bei48482_FM

and so the group velocity turns out to be

g (3.16)

The de Broglie wave group associated with a moving body travels with the same

velocity as the body.

The phase velocity pof de Broglie waves is, as we found earlier,

p (3.3)

This exceeds both the velocity of the body and the velocity of light c, since c.

However, phas no physical significance because the motion of the wave group, not

the motion of the individual waves that make up the group, corresponds to the mo-

tion of the body, and gcas it should be. The fact that pcfor de Broglie waves

therefore does not violate special relativity.

Example 3.3

An electron has a de Broglie wavelength of 2.00 pm 2.00  10 ^12 m. Find its kinetic energy

and the phase and group velocities of its de Broglie waves.

Solution

(a) The first step is to calculate pcfor the electron, which is

pc6.20  105 eV

620 keV

The rest energy of the electron is E 0 511 keV, so

KE EE 0 E^20 (pc)^2 E 0 (511 keV)^2 (620keV)^2 511 keV

803 keV 511 keV 292 keV

(b) The electron velocity can be found from

E

to be

c
1  c
1  

2

0.771c

Hence the phase and group velocities are respectively

p1.30c

g0.771c

c^2



0.771c

c^2





511 keV

803 keV

E^20

E^2

E 0



 1 ^2 c^2

(4.136  10 ^15 eV s)(3.00  108 m/s)



2.00  10 ^12 m

hc





c^2









k

De Broglie phase

velocity

De Broglie group

velocity

Wave Properties of Particles 103

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