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Chapter 5: Q16CQ (page 186)

The quantized energy levels in the infinite well get further apart as n increases, but in the harmonic oscillator they are equally spaced.

  1. Explain the difference by considering the distance 鈥渂etween the walls鈥� in each case and how it depends on the particles energy
  2. A very important bound system, the hydrogen atom, has energy levels that actually get closer together as n increases. How do you think the separation between the potential energy 鈥渨alls鈥� in this system varies relative to the other two? Explain.

Short Answer

Expert verified

As energy increases, the energy levels move closer and hence wavelength becomes smaller.

Step by step solution

01

 Energy of harmonic oscillator

  1. The energy spacing is equal to n+12h蝇. The ground state energy is larger than zero in a harmonic oscillator as n increases, the walls become further apart. But the energy of the oscillator is limited to certain values and hence allowed quantized energy levels are equally spaced.
    Higher energy states have higher total energies and hence the classical limits to amplitude of the displacement will be larger for these states. Thus, they have shorter wavelengths.
02

Explanation

(b) Since energy varies as n increases, the walls move apart and so the energy levels come closer to each other.Thus, the energy gap will be smaller and energy increases faster than the harmonic oscillator

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Most popular questions from this chapter

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.)

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.

Consider the wave function that is a combination of two different infinite well stationary states the nth and the mth

蠄x,t=12蠄nxe-iEn/鈩�t+12蠄me-iEm/鈩�t

  1. Show that the 蠄x,tis properly normalized.
  2. Show that the expectation value of the energy is the average of the two energies:E炉=12En+Em
  3. Show that the expectation value of the square of the energy is given by .
  4. Determine the uncertainty in the energy.

A classical particle confined to the positive x-axis experiences a force whose potential energy is-

U(x)=1x2-2x+1

a) By finding its minimum value and determining its behaviors at x=0and role="math" localid="1660119698069" x=鈭�, sketch this potential energy.

b) Suppose the particle has energy of 0.5J. Find any turning points. Would the particle be bound?

c) Suppose the particle has the energy of 2.0J. Find any turning points. Would the particle be bound?

Explain to your friend, who is skeptical about energy quantization, the simple evidence provided by distinct colors you see when you hold a CD (serving as grating) near a fluorescent light. It may be helpful to contrast this evidence with the spectrum produced by an incandescent light, which relies on heating to produce a rather nonspecific blackbody spectrum.
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