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Chapter 5: Q27P (page 239)
(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%?
(a) Error of Stirling's approximation for z =10 is 13.75%.
(b) Error of Stirling's approximation is less than 1% for z = 90.
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Suppose you have three particles, and three distinct one-particle stateare available. How many different three-particle states can be constructed (a) if they are distinguishable particles, (b) if they are identical bosons, (c) if they are identical fermions? (The particles need not be in different states would be one possibility, if the particles are distinguishable.)
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).
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) If and are orthogonal, and both normalized, what is the constant A in Equation 5.10?
(b) Ifrole="math" localid="1658225858808" (and it is normalized), what is A ? (This case, of course, occurs only for bosons.)
(a) Calculatefor the state(Equation 5.30). Hint: Dointegral
first, using spherical coordinates, and setting the polar axis along, so
that
(5.30).
Theintegral is easy, but be careful to take the positive root. You鈥檒l have to
break theintegral into two pieces, one ranging from 0 to,the other from
Answer: 5/4a.
(b) Use your result in (a) to estimate the electron interaction energy in the ground state of helium. Express your answer in electron volts, and add it to(Equation 5.31) to get a corrected estimate of the ground state energy. Compare the experimental value. (Of course, we鈥檙e still working with an approximate wave function, so don鈥檛 expect perfect agreement.)
(5.31).
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