Question

QUESTION (1) BOSE GAS: This one is to help you with the various manipulations. 1(a) The energy of a spinless 3-dimensional Bose gas is given by U = ig;P;E;, where g; is the degeneracy of the j-th state, P; the probability of occupation of this state, and E; the energy of this state. Rewrite this in the form of an energy integral, incorporating both the density of states, and the Bose distribution function, as functions of energy. 1(b) Now show that the energy of a system of unit volume is given by the expression U = (20kp7y8/2kB] [ die (0.1) Joel - 1 and write a final explicit expression for this energy using the result that the integral in this equation is given in terms of the Riemann zeta function ((z) and the Gamma function I(z) by $(2+1) = f(x+1) Šºdien (0.2) where z is real, and you can use the result that for integer n, T(n +1/2) = 200): 1/2. 1(c) Finally, using your result for U(T), or by some other means, find an explicit expression for the specific heat Cv (T) of this system (using T(n + 1) = n! for integer n).

Answer 1

Dos- Deneily of state to .. E = ak2 dE= zakolik = dk de de - 29K - rajta No. of Energy level per unite encagy per unit volume called density of etale . for BB No of energy level in the momentum range p. f p + dp 9. (b) db = ? V yap?d? þ= tok dp = hor g(K) dr = v us (tk) ² h dk - 2 x urt krok 40 t = 813 gekdk - U K²dk 1 goede - greide 1 g (f) = 9992a that v ft of called state devity of in 3D drah 730