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Chapter 9: Q11CQ (page 403)

The Fermi energy in a quantum gas depends inversely on the volume, Basing your answer on Simple Chapter 5 type quantum mechanics (not such quaint notions as squeezing classical particles of finite volume into a container too small). Explain why.

Short Answer

Expert verified

The description of the quantum gas as particles contained in an infinite well whose allowable energies correspond to those of a free particle leads in the dependency of the Fermi energy on volume in a quantum gas:

En=π2h22mL2n2

The allowed energies depend on the dimension Lof the 3-D well.

Step by step solution

01

Fermi energy

When referring to a quantum system of non-interacting fermions at absolute zero temperature, the term "Fermi energy" in quantum mechanics often refers to the energy difference between the highest and lowest occupied single-particle states.

02

Inverse Relationship between Energy and Wavelength

The Fermi energy can be considered to be inversely related to volume because the particles can be looked at as waves bounded by the walls of the container.

Because the particles can be viewed as waves bounded by the container's walls, the Fermi energy is inversely proportional to volume. The waves that are bounded inside the container get smaller as the container gets smaller. When the wavelength of something shrinks, its energy rises. As a result of the inverse relationship between energy and wavelength, it can help explain why Fermi energy is inversely related to volume.

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Most popular questions from this chapter

Heat capacity (at constant volume) is defined as∂U/∂T. (a) Using a result derived in Example 9.6. obtain an expression for the heat capacity per unit volume, inJ/K⋅mi3, of a photon gas. (b) What is its value at300K?

Consider a room divided by imaginary lines into three equal parts. Sketch a two-axis plot of the number of ways of arranging particles versus NleftandNrightfor the caseN=1023, Note that Nmiddleis not independent, being of courseN−Nnght−NleftYour axes should berole="math" localid="1658331658925" NleftandNright, and the number of ways should be represented by density of shading. (A form for numbers of ways applicable to a three-sided room is given in Appendix I. but the question can be answered without it.)

We based the exact probabilities of equation (9-9) on the claim that the number of ways of addingN distinct nonnegative integer quantum numbers to give a total ofM is{M+N-1)!/M!(N-1)!. Verify this claim (a) for the caseN=2,M=5and(b)for the case.

N=5,M=2

The electromagnetic intensity thermally radiated by a body of temperature Tis given by I=σT4whereσ=5.67×10−8W/m2⋅K4

This is known as the Stefan-Boltzmann law. Show that this law follows from equation (9-46). (Note: Intensity, or power per unit area, is the product of the energy per unit volume and distance per unit time. But because intensity is a flow in a given direction away from the blackbody, the correct speed is not c. For radiation moving uniformly in all directions, the average component of velocity in a given direction is14c .)

We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy ! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system of11oscillators sharing a total energy of just5hω0 . In the symbols of Section 9.3. N=11andM=5 .

  1. Using equation(9-9) , calculate the probabilities ofn , being0,1,2, and3 .
  2. How many particlesNn , would be expected in each level? Round each to the nearest integer. (Happily. the number is still 11. and the energy still5hω0 .) What you have is a distribution of the energy that is as close to expectations is possible. given that numbers at each level in a real case are integers.
  3. Entropy is related to the number of microscopic ways the macro state can be obtained. and the number of ways of permuting particle labels withN0 ,N1,N2 , and N3fixed and totaling11 is11!(N0!N1!N2!N3!) . (See Appendix J for the proof.) Calculate the number of ways for your distribution.
  4. Calculate the number of ways if there were6 particles inn=0.5 inn=1 and none higher. Note that this also has the same total energy.
  5. Find at least one other distribution in which the 11 oscillators share the same energy, and calculate the number of ways.

  6. What do your finding suggests?
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