Reference no: EM131316575

1. 3 particles in a 1 dimenstional box:-

a. A 1D particle in a box has energy levels n with energy E_n = (h²Π²n²) / (2mL²) where n is an integer from 1 to ∞. For an open system at k_bT = (h²Π²) / (2mL²), estimate what values of the chemical potential µ lead to on average three particles in the box for the case of fermions and bosons.

2. Critical temperature of a spin band

a. Consider a ferromagnetic Ising model on a rectangle with length N spins and width f(N) spins. Show that for f(N)=constant, that T_c =0

b. We know for f(N)=N that T_c > 0. For what m, such that f(N) = N^m, is T_c > 0? Explain why.

c. For what b, such that f(N) = (log(N))^b, will T_c > 0. Explain why.

3. Damping vibrations
Find a copy of Tuckerman and Berne, J. Chem. Phys. 98, 7301 (1993)

a. Explain why the Generalized Langevin Equation and the Molecular Dynamics simulation have good agreement for this problem.

b. Table 1 shows that the vibrational relaxation rates decrease with frequency above ω=50. Estimate the relaxation rates for ω=10, 20, 30 and 40.

c. They note that for high frequencies < R⁴(0)> i is greater than 3<hR²(0)>² which is the prediction for a Gaussian random process. Prove this prediction.

d. The authors conjecture that the non-Gaussian behavior is because the strong collisions should be treated as a Poisson process. Argue in favor or against this prediction.