Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Q11CQ (page 223)

The diagram below plots ω(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.

Short Answer

Expert verified

The values of phase and group velocities depend on the central value of k under consideration.

Step by step solution

01

Definition of group and phase velocities

Group velocity is defined as the velocity of the whole envelope of a wave in space.

Phase velocity is defined as the velocity of a phase or part of a wave in space.

02

Explanation and conclusion

For small (central) values of k, the slope of the curve, or tangent line, dω/dk is tany, and therefore the slope of a line from the origin, ω/k is relatively large. That is, the group velocity is smaller than the phase velocity. When the central value of k is large, these two quantities are equal, and so are the phase and group velocities.

Hence, the values of phase and group velocities depend on the central value of k under consideration.

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.

Start your free trial

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

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider a particle of mass m inside the well as shown in the figure. If bound, its lowest energy state would of course be the ground state, but would it be bound? Assume that for a while, it at least occupies the ground state, which is much lower thanU0, and the barriers qualify as wide. Show that a rough average time it would remain bound is given by:τ=mW42000hL2σ2eα whereσ=L8mU0h.

For a general wave pulse neither E nor p (i.eneither ωnor k)areweIldefined. But they have approximate values E0andp0. 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 uncertaintiesarereasonably small, and the particle may be said to have energy E0and momentum p0.Usingtherelativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater thancand given by

vphase=c2uparticle

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

For particles incident from the left on the potential energy shown below, what incident energies E would imply a possibility of later being found infinitely far to the right? Does your answer depend on whether the particles behave classically or quantum-mechanically?

A beam of particles of energy incident upon a potential step ofU0=(3/4),is described by wave function:ψinc(x)=eikx

The amplitude of the wave (related to the number of the incident per unit distance) is arbitrarily chosen as 1.

  1. Determine the reflected wave and wave inside the step by enforcing the required continuity conditions to obtain their (possibly complex) amplitudes.
  2. Verify the explicit calculation of the ratio of reflected probability density to the incident probability density agrees with equation (6-7).

In the E>Uopotential barrier, there should be no reflection when the incident wave is at one of the transmission resonances. Prove this by assuming that a beam of particles is incident at the first transmission resonance, E=Uo+(π2h2/2mL2), and combining continuity equations to show thatB=0. (Note: k’ is particularly simple in this special case, which should streamline your work.)
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Kinematics Physics

Read Explanation

Solid State Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Modern Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free