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Chapter 6: Q7CQ (page 223)

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?

Short Answer

Expert verified

The answer is electron.

Step by step solution

01

Definition of quantum tunneling

The quantum phenomenon, in which a particle can penetrate a barrier and pass through it even though it is forbidden classically, is known as quantum tunneling.

02

Explanation and conclusion

An electron would tunnel more readily because it has less mass as compared to the proton, provided all other things are kept the same.

Less mass would imply less momentum and quicker exponential decay and which in turn would result in a larger wavelength. The larger the wavelength, the more classical the behavior of the particle.

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Most popular questions from this chapter

As we learned in example 4.2, in a Gaussian function of the form蠄(x)伪e-(x2/2蔚2)is the standard deviation or uncertainty in position.The probability density for gaussian wave function would be proportional to蠄(x)squared:e-(x2/2蔚2). Comparing with the timedependentGaussian probability of equation (6-35), we see that the uncertainty in position of the time-evolving Gaussian wave function of a free particle is given by

.螖x=蔚1+h2t24m2蔚4 That is, it starts atand increases with time. Suppose the wave function of an electron is initially determined to be a Gaussian ofuncertainty. How long will it take for the uncertainty in the electron's position to reach5鈥�m, the length of a typical automobile?

Show that 蠄(x)=A'eikx+B'e-ikxis equivalent to 蠄(x)=Asinkx+Bcoskx, provided that A'=12(B-iA)B'=12(B+iA).

Question: Obtain equation (6.18) from(6.16) and (6.17).

Verify that the reflection and transmission probabilities given in equation (6-7) add to 1.

Given the situation of exercise 25, show that

(a) as Uo鈫�鈭�, reflection probability approaches 1 and

(b) as L鈫�0, the reflection probability approaches 0.

(c) Consider the limit in which the well becomes infinitely deep and infinitesimally narrow--- that is Uo鈫�鈭�and data-custom-editor="chemistry" L鈫�0but the product U0L is constant. (This delta well model approximates the effect of a narrow but strong attractive potential, such as that experienced by a free electron encountering a positive ion.) Show that reflection probability becomes:

R=[1+2h2EmUoL2]-1
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