Exercise #8

a) The total number of particle operator $\hat{N}$ is the infinitesimal generator of a rotation in Fock space $R_{\theta}=\exp(-i\hat{N}\theta)$. Prove that under this rotation the creation/destruction operators behave as

$a \rightarrow a \exp -i\theta$

$a^\dagger \rightarrow a^\dagger \exp i\theta$

and the field operators

$\Psi(x) \rightarrow \Psi(x) \exp -i\theta$

$\Psi^\dagger(x) \rightarrow \Psi^\dagger(x) \exp i\theta$

b) Using the previous results prove that the total number of particle is conserved for an Hamiltonian that contains both single-particle and two-particle interactions.

c) Consider the perfect Fermi and Bose gas with a general single particle dispersion $\epsilon(p)= a |p|^b$. Determine:

  • the density of the states $N(\epsilon)=\frac{V}{h^3}\int d^3 p \delta(\epsilon -\epsilon(p))$

  • the termodynamic quantities PV and $<N>$ in the Grand Canonical ensemble.

  • Obtain the classical perfect gas in the zero fugacity limit when $a=1/2 m$ and $b=2$.