.
Phase velocity = -Group Velocity
The envelope moves in the opposite
direction of the component waves.
Phase velocity > Group VelocityThe component waves move more
quickly than the envelope.
Phase velocity < Group Velocity
The component waves move more
slowly than the envelope.
Group Velocity = 0
The envelope is stationary while the
component waves move through it.
Phase velocity = 0
Now only the envelope moves over
stationary component waves.
Phase and Group Velocities: Dispersion
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Phase and Group Velocities: Dispersion
Dispersion occurs when the phase
velocity vp depends on k(or),
i.e. group velocity does not equal thephase velocity.
pg
pvv
dk
dv 0 or
= phase velocity vp
Diagram shows a wave packet with a
group velocity less than the phase
velocity, i.e. vg< vp.
= group velocity vg
For detailed explanation of fig See Modern Physics