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A system of non-interacting Fermi particles with Fermi energy (E_F) has the density of states proportional to √E, where E is the energy of a particle. The average energy per particle at temperature T=0 is?
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Calculation of Average Energy per Particle at T=0
Density of States:
- Given density of states: g(E) ∝ √E
- Density of states can be written as: g(E) = C√E, where C is a constant
Number of Particles:
- Number of particles up to Fermi energy E_F can be calculated by integrating the density of states up to E_F:
N = ∫{0}^{E_F} g(E) dE = C ∫{0}^{E_F} √E dE
N = C [ (2/3)E_F^(3/2) ]
Fermi Energy:
- Fermi energy E_F is related to the number of particles by: N = (2/3)C E_F^(3/2)
- Solving for Fermi energy: E_F = (3N / 2C)^(2/3)
Average Energy per Particle:
- The average energy per particle at T=0 can be calculated by integrating the energy up to Fermi energy and dividing by the number of particles:
⟨E⟩ = (1/N) ∫{0}^{E_F} E g(E) dE
⟨E⟩ = (1/N) ∫{0}^{E_F} E C√E dE
⟨E⟩ = C/N ∫{0}^{E_F} E^(3/2) dE
⟨E⟩ = C/N [ (2/5)E_F^(5/2) ]
Substitute E_F:
- Substitute E_F in terms of N and C:
⟨E⟩ = C/N [ (2/5) (3N / 2C)^(5/3) ]
Final Calculation:
- Simplifying the expression:
⟨E⟩ = (2/5) (3N / 2C)^(2/3)
Therefore, the average energy per particle at T=0 for a system of non-interacting Fermi particles with Fermi energy E_F and density of states proportional to √E is given by ⟨E⟩ = (2/5) (3N / 2C)^(2/3).
Density of States:
- Given density of states: g(E) ∝ √E
- Density of states can be written as: g(E) = C√E, where C is a constant
Number of Particles:
- Number of particles up to Fermi energy E_F can be calculated by integrating the density of states up to E_F:
N = ∫{0}^{E_F} g(E) dE = C ∫{0}^{E_F} √E dE
N = C [ (2/3)E_F^(3/2) ]
Fermi Energy:
- Fermi energy E_F is related to the number of particles by: N = (2/3)C E_F^(3/2)
- Solving for Fermi energy: E_F = (3N / 2C)^(2/3)
Average Energy per Particle:
- The average energy per particle at T=0 can be calculated by integrating the energy up to Fermi energy and dividing by the number of particles:
⟨E⟩ = (1/N) ∫{0}^{E_F} E g(E) dE
⟨E⟩ = (1/N) ∫{0}^{E_F} E C√E dE
⟨E⟩ = C/N ∫{0}^{E_F} E^(3/2) dE
⟨E⟩ = C/N [ (2/5)E_F^(5/2) ]
Substitute E_F:
- Substitute E_F in terms of N and C:
⟨E⟩ = C/N [ (2/5) (3N / 2C)^(5/3) ]
Final Calculation:
- Simplifying the expression:
⟨E⟩ = (2/5) (3N / 2C)^(2/3)
Therefore, the average energy per particle at T=0 for a system of non-interacting Fermi particles with Fermi energy E_F and density of states proportional to √E is given by ⟨E⟩ = (2/5) (3N / 2C)^(2/3).
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A system of non-interacting Fermi particles with Fermi energy (E_F) has the density of states proportional to √E, where E is the energy of a particle. The average energy per particle at temperature T=0 is?
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A system of non-interacting Fermi particles with Fermi energy (E_F) has the density of states proportional to √E, where E is the energy of a particle. The average energy per particle at temperature T=0 is? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A system of non-interacting Fermi particles with Fermi energy (E_F) has the density of states proportional to √E, where E is the energy of a particle. The average energy per particle at temperature T=0 is? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A system of non-interacting Fermi particles with Fermi energy (E_F) has the density of states proportional to √E, where E is the energy of a particle. The average energy per particle at temperature T=0 is?.
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