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Q44E

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Modern Physics
Found in: Page 228
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Short Answer

For a general wave pulse neither E nor p (i.eneither ωnor k) are weIldefined. But they have approximate values E0 and p0. Although it comprisemany plane waves, the general pulse has an overall phase velocity corresponding to these values.

vphase=ω0k0=E0/hp0/h=E0p0

If the pulse describes a large massive particle. The uncertainties are reasonably small, and the particle may be said to have energy E0and momentum p0.Using the relativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater than c and given by

vphase=c2uparticle

Note that the phase velocity is greatest for particles whose speed is least.

The phase velocity of the particle is obtained as vphase=c2uparticle.

See the step by step solution

Step by Step Solution

Step 1: Concept

Write the expression for general wave pulse has overall phase velocity.

Vphase=E0p0 …… (1)

Here E0,p0, are the energy and momentum of the particle if pulse describes as large massive particle.

Step 2: Determine phase velocity

Write the relativistic expression for energy.

E0=γ0mc2

Write the relativistic expression for momentum.

p0=γ0muparticle

Here γM, is Lorentz factor, c is light speed, u is particle speed.

Substitute these values in the equation

Vphase=E0p0 =γnmc2γMmuparticle =c2uparticle

Thus, using relativistic expression, phase velocity of the particle is obtained as .

Vphase=c2uparticle

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