Quantum Mechanics:Wave packet, phase velocity and group

7/28/2019 Quantum Mechanics:Wave packet, phase velocity and group

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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


19/54t = 1

vp>vg Normal Dispersion


20/54t = 2



21/54t = 3



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t = 4



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t = 5



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t = 6

Normal Dispersionvp>vg


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t = 7



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t = 8



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t = 9



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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



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vp < vg

t = 2



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vp< vg

t = 3



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vp < vg

t = 4



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vp < vg

t = 5



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vp < vg

t = 6



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vp < vg

t = 7



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vp< vg

t = 8



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vp < vg

t = 9



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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


v vdk

d


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vp= vg

t = 2


v vdk

d


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vp= vg

t = 3


v vdk

d


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vp= vg

t = 4


v vdk

d


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vp= vg

t = 5


v vdk

d


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vp= vg

t = 6


v vdk

d


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vp= vg

t = 7


v vdk

d


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vp= vg

t = 8


v vdk

dv


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vp= vg

t = 9


v vdk

dv


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vp= vg

t = 10


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

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Quantum Mechanics:Wave packet, phase velocity and group