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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

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鈭�12), wheren=1,2,3.....(using ann鈭�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=鈭�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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If a particle in a stationary state is bound, the expectation value of its momentum must be 0.

(a). In words, why?

(b) Prove it.

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Using equation (23), find the energy of a particle confined to a finite well whose walls are half the height of the ground-state infinite well energy, . (A calculator or computer able to solve equations numerically may be used, but this happens to be a case where an exact answer can be deduced without too much trouble.)
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