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Suppose your professor discovers dark matter, and it turns out to be a new type of subatomic particle which can share a state with one other particle – but no more – of the same type. In other words, the number of these (indistinguishable) particles in any given state can be 0, 1, or 2. (a) Derive the distribution function (analogous to the Bose-Einstein or Fermi-Dirac distribution) for the average occupancy of a state for particles of this type. It should be a function of e, µ, and kT. (b) What is the value of this distribution function at e = u? What value does the function approach for very large e? For very small e, under the assumption that u >> kT? (c) Suppose the "state" of the particle is defined only by its energy (do not consider spin, charge, etc.) Furthermore, suppose these particles are found in a system where the quantized energy levels are evenly spaced1 eV apart, and the lowest level has energy 0. What is the lowest possible energy of the system, if there are 10 of these particles in it? (d) Suppose the "macrostate" of the system in part (c) is defined by the number the total number of energy units q in excess of the lowest possible energy (where each energy unit is 1 eV). What is the entropy of the macrostate defined by N=10, q=5? hese particles N, and

Suppose your professor discovers dark matter, and it turns out to be a new type of subatomic particle which can share a state with one other particle – but no more – of the same type. In other words, the number of these (indistinguishable) particles in any given state can be 0, 1, or 2. (a) Derive the distribution function (analogous to the Bose-Einstein or Fermi-Dirac distribution) for the average occupancy of a state for particles of this type. It should be a function of e, µ, and kT. (b) What is the value of this distribution function at e = u? What value does the function approach for very large e? For very small e, under the assumption that u >> kT? (c) Suppose the "state" of the particle is defined only by its energy (do not consider spin, charge, etc.) Furthermore, suppose these particles are found in a system where the quantized energy levels are evenly spaced1 eV apart, and the lowest level has energy 0. What is the lowest possible energy of the system, if there are 10 of these particles in it? (d) Suppose the "macrostate" of the system in part (c) is defined by the number the total number of energy units q in excess of the lowest possible energy (where each energy unit is 1 eV). What is the entropy of the macrostate defined by N=10, q=5? hese particles N, and

Modern Physics
Modern Physics
3rd Edition
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Chapter7: Tunneling Phenomena
Section: Chapter Questions
Problem 16P
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Suppose your professor discovers dark matter, and it turns out to be a new type of subatomic particle which can share a state with one other particle – but no more – of the same type. In other words, the number of these (indistinguishable) particles in any given state can be 0, 1, or 2. (a) Derive the distribution function (analogous to the Bose-Einstein or Fermi-Dirac distribution) for the average occupancy of a state for particles of this type. It should be a function of e, µ, and kT. (b) What is the value of this distribution function at e = u? What value does the function approach for very large e? For very small e, under the assumption that u >> kT? (c) Suppose the "state" of the particle is defined only by its energy (do not consider spin, charge, etc.) Furthermore, suppose these particles are found in a system where the quantized energy levels are evenly spaced1 eV apart, and the lowest level has energy 0. What is the lowest possible energy of the system, if there are 10 of these particles in it? (d) Suppose the "macrostate" of the system in part (c) is defined by the number the total number of energy units q in excess of the lowest possible energy (where each energy unit is 1 eV). What is the entropy of the macrostate defined by N=10, q=5? hese particles N, and
Transcribed Image Text:Suppose your professor discovers dark matter, and it turns out to be a new type of subatomic particle which can share a state with one other particle – but no more – of the same type. In other words, the number of these (indistinguishable) particles in any given state can be 0, 1, or 2. (a) Derive the distribution function (analogous to the Bose-Einstein or Fermi-Dirac distribution) for the average occupancy of a state for particles of this type. It should be a function of e, µ, and kT. (b) What is the value of this distribution function at e = u? What value does the function approach for very large e? For very small e, under the assumption that u >> kT? (c) Suppose the "state" of the particle is defined only by its energy (do not consider spin, charge, etc.) Furthermore, suppose these particles are found in a system where the quantized energy levels are evenly spaced1 eV apart, and the lowest level has energy 0. What is the lowest possible energy of the system, if there are 10 of these particles in it? (d) Suppose the "macrostate" of the system in part (c) is defined by the number the total number of energy units q in excess of the lowest possible energy (where each energy unit is 1 eV). What is the entropy of the macrostate defined by N=10, q=5? hese particles N, and
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