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(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T...
Consider a two-dimensional non-interacting and non-relativistic gas of N spin-1/2 fermions at T 0 in a...
Consider a two-dimensional non-interacting and non-relativistic gas of N spin-1/2 fermions at T 0 in a box of area A. (a) Find the Fermi energy εF. (b) Show that the total energy is given by E- NE. 2
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-partic...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
Question 14. Consider the following 3-state model: a system of 21 non-interacting particles that can occupy...
Question 14. Consider the following 3-state model: a system of 21 non-interacting particles that can occupy one of three energy levels. This is analogous to electrons filling a number of orbitals. The energies associated with the states are E0, El and E2 respectively. Consider no external constraints on the system (i.e. thermal or pressure baths) and that the system is isolated a) What is the value of entropy change associated with the states of E0 = 21, El=E2=0. b) What...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
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...
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...
2. Consider an isolated system consisting of a large number N of very weakly interacting localized...
2. Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spin 1 2. Each particle has a rnagnetic mioment μ which can point parallel or anti-parallel to an applied field H. The energy E of the systern is then E =-(ni-n2):1H, antiparallel to H. (a) Consider the energy range between E and E+δΕ where δΕ < E but is microscopically large so that δΕ μΗ. What is the total number of states...
1- 5. Two particles each of mass m are fixed at the end of a rigid...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
1- 5. Two particles each of mass m are fixed at the end of a rigid...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
Consider a two-dimensional non-interacting and non-relativistic gas of N spin-1/2 fermions at T 0 in a box of area A. (a) Find the Fermi energy εF. (b) Show that the total energy is given by E- NE. 2
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
Question 14. Consider the following 3-state model: a system of 21 non-interacting particles that can occupy one of three energy levels. This is analogous to electrons filling a number of orbitals. The energies associated with the states are E0, El and E2 respectively. Consider no external constraints on the system (i.e. thermal or pressure baths) and that the system is isolated a) What is the value of entropy change associated with the states of E0 = 21, El=E2=0. b) What...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
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...
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...
2. Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spin 1 2. Each particle has a rnagnetic mioment μ which can point parallel or anti-parallel to an applied field H. The energy E of the systern is then E =-(ni-n2):1H, antiparallel to H. (a) Consider the energy range between E and E+δΕ where δΕ < E but is microscopically large so that δΕ μΗ. What is the total number of states...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
1- 5. Two particles each of mass m are fixed at the end of a rigid rod of length 2a. This rod lies in the xy plane and is free to rotate in that plane about an axis passing through the midpoint of the rod and perpendicular to it (that is, parallel to the z-axis). Neglect the inertial properties of the rod in the rest of this question z-axis 1. Derive the classical expression for the kinetic energy of the...
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