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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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(a) If and are orthogonal, and both normalized, what is the constant A in Equation 5.10?
(b) Ifrole="math" localid="1658225858808" (and it is normalized), what is A ? (This case, of course, occurs only for bosons.)
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 鈥渃old,鈥� 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.
Suppose you had three particles, one in state, one in state, and one in state. Assuming , and are orthonormal, construct the three-particle states (analogous to Equations 5.15,5.16, and 5.17) representing
(a) distinguishable particles,
(b) identical bosons, and
(c) identical fermions.
Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be completely antisymmetric, in the same sense. Comment: There's a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is , etc., whese second row is , etc., and so on (this device works for any number of particles).
(a) Show that for bosons the chemical potential must always be less than the minimum allowed energy. Hint:cannot be negative.
(b) In particular, for the ideal bose gas, for allT. Show that in this casemonotonically increases asTdecreases, assumingNandVare held constant.
Hint: Study Equation5.108, with the minus sign.
(c) A crisis (called Bose condensation) occurs when (as we lowerT )role="math" localid="1658554129271" hits zero. Evaluate the integral, for渭=0, and obtain the formula for the critical temperatureTc at which this happens. Below the critical temperature, the particles crowd into the ground state, and the calculational device of replacing the discrete sum (Equation5.78) by a continuous integral (Equation5.108) losesits validity 29.
Hint:role="math" localid="1658554448116"
where 螕 is Euler's gamma function and 味 is the Riemann zeta function. Look up the appropriate numerical values.
(d) Find the critical temperature for 4He. Its density, at this temperature, is 0.15 gm / cm3. Comment: The experimental value of the critical temperature in 4He is 2.17 K. The remarkable properties of 4He in the neighborhood of Tc are discussed in the reference cited in footnote 29.
(a) Calculatefor the state(Equation 5.30). Hint: Dointegral
first, using spherical coordinates, and setting the polar axis along, so
that
(5.30).
Theintegral is easy, but be careful to take the positive root. You鈥檒l have to
break theintegral into two pieces, one ranging from 0 to,the other from
Answer: 5/4a.
(b) Use your result in (a) to estimate the electron interaction energy in the ground state of helium. Express your answer in electron volts, and add it to(Equation 5.31) to get a corrected estimate of the ground state energy. Compare the experimental value. (Of course, we鈥檙e still working with an approximate wave function, so don鈥檛 expect perfect agreement.)
(5.31).
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