Consider a non-interacting gas of N spinless Bosons confined to two dimension (e.g. on a surface) of area 142. We want to calculate if Bose-Einstein Condensation is possible for both massive and massless Bosons in 2D. First consider Bosons with mass, such that the allowed quantum energies are given by h2T2n? 2 (a) (2 points) Show that the density of states for a spinless Boson with mass in 2D is mL2 2nh2 (b) (3 points) Following the derivation in K2 and the notes for 3D, write the integral expression for the total number of particles in all excited states Nex in 2D. Comparing your expression with the 3D one, discuss why Bose-Einstein Condensation (i.e. where Nex N) cannot occur in 2D for Bosons with mass. Now consider massless Bosons, such that the allowed quantum energies are now given by nrhc 2 (c) (3 points) Determine the density of states for a massless spinless Boson in 2D. (d) (2 points) As in part (b), write the integral expression for the total number of particles in all excited states Nex in 2D. Comparing your expression with the 3D one and comment if Bose-Einstein Condensation (i.e. where Nex Some possibly useful integrals: 1 N) can occur in this case. 00, 0 1 0 da; 1 2 6