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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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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).
(a) Figure out the electron configurations (in the notation of Equation
5.33) for the first two rows of the Periodic Table (up to neon), and check your
results against Table 5.1.
(5.33).
(b) Figure out the corresponding total angular momenta, in the notation of
Equation 5.34, for the first four elements. List all the possibilities for boron,
carbon, and nitrogen.
(5.34).
Suppose you could find a solutionto the Schrödinger 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 Schrödinger equation, with the same energy.
role="math" localid="1658219144812" (5.24).
role="math" localid="1658219153183" (5.25).
Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let σ be the number of free electrons per unit area.
(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,.
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