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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

A beam of particles of energy E incident upon a potential step ofU0=(5/4)E is described by wave function:蠄inc(x)=eikx

  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 the ratio of reflected probability density to the incident probability density is 1.

The diagram below plots 蠅(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.

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?

Solving the potential barrier smoothness conditions for relationships among the coefficients A,B鈥塧苍诲鈥�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鈥� 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鈥檛 difficult 鈥� just different.

(b) Obtain the smoothness conditions, and from these findR鈥塧苍诲鈥�T.

(c) Do the results make sense in the limitL鈫�鈭�?

For particles incident from the left on the potential energy shown below, what incident energies E would imply a possibility of later being found infinitely far to the right? Does your answer depend on whether the particles behave classically or quantum-mechanically?
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