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Chapter 5: Q15P (page 223)
Find the average energy per free electron , as a fraction of the
Fermi energy. Answer:
The average energy per free electron is
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Imagine two noninteracting particles, each of mass m, in the infinite square well. If one is in the state(Equation 2.28 ), and the other in state , calculate localid="1658214464999" , assuming (a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions.
Show that most of the energies determined by Equation 5.64are doubly degenerate. What are the exceptional cases? Hint: Try it for N=1,2,3,4.... , to see how it goes. What are the possible values of cos(ka)in each case?
(a)Use Equation5.113 to determine the energy density in the wavelength range. Hint: set, and solve for
(b)Derive the Wien displacement law for the wavelength at which the blackbody energy density is a maximum
You'll need to solve the transcendental equation, using a calculator (or a computer); get the numerical answer accurate to three significant digits.
(a) Write down the Hamiltonian for two noninteracting identical particles in the infinite square well. Verify that the fermion ground state given in Example 5.1 is an eigenfunction of H, with the appropriate eigenvalue.
(b) Find the next two excited states (beyond the ones in Example 5.1) - wave functions and energies - for each of the three cases (distinguishable, identical bosons, identical fermions).
The density of copper isand its atomic weight is
(a) Calculate the Fermi energy for copper (Equation 5.43). Assume d = 1, and give your answer in electron volts.
(5.43).
(b) What is the corresponding electron velocity? Hint: SetIs it safe to assume the electrons in copper are nonrelativistic?
(c) At what temperature would the characteristic thermal energyrole="math" localid="1656065555994" is the Boltzmann constant and T is the Kelvin temperature) equal the Fermi energy, for copper? Comment: This is called the Fermi temperature,
. As long as the actual temperature is substantially below the Fermi temperature, the material can be regarded as “cold,” with most of the electrons in the lowest accessible state. Since the melting point of copper is 1356 K, solid copper is always cold.
(d) Calculate the degeneracy pressure (Equation 5.46) of copper, in the electron gas model.
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