NTHU-Qubit

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====== HW6 ======

** Problem 27.1 (R.H. Swendsen): Bosons in two dimensions **

Consider an ideal boson gas in two dimensions. The $N$ particles in the gas each have mass $m$ and are confined to a box of dimensions $L\times L$.
  - Calculate the density of states for the two-dimensional, ideal Bose gas.
  - Calculate the Einstein temperature in two dimensions in terms of the given parameters and a dimensionless integral. You do have to evaluate the dimensionless integral.

** Problem 27.2 (R.H. Swendsen): Bosons in four dimensions **

Consider an ideal boson gas in four dimensions. The $N$ particles in the gas each have mass $m$ and are confined to a box of dimensions $L\times L\times L\times L$.
  - Calculate the density of states for the four-dimensional, ideal Bose gas.
  - Calculate the Einstein temperature in four dimensions in terms of the given parameters and a dimensionless integral. You do not have to evaluate the dimensionless integral.
  - Below the Einstein temperature, calculate the occupation number of the lowest energy, single-particle state as a function of temperature, in terms of the Einstein temperature and the total number of particles.

** Problem 27.3 (R. H. Swendsen): Energy of an ideal Bose gas in four dimensions **

Again consider an ideal boson gas in four dimensions. The $N$ particles in the gas each have mass $m$ and are confined to a box of dimensions $L\times L\times L\times L$. Calculate the energy and the specific heat as functions of temperature below the Einstein temperature.

** Problem 28.3 (R. H. Swendsen): Ideal Fermi gas in two dimensions **

Consider an ideal Fermi gas in two dimensions. It is contained in an area of dimensions $L\times L$. The particle mass is $m$.
  - Calculate the density of states.
  - Using your result for the density of states, calculate the number of particles as a function of the chemical potential at zero temperature. ($\mu(T=0)=\epsilon_F$, the Fermi energy.)
  - Calculate the Fermi energy as a function of the number of particles.
  - Again using your result for the density of states, calculate the total energy of the system at zero temperature as a function of the Fermi energy, $\epsilon_F$.
  - Calculate the energy per particle as a function of the Fermi energy $\epsilon_F$.
NTHU-Qubit

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