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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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Certain cold stars (called white dwarfs) are stabilized against gravitational collapse by the degeneracy pressure of their electrons (Equation 5.57). Assuming constant density, the radius R of such an object can be calculated as follows:
(a) Write the total electron energy (Equation 5.56) in terms of the radius, the number of nucleons (protons and neutrons) N, the number of electrons per nucleon d, and the mass of the electron m. Beware: In this problem we are recycling the letters N and d for a slightly different purpose than in the text.
(b) Look up, or calculate, the gravitational energy of a uniformly dense sphere. Express your answer in terms of G (the constant of universal gravitation), R, N, and M (the mass of a nucleon). Note that the gravitational energy is negative.
(c) Find the radius for which the total energy, (a) plus (b), is a minimum.
(Note that the radius decreases as the total mass increases!) Put in the actual numbers, for everything except , using d=1/2 (actually, decreases a bit as the atomic number increases, but this is close enough for our purposes). Answer:
(d) Determine the radius, in kilometers, of a white dwarf with the mass of the sun.
(e) Determine the Fermi energy, in electron volts, for the white dwarf in (d), and compare it with the rest energy of an electron. Note that this system is getting dangerously relativistic (seeProblem 5.36).
Check the equations 5.74, 5.75, and 5.77 for the example in section 5.4.1
Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation
(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.
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).
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