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Chapter 6: Q44E (page 228)

For a general wave pulse neither E nor p (i.eneither 蝇nor 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=蝇0k0=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.

Short Answer

Expert verified

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

Step by step solution

01

Concept

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

Vphase=E0p0 鈥︹�� (1)

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

02

Determine phase velocity

Write the relativistic expression for energy.

E0=纬0mc2

Write the relativistic expression for momentum.

p0=纬0muparticle

Here 纬M, is Lorentz factor, c is light speed, u is particle speed.

Substitute these values in the equation

Vphase=E0p0=纬nmc2纬Mmuparticle=c2uparticle

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

Vphase=c2uparticle

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Most popular questions from this chapter

The equations for Rand T in the E>U0barrier essentially the same as light through a transparent film. It is possible to fabricate a thin film that reflects no light. Is it possible to fabricate one that transmits no light? Why? Why not?

Given the same particle energy and barrier height and width, which would tunnel more readily: a proton or an electron? Is this consistent with the usual rule of thumb governing whether classical or non-classical behavior should prevail?

Exercise 39 gives the condition for resonant tunneling through two barriers separated by a space of width 2 s, expressed in terms of a factor 尾given in Exercise 30. (a) Suppose that in some system of units, k and 伪 are both2蟺. Find two values of 2s that give resonant tunneling. What are these distances in terms of wavelengths of蠄? Is the term resonant tunneling appropriate?(b) Show that the condition has no solution if s = 0 and explain why this must be so. (c) If a classical particle wants to surmount a barrier without gaining energy, is adding a second barrier a good solution?

Exercise 54 gives a rough lifetime for a particle trapped particle to escape an enclosure by tunneling.

(a) Consider an electron. Given thatW=100鈥塶尘,鈥�L=1鈥塶尘鈥夆�塧nd鈥夆��U0=5鈥塭痴, first verify that theEGS<<U0assumption holds, then evaluate the lifetime.

(b) Repeat part (a), but for a0.1碌gparticle, withW=1nm,鈥�L=1碌m, and a barrier heightU0that equals the energy the particle would have if its speed were just1鈥尘尘辫别谤测别补谤.

In the wide-barrier transmission probability of equation (6-18), the coefficient multiplying the exponential is often omitted. When is this justified, and why?
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