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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 “between the walls” 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 “walls” 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+12hω. 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

What is the product ofΔxandΔp(obtained in Exercise 83 and 85)? How does it compare with the minimum theoretically possible? Explain.

Determine the particle’s most probable position.

In a study of heat transfer, we find that for a solid rod, there is a relationship between the second derivative of the temperature with respect to position along the rod and the first with respect to time. (A linear temperature change with position would imply as much heat flowing into a region as out. so the temperature there would not change with time).

∂2T(x,τ)∂x2=β∂T(x,τ)∂τδx

(a) Separate variables this assume a solution that is a product of a function of xand a function of tplug it in then divide by it, obtain two ordinary differential equations.

(b) consider a fairly simple, if somewhat unrealistic case suppose the temperature is 0 atx=0and, and x=1 positive in between, write down the simplest function of xthat (1) fits these conditions and (2) obey the differential equation involving x.Does your choice determine the value, including sign of some constant ?

(c) Obtain the fullT(x,t)for this case.

Whereas an infinite well has an infinite number of bound states, a finite well does not. By relating the well heightU0 to the kinetic energy and the kinetic energy (through λ) to n and L. Show that the number of bound states is given roughly by8ml2U0/h2 (Assume that the number is large.)

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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