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Lecture 19 — The Canonical Ensemble Chapter 6, Friday February 22 nd

Lecture 19 — The Canonical Ensemble Chapter 6, Friday February 22 nd. Finish discussion of equipartition theorem Identical particles and quantum statistics Symmetry and antisymmetry Bosons Fermions Implications for statistics. Reading: All of chapters 5 and 6 (pages 91 - 157)

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Lecture 19 — The Canonical Ensemble Chapter 6, Friday February 22 nd

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  1. Lecture 19 — The Canonical Ensemble Chapter 6, Friday February 22nd • Finish discussion of equipartition theorem • Identical particles and quantum statistics • Symmetry and antisymmetry • Bosons • Fermions • Implications for statistics Reading: All of chapters 5 and 6 (pages 91 - 157) Assigned problems, Ch. 5: 8, 14, 16, 18, 22 Homework 5 due today Assigned problems, Ch. 6: 2, 4, 6, 8, (+1) Homework 6 due next Friday

  2. More on the equipartition theorem Classical uncertainty: Where is the particle? V(x) V = ∞ V = ∞ V = 0 W = ∞ S = ∞ x x = L

  3. More on the equipartition theorem: phase space Area h Cell: (x,px) dpx px dx x

  4. Examples of degrees of freedom:

  5. Quantum statistics and identical particles Indistinguishable events 1. 1. Heisenberg uncertainty principle h 2. 2. The indistinguishability of identical particles has a profound effect on statistics. Furthermore, there are two fundamentally different types of particle in nature: bosons and fermions. The statistical rules for each type of particle differ!

  6. Bosons • This wave function is also symmetric with respect to exchange, but it is not normalized. Note that there are 3! terms (permutations), i.e. 3P1. • In general, for N particles, there will be N! (or NP1) terms in the wave function, i.e. A LOT! This wave function is symmetric with respect to exchange.

  7. Bosons • Easier way to describe N particle system: • The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wave function fi. • For bosons, these occupation numbers can be zero or ANY positive integer.

  8. Fermions • It turns out that there is an alternative way to write down this wave function which is far more intuitive: This wave function is antisymmetric with respect to exchange.

  9. Fermions This wave function is antisymmetric with respect to exchange. • It turns out that there is an alternative way to write down this wave function which is far more intuitive: • The determinant of such a matrix has certain crucial properties: • It changes sign if you switch any two labels, i.e. any two rows. • It is ZERO if any two columns are the same. • Thus, you cannot put two Fermions in the same single-particle state!

  10. Fermions This wave function is antisymmetric with respect to exchange. • As with bosons, there is an easier way to describe N particle system: • The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wave function fi. • For Fermions, these occupation numbers can be ONLY zero or one.

  11. Fermions • As with bosons, there is an easier way to describe N particle system: • The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wave function fi. • For Fermions, these occupation numbers can be ONLY zero or one. 2e e 0

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