For a general wave pulse neither E nor p (i.eneither $\mathit{\omega }$nor k)areweIldefined. 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 uncertaintiesarereasonably small, and the particle may be said to have energy ${\mathit{E}}_{\mathbf{0}}$and momentum ${\mathit{p}}_{\mathbf{0}}$.Usingtherelativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater thancand 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.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!