Quantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids, superconductors

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1 BYU PHYS 73 Statistical Mechanics Chapter 7: Sethna Professor Manuel Berrondo Quantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids, superconductors Charaterstics: discrete energy spectrum non-commutativity e β(k+v ) e βk e βv Bose-Einstein and Fermi-Dirac statistics (bosons and fermions) Tools: Q.M. S.M. wave functions (bras, kets) mixed states density matrix (ket-bra) operators (Hamiltonian) traces permutation symmetry

2 Quantum ensembles formed from mixed states, i.e., incoherent mixtures of wave functions (instead of superposition of wave functions as in Q.M.) Operator Â, wave functions {Ψ n (Q)} n=,... Expectation value of Â: Q.M.(pure state n),  pure ˆ = ψn (Q) Âψ n (Q) d 3n Q n Ensemble, in addition, p n = weight of expectation value S.M.  = ˆ p n ψn (Q) Âψ n (Q) d 3n Q n Bra-ket notation:  = n p n ψ n  ψ n Separate dynamics from operator: ρ = n p n ψ n ψ n density matrix 2

3 Using identity, I = α φ α φ α, for any complete set { φ α },  = nα p n ψ n φ α φ α  ψ n = nα p n φ α  ψ n ψ n φ α = α φ α Âρ φ α Also valid in QM for ρ n = ψ n ψ n. ( ) ( ) ( ) Tr ρ n  = Tr ψ n ψ n  = Tr ψ n  ψ n = ψ n  ψ n For pure states ρ n is idempotent: (if and only if) constraint: Tr (ρ) = Tr (ρ) = Tr n p n ψ n ψ n = n Tr ψ n ψ n p n = n p n = 3

4 Quantum Canonical Ensemble: p n = e βɛ n Z Boltzmann factor n ket corresponding to ɛ n (labels the quantum states), ρ = e βɛ n n n Z n and Z = e βɛ n = n e βh n = Tr ( e βh) n n where n are eigenkets of H. Partition function: Z = Tr ( e βh) independent of n representation e βh = n e βh n n = n e βɛ n n n where n n is the projector to the n th state. The density matrix is, ρ = e βh Tr (e βh ) and the entropy is S = k B Tr (ρ ln ρ) 4

5 BYU PHYS 73 Statistical Mechanics Chapter 7: Sethna Professor Manuel Berrondo Time evolution of ρ (t) when ψ = ψ (t), using TDSE for ρ n = ψ n ψ n, ρ n t = ψ n ψ n + ψ n ψ n t t ψ ( ψ ) where t ψ n = i H ψ n and t ψ n = i ψ n H for H hermitian H =H. (quantum) Liouville equation Dening as A B = [A, B], then antisymmetric nonassociative distributive satises Jacobi identity ρ (t) =e i t H ρ (0) Expansion: multiple nested commutators w/h 5

6 Applications Quantum SHO: ɛ n = ω ( ) n + 2 n = 0,,... Z = Tre βh = e βɛ n = e β (n+ 2) ω n=0 n ( = e ) α 2 e α n e α 2 = e = α 2 sinh ( ) α where α = β ω = ω k B T n Z = ( 2 sinh ω 2k B T compares E with T. ) 2 Average energy: ln Z = α 2 ln ( e α ) E = ln Z ( β = ω 2 + ) e α E = ω + e ω k BT }{{ 2 } n n = e ω/k BT and c V = E ( ) 2 ω T = k e ω/k BT B ( k B T ) e ω/k B T 2 k B as T 6

7 Noninteracting quantum particles. - Quantum Statistics Bosons Ψ b (,..., N) = Ψ b (P (,..., N)) where P (...) is a permutation. Example: N = P ( 2 3) = even odd Ψ B (, 2, 3) = [Ψ (, 2, 3) + Ψ (3,, 2) + Ψ (2, 3, ) ± Ψ (2,, 3) ± Ψ (, 3, 2) ± Ψ (3, 2, )] 3! (+) bosons ( ) fermions Ψ f (,..., N) = ( ) p Ψ f (P (,..., N)), where p is the parity of the permutation. 7

