Question

Consider a non-interacting Bose gas of N atoms of mass m and zero spin which are restricted to motion on a flat surface of area A, causing the system to be two-dimensional. a. Write an equation which fixes the chemical potential μ of the atoms as a function of N,A, and the temperature T when the system is 'normal'. b. Show that the integral which arises in the expression for μ diverges as μ→0−. From this observation deduce that there always exists a negative value of μ, which satisfies the equation. c. Does Bose condensation exist for a non-interacting Bose gas in two dimensions?

Answer 1

The equation involving the chemical potential μ for a non-interacting Bose gas in two dimensions can be constructed, but Bose-Einstein condensation does not occur in such a system as the integral expression for μ diverges.

Step-by-step explanation:

Regarding a non-interacting Bose gas in two dimensions, the equation to fix the chemical potential μ as a function of the number of atoms N, the area A, and the temperature T in the normal phase can be derived using the grand canonical ensemble. The resultant integral expression for μ diverges as μ approaches 0-. This divergence implies that there must exist a negative chemical potential μ that satisfies the integral equation for any finite number of particles N. However, for a non-interacting Bose gas in two dimensions, Bose-Einstein condensation does not occur due to the divergence of the density of states as the energy approaches zero, preventing the macroscopic occupation of the ground state necessary for condensation.