Consider an isolated system of $N$ identical, indistinguishable, and non-interacting point particles, in \underline{two dimensions}. Each particle is of mass $m$. The particles are confined to an area $A$. Let $\widehat{\Omega}(E,A,N)$ denote the number of micro states of the (macroscopic) system with energy less than or equal to $E$.
- Show that\footnote{Hint : The microstate of a single particle in two dimension, is specified a string of four numbers, two for position and two for momentum. The microstate of $N$ particles is specified by an ordered string of $4N$ numbers.}, $$\widehat{\Omega}(E,A,N) = \frac{1}{h^{2N}}\ \frac{A^N}{N!}\ \frac{(2\pi mE)^{N}}{\Gamma\left(N+1\right) }$$
- Derive an expression for the density of states of a single particle.
- Carry out quantum-counting of micro states of a single particle confined to a two dimensional box of length $L$ and show that the resulting expression is the same as the one obtained by classical Boltzmann counting.
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