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Suppose there is a two state system with energies ε_0 = 0 and ε_1 = kT that follows the Boltzmann distribution. What is the partition function?
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- 1) Consider a system consisting of two particles, each of which can occupy any one of three single-particle states, with energy 0, e and 2e. a) How many states are there if the particles are bosons? Write down the partition function for this system? b) function for this system? How many states are there if the particles are fermions? Write down the partitionTwo nonintercating gaseous particles form an open system. There are three nondegenerate single particle states of energies Є, 2Є, 3Є in the system. Find the canonical partition functions Z0, Zl, Z2 and the grand canonical partition function Z if the particles are (a) fermions (b) bosons What is the average number of particles per state in each case? Calculate the internal energy U? (related to statistical physics)Define the Partition Function using Boltzmann factors ?
- Statistical Mechanics. We have a system of N bosons with zero spin. Each boson can have two states of energies 0 and E. Let μ be the chemical potential of the system and suppose that N >> 1. a) Determine the temperature T so that the mean population of the ground state is twice that of the excited state of energy E. Express kT only in terms of N and E. b) What would be the corresponding temperature T′ if the particles obeyed Boltzmann statistics? Compare both results and discuss them physically.Consider a hypothetical atom that has just two states: a ground state with energy zero and an excited state with energy 2 eV. Draw a graph of the partition function for this system as a function of temperature, and evaluate the partition function numerically at T = 300 K, 3000 K, 30,000 K, and 300,000 K.Use Boltzmann distribution to solve this problem.A system consists of 3, 000 particles that can only occupy two energy levels: a nondegen-erate ground state of 0.052 eV and a threefold degenerate excited state at 0.156 eV. IfT = 900 K,(a) find the number of particles at each energy level.(b) what is the total energy of the system?
- Consider a model thermodynamic assembly in which the allowed one-particle states have energies 0, ?, 2?, 3?, 4?,5?,6?,.... The assembly has three particles and a total energy of 7?. Identify the possible particle number distributions and calculate the average distribution of the three particles in the energy states when the particles are (a) localized and distinguishable (b) gaseous bosons (c) gaseous fermionsWrite Boltzmann’s equation for the one-particle distribution function f (r, k, t).A system has N weakly interacting particles with energy levels- E0, 0, E0, where E0 is positive.(a) What is the single particle partition function ofthe system(b) What is the energy of the system as a function of temperature?(c) What is the energy at low temperatures?(d) what is the energy at hightemperatures? What does it mean?
- please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L where n., ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as ("", ""), show that the density of states g(k), defined such that the number of states with |k| between k and k +dk is given by g(k)dk, is Ak 9(k) = 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 oq = ng = mk„T/2nh?. Hence show that the average…A continuous-time Markov chain (CTMC) has the following Q () matrix (all rates are transition/second) 9 19 3 18 Q=() = !! 27 39 0 32 3 Given that the process is in state 3, the probability to move next to state 2 is O0.39796 O0.5 O0.2 O0.27551Problem 4. Consider two indistinguishable, noninteracting spin-1/2 fermions in a one- dimensional infinite square well potential of length L. (a) What is the ground-state energy of the two-particle system? (b) What is the ground-state quantum state vector? (c) What is the first excited state energy of the two-particle system? (d) What are the quantum state vectors of the first excited state? (e) What is the degeneracy of the first excited state? (f) Discuss qualitatively how the excited-state energies change if we consider the particles to be interacting through the Coulomb potential.