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Chapter 6: Q1CQ (page 223)

Could the situation depicted in the following diagram represent a particle in a bound state? Explain.

Short Answer

Expert verified

The answer is no.

Step by step solution

01

Definition of bound state

The bound state of a particle is defined as the quantum state in which a particle has a tendency to remain localized in space when subjected to a potential barrier.

02

Explanation and conclusion

The particle within any barrier would hit walls back and forth. But in case of this potential, the particle will keep on bouncing back and forth and eventually get out of the barrier.

Hence, this diagram does not depict a bound state.

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Most popular questions from this chapter

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

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

For the E>U0 potential barrier, the reflection, and transmission probabilities are the ratios:

R=B*BA*AT=F*FA*A

Where A, B, and F are multiplicative coefficients of the incident, reflected, and transmitted waves. From the four smoothness conditions, solve for B and F in terms of A, insert them in R and T ratios, and thus derive equations (6-12).

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?

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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