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Lecture 21 — Identical Particles Chapter 6, Wednesday February 27 th

Lecture 21 — Identical Particles Chapter 6, Wednesday February 27 th. Review of Lecture 19 Calculating partition function for identical particles Dilute and dense gases Identical particles on a lattice Spin and rotation in diatomic molecules (if time).

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Lecture 21 — Identical Particles Chapter 6, Wednesday February 27 th

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  1. Lecture 21 — Identical Particles Chapter 6, Wednesday February 27th • Review of Lecture 19 • Calculating partition function for identical particles • Dilute and dense gases • Identical particles on a lattice • Spin and rotation in diatomic molecules (if time) Reading: All of chapter 6 (pages 128 - 142) Assigned problems, Ch. 6: 2, 4*, 6, 8, (+1) Homework 6 due on Friday 29th 1 more homework before spring break (Fri.) Exam 2 on Wed. after spring break

  2. Bosons • Wavefunction symmetric with respect to exchange. There are N! terms. • Another way to describe an N particle system: • The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunction fi. • For bosons, occupation numbers can be zero or ANY positive integer.

  3. Fermions • Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.: • 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 antisymmetricwith respect to exchange • It is ZERO if any two columns are the same. • Thus, you cannot put two Fermions in the same single-particle state!

  4. Fermions 2e e 0 • As with bosons, there is another way to describe N particle system: • For Fermions, these occupation numbers can be ONLY zero or one.

  5. Bosons • For bosons, these occupation numbers can be zero or ANY positive integer.

  6. A more general expression for Z • Terms due to double occupancy – correctly counted. • Terms due to single occupancy – double counted. • First consider just two particles, and make a guess: VERY IMPORTANT: if the two particles are distinguishable, the counting is fine, i.e. y(x1,x2) and y(x1,x2) represent distinct quantum states. The states are indistinguishable if the particles are identical.

  7. A more general expression for Z • Terms due to double occupancy – under counted. • Terms due to single occupancy – correctly counted. • What if we divide by 2 (actually, 2!): • SO: we fixed one problem, but created another. Which is worse? • Consider the relative importance of these terms....

  8. A more general expression for Z • What if we divide by 2 (actually, 2!): • Dilute gases (what does dilute mean in this context?): • Particle spacing large compared to average de-Broglie wavelength. • Energy levels are sparsely occupied.

  9. Dense versus dilute gases Dilute: classical, particle-like Dense: quantum, wave-like lD • Either low-density, high temperature or high mass • de Broglie wave-length • Low probability of multiple occupancy • Either high-density, low temperature or low mass • de Broglie wave-length • High probability of multiple occupancy lD (mT)-1/2 lD (mT)-1/2

  10. A more general expression for Z • What if we divide by 2 (actually, 2!): • Dilute gases (what does dilute mean in this context?): • Particle spacing large compared to average de-Broglie wavelength. • Energy levels are sparsely occupied. • In the dilute limit, the error associated with doubly occupied states turns out to be inconsequential.

  11. A more general expression for Z • Therefore, for N particles in a dilute gas: and • VERY IMPORTANT: this is completely incorrect if the gas is dense. • If the gas is dense, then it matters whether the particles are bosonic or fermionic, and we must fix the error associated with the doubly occupied terms in the expression for the partition function. • Problem 8 and Chapter 10.

  12. Identical particles on a lattice Localized → Distinguishable Delocalized → Indistinguishable

  13. Spin } Symmetric Antisymmetric

  14. Diatomic molecules: ortho and para 1H2

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