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Equation (3.10) represents a wave of angular frequency and wave number k

that has superimposed upon it a modulation of angular frequency ^12 and of wave

number ^12 k.

The effect of the modulation is to produce successive wave groups, as in Fig. 3.4.

The phase velocity pis

Phase velocity p (3.11)

and the velocity gof the wave groups is

Group velocity g (3.12)

When and khave continuous spreads instead of the two values in the preceding

discussion, the group velocity is instead given by

Group velocity g (3.13)

Depending on how phase velocity varies with wave number in a particular situa-

tion, the group velocity may be less or greater than the phase velocities of its member

waves. If the phase velocity is the same for all wavelengths, as is true for light waves

in empty space, the group and phase velocities are the same.

The angular frequency and wave number of the de Broglie waves associated with a

body of mass mmoving with the velocity are

 2 

 (3.14)

k 

 (3.15)

Both and kare functions of the body’s velocity .

The group velocity gof the de Broglie waves associated with the body is

g

Now 



2 m



h(1^2 c^2 )^3 ^2

dk



d

2 m



h(1^2 c^2 )^3 ^2

d



d

dd



dkd

d



dk

2 m



h 1 ^2 c^2

Wave number of

de Broglie waves

2 m



h

2 





2 mc^2



h 1 ^2 c^2

Angular frequency of

de Broglie waves

2 mc^2



h

d



dk





k





k

Wave Properties of Particles 101

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