Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 35C / b

Table of contents
- Solution b

Solution

At zero temperature all states up to

E_{F}

are occupied. Writing

g = \frac{A m}{π ℏ ^{2}},

we find

N = \int_{0}^{E_{F}} g d E = g E_{F}

and

E_{tot} = \int_{0}^{E_{F}} E g d E = \frac{1}{2} g E_{F}^{2} = \frac{1}{2} N E_{F} .

Because

g = C A

with

C = m / (π ℏ^{2})

, fixed particle number gives

E_{F} = N / (C A)

and

E_{tot} = \frac{N ^{2}}{2 C A} .

The two-dimensional degeneracy pressure of a two-dimensional Fermi gas is therefore

p = - (\frac{\partial E _{tot}}{\partial A})_{N} = \frac{N ^{2}}{2 C A ^{2}} = \frac{N E _{F}}{2 A} .

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