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Phase and Group Velocities of the de Broglie Wave

The group of wave need not have the same velocity as the individual one.

 

The amplitude of the de Broglie wave is associated with the moving body represents the probability of finding a body at a particular time and space. The wave equation y = A cos (rot- kx) does not represent the de Broglie wave. The de Broglie wave is represented by the  Combination of several such waves. Thus, the de Broglie waves can be obtained by the superposition of the several waves. Hence the mathematical expression of the de Broglie waves is obtained by the superposition of number of waves or group of waves or wave group of  The diagramatic representation of the de Broglie wave may be shown in the form of wave packet or wave group.

Let us find the velocity of the wave packet or de Broglie wave. We assume that the wave packet (or group) is formed due to combination of the two waves having equal amplitude and different their angular frequency by dw and wave velocity dk. The waves are represented mathematically as

 

y1= A cos (wt – kx)

y2 =A cos [ (w +dw ) t – (k +dk) x]

 

where k = 2π /λ  , w 2 π v.

 

According to the superposition principle, the resultant displacement Y at any time t at any point x is the sum of two displacements

Y =y1 +y2

= A  [cos  (wt –kx ) +cos { { w +dw )t – (k +dk ) x}]

= 2 cos [ (2w + dw )t – (2k +dk ) x ] / 2 , cos [dw t –dk x ] / 2

 

dw and dk are very small as compared to w and k

2w + dw = 2w and 2k + dk = 2k

cos

Hence Y = 2A cos (wt-kx) cos  ( dw t / 2 – dk x /2)

 

This equation represents a wave of angular frequency w and wave number k moving in the same direction superimposed by a modulated wave of angular frequency dw and wave vector dk. Thus, the superposition of the two waves results a new waves and successively they form a new wave.

 

Equation (1) describes the de Broglie wave or wave packet or wave group.

Form Equation (1), two types of velocities are defined as :

The phase velocity of the de Broglie waves is defined by

Vp = w / k

and the group velocity of the de Broglie wave is defined as

v g = dw /dk

The group velocity of the de Broglie wave depends upon the manner in which the phase velocity of the medium varies or constant.

The phase velocity is defined by

Vp =w/k =2π v / 2π /λ = v

Vp =vλ = v

where v = velocity of wave.

 

This shows that the phase velocity of the de Broglie wave is same as the wave-velocity.  in other words, the phase velocity is also called wave velocity.

 

Let us consider a de Broglie wave associated with a moving particle of a rest mass m 0 and velocity v. The angular frequency of the de Broglie wave is given by

 

W =2 πv

W =2π E/ h

Similarly the wave vector k is given by

 

K= 2 π / λ

 

Using de Broglie relation  λ =h /p

K = 2 π p /h

K = 2 π mv / h

K = 2 π v /h

The phase velocity of the de Broglie wave

Vp=w/k

 

Substituting the value of w and k from equations (5) and (6), we get

V p=2 π c2 /h , m o / (1- v2 /c2 )1/2  . h ( 1-v2 /c2 ) 1/2 / 2 π vm   o

 

V p=c2 /v =>  v p =(c/v )c

 

C /v  >> 1 , for material particle

V p > c

 

This shows the phase velocity of the de Broglie wave is greater than the velocity of light.

The group velocity of the de Broglie wave is defined as

Vg = dw /dk

 

9.4.1 Relation between the Phase Velocity and Group Velocity of the de  broglie Wave

 

The wave velocity is given by v = vλ

= 2πv λ/ 2π

v=w/k

v=w/k = v p

 

where w = 2πv = angular frequency and 2π / λ wave number.

 

This shows that the phase velocity of the de Broglie wave is equal to the wave velocity

and hence phase velocity is also known as wave velocity

 

Phase velocity is defined by

 

V p = w/k  => w =kv p

 

The group velocity of a de Broglie wave is defined by

 

Vg=dw / dk

 

Putting the value of w = kv P in equation (2), we get

Vg = d / dk (kv p)

Vg=v p +k dv p /dk

Putting the value of k

Now                              d (1 / λ) = – 1 /λ2

 

Vg =vp – λ dv p /d λ

 

This is a relation between the phase velocity and group velocity for a dispersive medium.

 

For normal dispersion the quantity dv p /dλ  is a positive quantity. Therefore for normal  dispersion, group velocity is less than the phase velocity. For anomalous dispersion, the quantity Dv P / dλ is a negative quantity, the group velocity is greater than the phase velocity.

 

For non-dispersive medium vP =w /k =constant .. dv p d =0 . Hence vg = v p  · Hence for

nondispersive medium the group velocity is equal to the phase velocity. For electromagnetic waves in vacuum, the speed of light (c) is constant. Therefore group velocity vg and the phase velocity v, of the light radiations are same.