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Question 1 A system of N identical non-interacting magnetic ions of spin Y%, has energy u...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with energies e e+ is positive and e. is negative. The spin partition function for a collection of N non-interacting spins is kTexp Show All Your Work! a) Derive the expression for b) As the temperature, T, of the system decreases, the average energy decreases. Why is that? (E)-- In(QCN, V, β)] Cv aP эт
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B),...
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the...
For a spin-1/2 particle in a magnetic field B, with energies and , (a) calculate the partition function. (b) Show that the mean energy of this particle is given by ̅ For a system of noninteracti...
For a spin-1/2 particle in a magnetic field B, with energies and , (a) calculate the partition function. (b) Show that the mean energy of this particle is given by ̅ For a system of noninteracting spins, (c) what is the total partition function and (d) mean energy? We were unable to transcribe this image2 2kT 2 2kT
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
Calculate the internal energy, entropy and the equation of states for a system composed of N...
Calculate the internal energy, entropy and the equation of states for a system composed of N indistinguishable and non-interacting particles with the partition function (9) given below (a, b and C are constants): q = C(kt)}(V – Nb) eVKY
2. A system consists of N very weakly interacting particles at a temperature T high enough...
2. A system consists of N very weakly interacting particles at a temperature T high enough that classical statistical mechanics is applicable. Each particle is fixed in space, has mass m, a. Calculate the heat capacity of this system of particles at this temperature in each of the i. The effective restoring force has magnitude κ x, where x is the displacement from and is free to perform one-dimensional oscillations about its equilibrium position. following cases: equilibrium. The effective restoring...
16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical...
16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical particles that could be either spinless bosons or spin-half fermions by putting a number of them inside a potential and measuring the energy levels of the system, but without being able to resolve the degeneracy of each level. Energy levels do not depend on the particles' spin. The following values of the energy are observed: No particles: 0 -5€ 1 particle: E, 2E, 5E...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with energies e e+ is positive and e. is negative. The spin partition function for a collection of N non-interacting spins is kTexp Show All Your Work! a) Derive the expression for b) As the temperature, T, of the system decreases, the average energy decreases. Why is that? (E)-- In(QCN, V, β)] Cv aP эт
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
Calculate the internal energy, entropy and the equation of states for a system composed of N indistinguishable and non-interacting particles with the partition function (9) given below (a, b and C are constants): q = C(kt)}(V – Nb) eVKY
2. A system consists of N very weakly interacting particles at a temperature T high enough that classical statistical mechanics is applicable. Each particle is fixed in space, has mass m, a. Calculate the heat capacity of this system of particles at this temperature in each of the i. The effective restoring force has magnitude κ x, where x is the displacement from and is free to perform one-dimensional oscillations about its equilibrium position. following cases: equilibrium. The effective restoring...
16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical particles that could be either spinless bosons or spin-half fermions by putting a number of them inside a potential and measuring the energy levels of the system, but without being able to resolve the degeneracy of each level. Energy levels do not depend on the particles' spin. The following values of the energy are observed: No particles: 0 -5€ 1 particle: E, 2E, 5E...
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