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Chapter 5: Q 9CQ (page 186)

A half-infinite well has an infinitely high wall at the origin and one of finite height U0 at x= L . Like the finite well, the number of allowed states is limited. Assume that it has two states, of energy E1 and E2 , where E2 is not much below U0. Make a sketch of the potential energy, then add plausible sketches of the two allowed wave functions on separate horizontal axes whose heights are E1 and E2 .

Short Answer

Expert verified

E1 and E2 are two states of energy in a half-infinite square well. At a distance of L from the left wall, the right "step" has a height of U0 .

Step by step solution

01

Given data

Halfway between an infinite square well (potentials on both sides are infinite) and the finite square well lies the half-infinite square well.

02

Graph for the data

A half-infinite square well with two states of energies and E1 and E2 . Here the right "step" has a height of U0 at a distance L from the left wall. First, we will draw the potential. This is pretty straightforward. "An infinite barrier exists at the origin, followed by a finite barrier at." x= L.

Hence, E1 and E2 are two states of energy in a half-infinite square well. At a distance of L from the left wall, the right "step" has a height of U0 .

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Most popular questions from this chapter

A bound particle of massdescribed by the wave function

ψ(x)=Axe-x2/2b2

What is the most probable location at which to find the particle?

An electron istrapped in a finite well. How "far" (in eV) is it from being free if the penetration length of its wave function into the classically forbidden region 1nm?

A student of classical physics says, "A charged particle. like an electron orbiting in a simple atom. shouldn't have only certain stable energies: in fact, it should lose energy by electromagnetic radiation until the atom collapses." Answer these two complaints qualitatively. appealing to as few fundamental claims of quantum mechanics as possible.

To determine the classical expectation value of the position of a particle in a box is L2 , the expectation value of the square of the position of a particle in a box isrole="math" localid="1658324625272" L23 , and the uncertainty in the position of a particle in a box isL12 .

In several bound systems, the quantum-mechanically allowed energies depend on a single quantum number we found in section 5.5 that the energy levels in an infinite well are given by, En=a1n2wheren=1,2,3.....andis a constant. (Actually, we known whata1is but it would only distract us here.) section 5.7 showed that for a harmonic oscillator, they areEn=a2(n−12), wheren=1,2,3.....(using ann−12with n strictly positive is equivalent towith n non negative.) finally, for a hydrogen atom, a bound system that we study in chapter 7,En=−a3n2, wheren=1,2,3.....consider particles making downwards transition between the quantized energy levels, each transition producing a photon, for each of these three systems, is there a minimum photon wavelength? A maximum ? it might be helpful to make sketches of the relative heights of the energy levels in each case.
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