L7A Quantum Statistics.pdf - Chapter 7A Quantum Statistics...

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Chapter 7A Quantum Statistics
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Outline of this Chapter 1. Bose Statistics 2. Fermi Statistics 3. Density of States and Degenerate Fermi gas 4. Blackbody Radiation 5. Debye Solids 6. Bose-Einstein Condensation Phys4050 Thermodynamics and Statistical Physics 2
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Classical Indistinguishable Particles Consider a system of ? indistinguishable particles. Phys4050 Thermodynamics and Statistical Physics 3 However, this works only when the probability of having more than one particle in the same state is negligible . ? = ? 1 ? 2 ? 3 ⋯ ? 𝑁 distinguishable ? = 1 ?! ? 1 𝑁 indistinguishable Our classical correction for indistinguishability was simply to divide the partition function by the number of permutations ?! :
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Classical Particle vs Quantum Particle This is true when the average number of available single-particle state is much greater than the number of particles: This is a comparison of the average volume occupied by a molecule ?/? to the quantum volume 𝑣 Q . Phys4050 Thermodynamics and Statistical Physics 4 ? 1 = ? 𝑣 Q ≫ ? ? ? ≫ 𝑣 Q or
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Classical Gas vs Quantum Gas As the average intermolecular distance is ҧ ? ≈ ?/? 1/3 , the classical approximation is invalid when the de Broglie wavelength ? Q = 𝑣 Q 1/3 is comparable to ҧ ? . Phys4050 Thermodynamics and Statistical Physics 5
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Quantum Particles Identical quantum particles are classified into two types according to the spin: Integral spin boson (photon, pion, helium- 4, etc…) Half-integral spin fermion (electron, proton, neutron, helium- 3, etc…) For fermions , no more than one particle can occupy a quantum state ( Pauli’s exclusion principle ). For bosons , any number of particles can occupy each quantum state. Phys4050 Thermodynamics and Statistical Physics 6
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