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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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Chlorine has two naturally occurring isotopes,and . Show that
the vibrational spectrum of HCIshould consist of closely spaced doublets,
with a splitting given by , where v is the frequency of the
emitted photon. Hint: Think of it as a harmonic oscillator, with , where
is the reduced mass (Equation 5.8 ) and k is presumably the same for both isotopes.
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)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.
Obtain equation 5.76 by induction. The combinatorial question is this: How many different ways you can put N identical balls into d baskets (never mind the subscriptfor this problem). You could stick all of them into the third basket, or all but one in the second basket and one in the fifth, or two in the first and three in the third and all the rest in the seventh, etc. Work it out explicitly for the casesand; by that stage you should be able to deduce the general formula.
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