Lecture 21 Lecture 21 ——Identical Particles Identical Particles Chapter 6, Chapter 6, Wednesday February 27Wednesday February 27thth•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: Reading: All of chapter 6 (pages 128 All of chapter 6 (pages 128 --142)142)Assigned problems, Assigned problems, Ch. 6Ch. 6: 2, 4*, 6, 8, (+1): 2, 4*, 6, 8, (+1)Homework 6 due on Friday 29thHomework 6 due on Friday 29th1 more homework before spring break (Fri.)1 more homework before spring break (Fri.)Exam 2 on Wed. after spring breakExam 2 on Wed. after spring breakBosonsBosons() ()() ()(){}()2,Bose 1 2 1 2 2 1 2,Bose 2 11,,2ij i jxxxxxxxxψφφ φφ ψ=+=()()()()()()()()()() ()()()() () () () () ()3,Bose 1 2 2 1 2 3 2 1 3231 321112 132,,ijk ijkijk ijkijk ijkxxx x x x x x xxxx xxxxxx xxxψφφφφφφφφφ φφφφφφ φφ φ=+++++• Wavefunction symmetric with respect to exchange. There are N! terms.• Another way to describe an N particle system:12311 2 2 33,,,iinnnEn n nψεεε=⋅⋅⋅⋅=+++⋅⋅⋅⋅• The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunctionφi.• For bosons, occupation numbers can be zero or ANY positive integer.FermionsFermions() ()() () (){}()2,Fermi 1 2 1 2 2 1 2,Fermi 2 11,,2ij ijxxxxxx xxψφφ φφ ψ=−=−• Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.:()1113,Fermi 1 2 3 2 2 2333() () (),, () () ()() () ()ijkijkijkxxxxxx x x xxxxφφφψφφφφφφ=• The determinant of such a matrix has certain crucial properties:1. It changes sign if you switch any two labels, i.e. any two rows.It is antisymmetric with respect to exchange2. It is ZERO if any two columns are the same.• Thus, you cannot put two Fermions in the same single-particle state!FermionsFermions• As with bosons, there is another way to describe N particle system:12311 2 2 33,,,iinnnEn n nψεεε= ⋅⋅⋅⋅=+ + +⋅⋅⋅⋅• For Fermions, these occupation numbers can be ONLY zero or one.0ε2ε/2/3/FermiBBBkT kT kTZe e eεεε−− −=+ +BosonsBosons11 2 2 33iEn n nεεε=+ + +⋅⋅⋅⋅• For bosons, these occupation numbers can be zero or ANY positiveinteger./ 2/3/4/Bose12BBBBkT kT kT kTZeeeeεεεε−−−−=+ + + +A more general expression for A more general expression for ZZ• First consider just two particles, and make a guess:()()()()()()3121312 1423 2524//2112/2/ 2///////222222jBiBBBBBBBBBBMMkTkTijkTkT kTkTkT kTkT kTkTZe eeeeeeeeeeεεεεεεεεε εεεε εεεε−−==−−−−+−+ −+−+ −+−+⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠= + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅∑∑• Terms due to double occupancy – correctly counted.• Terms due to single occupancy – double counted.VERY IMPORTANT:if the two particles are distinguishable, the counting is fine, i.e. ψ(x1,x2) and ψ(x1,x2) represent distinct quantum states. The states are indistinguishable if the particles are identical.A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):()()()()()()3121312 1423 2524//2112/2/ 2/111222//////12!jBiBBBBBBBBBBMMkTkTijkTkT kTkTkT kTkT kTkTZe eeeeeeeeeeεεεεεεεεε εεεε εεεε−−==−−−−+−+ −+−+ −+−+⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠= + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅∑∑• Terms due to double occupancy – under counted.• Terms due to single occupancy – correctly counted.SO:we fixed one problem, but created another. Which is worse?•Consider the relative importance of these terms....A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):()()()()()()3121312 1423 2524//2112/2/ 2/111222//////12!jBiBBBBBBBBBBMMkTkTijkTkT kTkTkT kTkT kTkTZe eeeeeeeeeeεεεεεεεεε εεεε εεεε−−==−−−−+−+ −+−+ −+−+⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠= + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅∑∑Dilute gases (what does dilute mean in this context?):• Particle spacing large compared to average de-Broglie wavelength.• Energy levels are sparsely occupied.Dense versus dilute gasesDense versus dilute gases•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 occupancyDilute: classical, particle-like Dense: quantum, wave-likeλDλD∝ (mT )−1/2λD∝ (mT )−1/2A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):()()()()()()3121312 1423 2524//2112/2/ 2/111222//////12!jBiBBBBBBBBBBMMkTkTijkTkT kTkTkT kTkT kTkTZe eeeeeeeeeeεεεεεεεεε εεεε εεεε−−==−−−−+−+ −+−+ −+−+⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠= + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅∑∑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.A more general expression for A more general expression for ZZ• Therefore, for N particles in a dilute gas:()1!NNZZN=()(){}1ln ln 1BFNkT Z N=− − +andVERY IMPORTANT:VERY IMPORTANT:this is completely incorrect if the gas is this is completely incorrect if the gas is densedense..••If the gas is dense, then it matters whether the particles are bIf the gas is dense, then it matters whether the particles are bosonic osonic or or fermionicfermionic, and we must fix the error associated with the doubly , and we must fix the error associated with the doubly occupied terms in the expression for the partition function.occupied terms in the expression for the partition function.••Problem 8 and Chapter 10.Problem 8 and Chapter 10.Identical particles on a latticeIdentical particles on a latticeLocalized Localized →→DistinguishableDistinguishable() ()11and lnNNBZZ FNkTZ==−DeDelocalized localized →→InIndistinguishabledistinguishable()() (){}11and ln ln 1!NNBZZFNkTZNN==−−+SpinSpin351222: , , ,....: 0, , 2 , 3 ,....FermionsBosons======12 sp