Recent content by Selveste

Problem statement: Consider a fermionic system with two states 1,2 with energy levels \epsilon_i, i=1,2 . Moreover, the number of particles in state i is n_i = 0,1 . Let the Hamiltonian of the system be H = \sum_{i=1}^2 \epsilon_i n_i + \sum_{i \neq j} U n_i n_j Here, U > 0 is a...

Since the integrand is spherically symmetric I use spherical coordinates \int d\vec{r} = \Omega_d\int_{0}^{\infty}dr r^{d-1} where \Omega_d is the solid angle i d dimensions. Since I am to plot as a function of 1/\beta U_0 (= k_bT/U_0) , I introduce a new varible \Theta = 1/\beta U_0...

Homework Statement Solve the Laplace equation in 2D by the method of separation of variables. The problem is to determine the potential in a long, square, hollow tube, where four walls have different potential. The boundary conditions are as follows: V(x=0, y) = 0 V(x=L, y) = 0 V(x, y=0) = 0...

Homework Statement An initial particle distribution n(r, t) is distributed along an infinite line along the z-axis in a coordinate system. The particle distribution is let go and spreads out from this line. a) How likely is it to find a particle on a circle with distance r from the z-axis at...

Thank you. So integrating aT^{\nu}dT = -\frac{\dot{Q}}{2\pi L}\frac{dr}{r} from r_1 to r I find \frac{a}{\nu +1}\left(T^{\nu +1} - T_1^{\nu +1}\right) = -\frac{\dot{Q}}{2\pi L}\left(\ln r - \ln r_1\right) \tag{1} which by setting r = r_2 gives the heat flow \dot{Q} = \frac{2\pi...

Homework Statement A cylinder has length L , inner diameter R_1 and outer diameter R_2 . The temperature on the inner cylinder surface is T_1 and on the outer cylinder surface T_2 . There is no temperature variation along the cylinders lenght-axis. Assume that the heat conductivity k...

Homework Statement A generalized TdS-equation for systems of several types of "work-parts" and varying number of particles in multiple components, is given by dU = TdS + \sum_{i}y_idX_i+\sum_{\alpha =1}^{c}\mu_\alpha dN_{\alpha} Thus, its natural to regard the internal energy U (an...