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Q44E

Expert-verifiedFound in: Page 228

Book edition
2nd Edition

Author(s)
Randy Harris

Pages
633 pages

ISBN
9780805303087

**For a general wave pulse neither E nor p (i.eneither ${\mathit{\omega}}$****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.**

${{\mathit{v}}}_{\mathbf{p}\mathbf{h}\mathbf{a}\mathbf{s}\mathbf{e}}{\mathbf{=}}\frac{{\mathbf{\omega}}_{\mathbf{0}}}{{\mathbf{k}}_{\mathbf{0}}}{\mathbf{=}}\frac{{\mathbf{E}}_{\mathbf{0}}\mathbf{/}\mathbf{h}}{{\mathbf{p}}_{\mathbf{0}}\mathbf{/}\mathbf{h}}{\mathbf{=}}\frac{{\mathbf{E}}_{\mathbf{0}}}{{\mathbf{p}}_{\mathbf{0}}}$

**If the pulse describes a large massive particle. The uncertainties ****are ****reasonably small, and the particle may be said to have energy ${{\mathit{E}}}_{{\mathbf{0}}}$and momentum ${{\mathit{p}}}_{{\mathbf{0}}}$.Using ****the ****relativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater than ****c ****and given by**

${{\mathit{v}}}_{\mathbf{p}\mathbf{h}\mathbf{a}\mathbf{s}\mathbf{e}}{\mathbf{=}}\frac{{\mathbf{c}}^{\mathbf{2}}}{{\mathbf{u}}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{i}\mathbf{c}\mathbf{l}\mathbf{e}}}$

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

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

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.

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 .

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

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