8 Sums in the partition function are restricted to B.E. or F.D. statistics Non-interacting (external potential V (r) for each partial, no V (r ij )) One-particle T.I.S.E. H = N j= ) ( 2 2m 2 j + V (r j ) = hψ k (r) = ɛ k ψ k (r) N j= h j a) Distiguishable (separation of variables) Ψ dist. (,..., N) = N ψ kj (r j ) j= b) Bosons Ψ (,..., N) = N p N ψ kj (P (r j )) j= (permanent) c) Fermions Ψ F (,..., N) = N p ( ) p = N N j= ψ kj (P (r j )) ψ (r ),..., ψ N (r ) ψ (r 2 ),..., ψ N (r 2 )... ψ (r N ),..., ψ N (r N ) N = N! for orthonormal ψ i ψ j = δ ij 8

9 Grand canonical ensemble E i = i th energy level for many-body non-interacting, N i, E i distributed: 2 ɛ 4 ɛ 4 orbital levels 2 ɛ 3 ɛ 3 3 ɛ 2 0 ɛ 2 4 ɛ ɛ n α ɛ α n α ɛ α α one-particle quantum numbers # particles in state i: N i = α ni α non-interacting E i = α ni αɛ α for i = ground state, st excited,... Ξ = i e β(e i µn i ) = i e β ( α ni αɛ α µ α ni α) for non interacting particles. Ξ = α Ξ α corresponding to each orbital α. 9

10 F.D. statistics: n α = {0, } Ξ α = e βn α(ɛ α µ) = + e β(ɛ α µ) n α =0 B.E. statistics: Ξ α n α =0 e βn α(ɛ α µ) = ( ) e β(ɛ nα α µ) n α = e β(ɛ α µ) so Φ kt ln Ξ = ±kt α ln ( e β(ɛ α µ) ) is the grand canonical quantum free energy. n α = Φ µ = occupation T n α B.E. = e β(ɛ α µ) n α F.D. f (ɛ α ) = e β(ɛ α µ) + finite 0

11 Density of states g (ɛ) or g (ω) for bosons, total number N: N = dn (ɛ) f (ɛ) dɛ F.D. f (ɛ α ) = dɛ α dn (ω) f (ω) dω B.E. dω N = ɛ α g α f (ɛ α ) g (ω) = Calculate g (ω) dn (ω) dω is the degeneracy, i.e. # states for a given ω. i) quantization condition π or 2π Periodic B.C. k = 2π L n n, n 2, n 3 quantum #'s ii) volume of (thick) shell: (in n space) 4πn 2 dn ω = kc iii) dispersion relation E = 2 k 2 2m { 2 iv) spin (polarization) degeneracy 2s +

12 For photons (black body) g (ω) dω = 2 4πn 2 dn dω dω µ = 0 k = 2π L n k2 dk = ( ) 3 2π n 2 dn L ω = kc n 2 dn = V dk = V ω 2 dω 8π 3k2 8π 3 c 3 g (ω) = V π 2 ω 2 c 3 #photons per (dω) = g (ω) f ( ω) = g (ω) e β ω Energy density: u (ω) = V ωg (ω) f ( ω) u (ω) dω = ω 3 dω π 2 c 3 e β ω RJ Planck 2

13 Bose Condensation g (ɛ) dɛ = 4πn 2 dn where ɛ = 2 k 2 ɛ = 2 2m 2m ( 2π L k = 2π L n (dispersion relation) ) 2 n 2 dɛ = 2 2m g (ɛ) dɛ = V m3/2 2π2 3 ɛdɛ and ( ) 2 2π 2ndn L N (µ) = ˆ 0 g (ɛ) dɛ e β(ɛ µ) Vary µ to get N V = ρ density. N (µ = 0) = g (ɛ) dɛ e βɛ = V ( ) λ 3ζ 3 2 where ζ ( 3 2 ) 2.62 and λ = h 2πmkB T ρ = 2.62 λ 3 Bose condensation: add more particles to ground state cong: ρ = ρ B.E. + ρ (µ = 0) ( critical temperature k B T c = h2 ρ ) 2 3 2πm 2.6 [ ( T ρ B.E. = ρ T c ) ] 3 2 3

14 Fermi Dirac: f (ɛ) = e β(ɛ µ) + and for s = 2, 2s + = 2, so g (ɛ) dɛ = 2 V m 3 2 2π2 h 3 ɛdɛ, and N (µ) = 0 g (ɛ) f (ɛ) dɛ µ for T = 0 Fermi energy = 0 N V = ɛ F 0 g (ɛ) V dɛ = (2ɛ 3 2 Fm) 3π 2 3 or ɛ F ρ 2 3 or N V = k3 F 3π 2 k F Fermi wave number 4