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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) Find the percent error in Stirling鈥檚 approximation for z = 10 ?
(b)What is the smallest integer z such that the error is less than 1%?
Suppose you could find a solutionto the Schr枚dinger equation (Equation 5.25), for the Hamiltonian in Equation 5.24. Describe how you would construct from it a completely symmetric function, and a completely anti symmetric function, which also satisfy the Schr枚dinger equation, with the same energy.
role="math" localid="1658219144812" (5.24).
role="math" localid="1658219153183" (5.25).
(a) Find the chemical potential and the total energy for distinguishable particles in the three dimensional harmonic oscillator potential (Problem 4.38). Hint: The sums in Equations5.785.78and5.795.79can be evaluated exactly, in this case鈭掆垝no need to use an integral approximation, as we did for the infinite square well. Note that by differentiating the geometric series,
You can get
and similar results for higher derivatives.
(b)Discuss the limiting caserole="math" localid="1658400905376" .
(c) Discuss the classical limit,role="math" localid="1658400915894" , in the light of the equipartition theorem. How many degrees of freedom does a particle in the three dimensional harmonic oscillator possess?
(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.
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