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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=蟺2h22mL2n2

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

Consider a simple thermodynamic system in which particles can occupy only two states: a lower state, whose energy we define as 0 , and an upper state, energyEu鈭�

(a) Cany out the sum (with only two states, integration is certainly not valid) giving the average particle energy E. and plot your result as a function of temperature.

(b) Explain qualitatively why it should behave as it does,

(c) This system can be used as a model of paramagnetic, where individual atoms' magnetic moments can either be aligned or anti aligned with an external magnetic field, giving a low or high energy, respectively. Describe how the average alignment or antialignment depends on temperature. Does it make sense'?

(a) Show that the number of photons per unit volume in a photon gas of temperature Tis approximately(2脳107K鈭�3m鈭�3)T3鈰�(Note:鈭�0鈭�x2(ex鈭�1)鈭�1dx鈮�2.40.)

(b)Combine this with a result derived in Example 9.6 to show that the average photon energy in a cavity at temperatureTis given byE炉鈮�2.7kBT.

The entropy of an ideal monatomic gas is(3/2)NkBlnE+,NkBlnV-NkBlnN to within an additive constant. Show that this implies the correct relationship between internal energy Eand temperature.

Calculate the Fermi energy for copper, which has a density of8.9脳103kg/m3and one conduction electron per atom. Is room temperature "cold"?

Equation (9-27) gives the density of states for a system of oscillators but ignores spin. The result, simply one state per energy change ofbetween levels, is incorrect if particles are allowed different spin states at each level, but modification to include spin is easy. From Chapter 8, we know that a particle of spinis allowedspin orientations, so the number of states at each level is simply multiplied by this factor. Thus,

D(E)=(2s+1)/h蝇0.

(a) Using this density of states, the definitionNh蝇0/(2s+1)=蔚1, and

N=鈭�0鈭�N(E)D(E)dE

calculate the parameterin the Boltzmann distribution (9-31) and show that the distribution can thus be rewritten as

N(E)Boltz=蔚kBT1eE/kBT

(b) Argue that ifkBT>>蔚,the occupation number is much less than 1 for all E.
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