1. Noninteracting Fermions Consider a system of noninteracting fermions such that there are VD(e)de = g(€)de single particle states with energies e, in the interval € ≤ €; < € + de, and V is the volume. That is, g(e) is the density of states, and D(e) is the density of states per volume. For this problem, the only results about ideal Fermi systems that you should use, are the following equations: BpV=-80 = == In In[1e−€₁)], a (1) (N) = Σ(Πα) = Σ 1 1+(-) a (2) where each sum over a is a sum over all single-particle states, is the grand potential, is the grand canonical partition function, and all other symbols have their usual meanings. To simplify the calculations, you may want to define the fugacity z = eß³μ In this problem, the results in parts (a), (b), and (c) should be applicable to any non-interacting fermion system. In particular, results for (b) and (c) should be written in terms of D(e), without substituting it for any specific case. Items (d), (e), (f) by contrast, refers to a specific case, and therefore D(e) should be replaced by its value for that specific case. (a) (b) (c) (d) (e) Calculate the entropy of a system of noninteracting fermions and show that S=-kg (na) (na) + (1 − (na)) In(1 − (na))] . a By using the above definition of D(e), convert the rhs of the Eqs. (1) and (2) into integrals over the single particle energy €. Starting from the formulas obtained in the previous part, write a formula for the pressure of a system of noninteracting fermions, in term of an integral containing the density of states per volume, D(e), and the Fermi function F(e) = 1/(exp(ẞ(€ − µ) + 1). In 3 dimensions, D(e) = 0 for € 0 and D(e) = √√2me for € >0. By using the expressions obtained in the two previous parts, show that for a system of free fermions of spin 1/2 in 3 dimensions at any temperature, (E) = pV. Show that the kinetic energy of a three-dimensional gas of N free electrons at OK is Eo NEF, where the Fermi energy & is the highest energy of an occupied state in a system of non-interacting fermions at T = 0. (f) Degenrate pressure. Derive a relation connecting the pressure p, the volume V, and the Fermi energy of an electron gas at OK.

1. Noninteracting Fermions Consider a system of noninteracting fermions such that there are VD(e)de = g(€)de single particle states with energies e, in the interval € ≤ €; < € + de, and V is the volume. That is, g(e) is the density of states, and D(e) is the density of states per volume. For this problem, the only results about ideal Fermi systems that you should use, are the following equations: BpV=-80 = == In In[1e−€₁)], a (1) (N) = Σ(Πα) = Σ 1 1+(-) a (2) where each sum over a is a sum over all single-particle states, is the grand potential, is the grand canonical partition function, and all other symbols have their usual meanings. To simplify the calculations, you may want to define the fugacity z = eß³μ In this problem, the results in parts (a), (b), and (c) should be applicable to any non-interacting fermion system. In particular, results for (b) and (c) should be written in terms of D(e), without substituting it for any specific case. Items (d), (e), (f) by contrast, refers to a specific case, and therefore D(e) should be replaced by its value for that specific case. (a) (b) (c) (d) (e) Calculate the entropy of a system of noninteracting fermions and show that S=-kg (na) (na) + (1 − (na)) In(1 − (na))] . a By using the above definition of D(e), convert the rhs of the Eqs. (1) and (2) into integrals over the single particle energy €. Starting from the formulas obtained in the previous part, write a formula for the pressure of a system of noninteracting fermions, in term of an integral containing the density of states per volume, D(e), and the Fermi function F(e) = 1/(exp(ẞ(€ − µ) + 1). In 3 dimensions, D(e) = 0 for € 0 and D(e) = √√2me for € >0. By using the expressions obtained in the two previous parts, show that for a system of free fermions of spin 1/2 in 3 dimensions at any temperature, (E) = pV. Show that the kinetic energy of a three-dimensional gas of N free electrons at OK is Eo NEF, where the Fermi energy & is the highest energy of an occupied state in a system of non-interacting fermions at T = 0. (f) Degenrate pressure. Derive a relation connecting the pressure p, the volume V, and the Fermi energy of an electron gas at OK.