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Chapter 5: Q34P (page 246)
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.
The Fermi energy for electrons in a two-dimensional infinite square well is
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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?
Chlorine has two naturally occurring isotopes,and . Show that
the vibrational spectrum of HCIshould consist of closely spaced doublets,
with a splitting given by , where v is the frequency of the
emitted photon. Hint: Think of it as a harmonic oscillator, with , where
is the reduced mass (Equation 5.8 ) and k is presumably the same for both isotopes.
Typically, the interaction potential depends only on the vectorbetween
the two particles. In that case the Schrodinger equation seperates, if we change variables from and(the center of mass).
(a)Show that
localid="1655976113066"
localid="1655976264171"
is the reduced mass of the system
(b) Show that the (time-independent) Schrödinger equation (5.7) becomes
(c) Separate the variables, letting Note that satisfies the
one-particle Schrödinger equation, with the total mass in place of m, potential zero, and energy while satisfies the one-particle Schrödinger equation with the reduced mass in place of m, potential V(r) , and energy localid="1655977092786" . The total energy is the sum: . What this tells us is that the center of mass moves like a free particle, and the relative motion (that is, the motion of particle 2 with respect to particle 1) is the same as if we had a single particle with the reduced mass, subject to the potential V. Exactly the same decomposition occurs in classical mechanics; it reduces the two-body problem to an equivalent one-body problem.
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).
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).
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