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Chapter 5: Q26P (page 239)
Use the method of Lagrange multipliers to find the rectangle of largest area, with sides parallel to the axes that can be inscribed in the ellipse. What is the maximum area?
The maximum area of an inscribed triangle can have is
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We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic domain by replacing the classical kinetic energy, ,with the relativistic formula, . Momentum is related to the wave vector in the usual way: . In particular, in the extreme relativistic limit,
(a) Replace n Equation 5.55 by the ultra-relativistic expression, , and calculatein this regime.
(5.55).
(b) Repeat parts (a) and (b) of Problem 5.35 for the ultra-relativistic electron gas. Notice that in this case there is no stable minimum, regardless of R; if the total energy is positive, degeneracy forces exceed gravitational forces, and the star will expand, whereas if the total is negative, gravitational forces win out, and the star will collapse. Find the critical number of nucleons, Nc , such that gravitational collapse occurs for is called the Chandrasekhar limit.
(c) At extremely high density, inverse beta decay,converts virtually all of the protons and electrons into neutrons (liberating neutrinos, which carry off energy, in the process). Eventually neutron degeneracy pressure stabilizes the collapse, just as electron degeneracy does for the white dwarf (see Problem 5.35). Calculate the radius of a neutron star with the mass of the sun. Also calculate the (neutron) Fermi energy, and compare it to the rest energy of a neutron. Is it reasonable to treat a neutron star non relativistic ally?
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?
Evaluate the integrals (Equation5.108 and 5.109) for the case of identical fermions at absolute zero. Compare your results with equations 5.43 and5.45. (Note for electrons there is an extra factor of 2 in Equations 5.108 and 5.109. to account for the spin degeneracy.)
a) Hund鈥檚 first rule says that, consistent with the Pauli principle, the state with the highest total spin (S) will have the lowest energy. What would this predict in the case of the excited states of helium?
(b) Hund鈥檚 second rule says that, for a given spin, the state with the highest total orbital angular momentum (L) , consistent with overall antisymmetrization, will have the lowest energy. Why doesn鈥檛 carbon have? Note that the 鈥渢op of the ladder鈥�is symmetric.
(c) Hund鈥檚 third rule says that if a subshellis no more than half filled,
then the lowest energy level has; if it is more than half filled, thenhas the lowest energy. Use this to resolve the boron ambiguity inProblem 5.12(b).
(d) Use Hund鈥檚 rules, together with the fact that a symmetric spin state must go with an antisymmetric position state (and vice versa) to resolve the carbon and nitrogen ambiguities in Problem 5.12(b). Hint: Always go to the 鈥渢op of the ladder鈥� to figure out the symmetry of a state.
(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.
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