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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) 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,.
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?
(a) Construct the completely anti symmetric wave function for three identical fermions, one in the state , one in the state ,and one in the state
(b)Construct the completely symmetric wave function for three identical bosons (i) if all are in state (ii) if two are in state and another one is role="math" localid="1658224351718" c) one in the state , one in the state ,and one in the state
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?
(a) Using Equations 5.59 and 5.63, show that the wave function for a particle in the periodic delta-function potential can be written in the form
(b) There is an exception; At the top of a band where z is an integer multiple of yields .
Find the correct wave function for the case. Note what happens toeach delta function.
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