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Q44E

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Found in: Page 228

### Modern Physics

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

# For a general wave pulse neither E nor p (i.eneither ${\mathbit{\omega }}$nor k) are weIldefined. But they have approximate values ${{\mathbf{E}}}_{{\mathbf{0}}}{\mathbf{}}{\mathbf{and}}{\mathbf{}}{{\mathbf{p}}}_{{\mathbf{0}}}$. Although it comprisemany plane waves, the general pulse has an overall phase velocity corresponding to these values.${{\mathbit{v}}}_{\mathbf{p}\mathbf{h}\mathbf{a}\mathbf{s}\mathbf{e}}{\mathbf{=}}\frac{{\mathbf{\omega }}_{\mathbf{0}}}{{\mathbf{k}}_{\mathbf{0}}}{\mathbf{=}}\frac{{\mathbf{E}}_{\mathbf{0}}\mathbf{/}\mathbf{h}}{{\mathbf{p}}_{\mathbf{0}}\mathbf{/}\mathbf{h}}{\mathbf{=}}\frac{{\mathbf{E}}_{\mathbf{0}}}{{\mathbf{p}}_{\mathbf{0}}}$If the pulse describes a large massive particle. The uncertainties are reasonably small, and the particle may be said to have energy ${{\mathbit{E}}}_{{\mathbf{0}}}$and momentum ${{\mathbit{p}}}_{{\mathbf{0}}}$.Using the relativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater than c and given by${{\mathbit{v}}}_{\mathbf{p}\mathbf{h}\mathbf{a}\mathbf{s}\mathbf{e}}{\mathbf{=}}\frac{{\mathbf{c}}^{\mathbf{2}}}{{\mathbf{u}}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{i}\mathbf{c}\mathbf{l}\mathbf{e}}}$Note that the phase velocity is greatest for particles whose speed is least.

The phase velocity of the particle is obtained as ${v}_{phase}=\frac{{c}^{2}}{{u}_{particle}}$.

See the step by step solution

## Step 1: Concept

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

${V}_{phase}=\frac{{E}_{0}}{{p}_{0}}$ …… (1)

Here ${E}_{0},{p}_{0}$, 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.

${E}_{0}={\gamma }_{0}m{c}^{2}$

Write the relativistic expression for momentum.

${p}_{0}={\gamma }_{0}m{u}_{particle}$

Here ${\gamma }_{M}$, is Lorentz factor, c is light speed, u is particle speed.

Substitute these values in the equation

${V}_{phase}=\frac{{E}_{0}}{{p}_{0}}\phantom{\rule{0ex}{0ex}}=\frac{{\gamma }_{n}m{c}^{2}}{{\gamma }_{M}m{u}_{particle}}\phantom{\rule{0ex}{0ex}}=\frac{{c}^{2}}{{u}_{particle}}$

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

${V}_{phase}=\frac{{c}^{2}}{{u}_{particle}}$