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Chapter 5: Q19P (page 229)
Find the energy at the bottom of the first allowed band, for the case , correct to three significant digits. For the sake of argument, assume eV.
The energy at the bottom of the first band is 0.345eV.
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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).
Imagine two non interacting particles, each of mass , in the one dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate assuming
(a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state.)
(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.
Discuss (qualitatively) the energy level scheme for helium if (a) electrons were identical bosons, and (b) if electrons were distinguishable particles (but with the same mass and charge). Pretend these 鈥渆lectrons鈥� still have spin 1/2, so the spin configurations are the singlet and the triplet.
(a) Suppose you put both electrons in a helium atom into the state;
what would the energy of the emitted electron be?
(b) Describe (quantitatively) the spectrum of the helium ion,.
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