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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

Particles of energy Eare incident from the left, where U(x)=0, and at the origin encounter an abrupt drop in potential energy, whose depth is -3E.

  1. Classically, what would the particles do, and what would happen to their kinetic energy?
  2. Apply quantum mechanics, assuming an incident wave of the formψinc=eikx, where the normalization constant has been given a simple value of 1, determine completely the wave function everywhere, including numeric values for multiplicative constants.
  3. What is the probability that incident particles will be reflected?

A particle moving in a region of zero force encounters a precipice---a sudden drop in the potential energy to an arbitrarily large negative value. What is the probability that it will “go over the edge”?

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

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→∞?

To obtain a rough estimate of the mean time required for uranium-238 to alpha-decay, let us approximate the combined electrostatic and strong nuclear potential energies by rectangular potential barrier half as high as the actual 35 Mev maximum potential energy. Alpha particles (mass 4 u) of 4.3 Mev kinetic energy are incident. Let us also assume that the barrier extends from the radius of nucleus, 7.4 fm to the point where the electrostatic potential drops to 4.3 Mev (i.e., the classically forbidden region). Because Uα(1/r), this point is 35/4.3 times the radius of the nucleus, the point at which U(r) is 35 Mev. (a) Use these crude approximations, the method suggested in Section 6.3, and the wide-barrier approximation to obtain a value for the time it takes to decay. (b) To gain some appreciation of the difficulties in a theoretical prediction, work the exercise “backward” Rather than assuming a value for U0, use the known value of the mean time to decay for uranium-238 and infer the corresponding value of U0, Retain all other assumptions. (c) Comment on the sensitivity of the decay time to the height of the potential barrier.
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