A monatomic particle at rest can be in either of two energy levels, separated by an energy ε. Consider a dilute gas of N such particles at a fixed temperature T. a) Write down the probability for a single particle to be moving with momentum h❘k and in the excited energy level. b) Derive the canonical partition function of a single particle. c) [You may use the Gaussian integral: ୮ dx x²e-x²y² = 4y3 and approximate the density of states in three dimensions as g(k)dk ≈ k²dk] Hence, or otherwise, find the internal energy per particle of this gas and comment on how it behaves at large temperature

3) Consider the collection of identical harmonic oscillators (as in the Einstein floor). The permitted energies of each oscilator (E = nhf (n=0, 1, 2.0, hf. 2hf and so on. a) Calculate the splitting function of a single harmonic oscitor. What is the splitting function of N oscilator? wwww wwwwww www www b) Obtain the average energy of the T-temperature N oscilator from the split function. c) Calculate the heat capacity of this system and T → 0 ve T → 0 in limits, what is the heat capacity of the system? Are these results in line with the experiment? Why? What's the right theory about that? w w d) Find the Helmholtz free energy of this system. www ww e) which gives the entropy of this system as a function of temperature. ww wd wwww ww

A particle with mass m is in the state mr2 tiat ¥(x,t) = Ae where A and a are positive real constants. 2h Normalise Y and calculate A.

one-dimensional A one-particle, system has the potential energy function V = V₁ for 0 ≤ x ≤ 1 and V = ∞ elsewhere (where Vo is a constant). a) Use the variation function = sin() for 0 ≤ x ≤ 1 and = 0 elsewhere to estimate the ground-state energy of this system. b) Calculate the % relative error.