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Chapter 6: Q10CQ (page 226)

Question: An electron bound in an atom can be modeled as residing in a finite well. Despite the walls. When many regularly spaced atoms are relatively close together as they are in a solid-all electrons occupy alltheatoms. Make a sketch of a plausible multi-atom potential energy and electron wave function.

Short Answer

Expert verified

Answer:

Finite well, multiple finite and Multi-atom system is described as follows.

Step by step solution

01

concept:

A finite potential well (also known as a finite square well) is a concept from quantum mechanics. It is an extension of an infinite potential well in which the particle is confined to a "box", but one that has "walls" of finite potential

02

 Step 2: Determine Finite well:

The wave exists as a traveling incident and reflected wave inside a finite well.

03

Determine multiple finite:

A particle can tunnel through a potential barrier between two or more finite when they are near to one another.

04

Determine the Multi-atom system:

In a multi-atom system, the potential energy function in the radial direction would be a superposition of the coulomb potential from each other.

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Most popular questions from this chapter

Solving the potential barrier smoothness conditions for relationships among the coefficients A,B and Fgiving the reflection and transmission probabilities, usually involves rather messy algebra. However, there is a special case than can be done fairly easily, through requiring a slight departure from the standard solutions used in the chapter. Suppose the incident particles’ energyEis preciselyU0.

(a) Write down solutions to the Schrodinger Equation in the three regions. Be especially carefull in the region0<x<L. It should have two arbitrary constants and it isn’t difficult – just different.

(b) Obtain the smoothness conditions, and from these findR and T.

(c) Do the results make sense in the limitL→∞?

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

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.

A ball is thrown straight up at 25ms-1. Someone asks “Ignoring air resistance. What is the probability of the ball tunneling to a height of1000m?” Explain why this is not an example of tunneling as discussed in this chapter, even if the ball were replaced with a small fundamental particle. (The fact that the potential energy varies with position is not the whole answer-passing through nonrectangular barriers is still tunnelirl8.)

Reflection and Transmission probabilities can be obtained from equations (6-12). The first step is substituting -iαfork'. (a) Why? (b) Make the substitutions and then use definitions of k and α to obtain equation (6-16).
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