5. Formulation of Quantum Statistics

5. Formulation of Quantum Statistics Quantum Mechanical Ensemble Theory: The Density Matrix Statistics of the Various Ensembles Examples Systems Composed of Indistinguishable Particles The Density Matrix & the Partition Function of a System of Free Particles

Advantage of using density matrix : Quantum & ensemble averaging are combined into one averaging.

Classical Statistical Mechanics (Probability) density function ( p,q,t ) : Caution: Some authors, e.g., Landau-Lifshitz, use a normalized version of  . Liouville’s theorem : Microcanonical ensemble: Canonical ensemble: Grand canonical ensemble:

Quantum Statistical Mechanics (To be Proved) Ensemble = phase space Classical mechanics : Ensemble = Hilbert space Quantum mechanics : PE = projection operator onto the N-D subspace of states with energy E. Microcanonical : Canonical : Grand canonical :

Pure State Density Operator Orthonormal basis { | n } is complete : Expectation value of f : Density operator for |   :

r-Representation  f is a 1-particle operator 

Mixed State Density Operator Averaged value of f : Orthonormal basis { | n } is complete : Density operator : Skip to ensembles Ex: Derive the quantum Liouville eq.

5.1. Quantum Mechanical Ensemble Theory: The Density Matrix Consider ensemble of N identical systems labelled by k = 1, 2,..., N. Each system is described by i = 1,2,..., N  k runs through all independent solutions of this Schrodinger eq. be the wave function of the kth system in the ensemble. Let Let be a set of complete orthonormal basis that spans the Hilbert space of H & satisfies the relevant B.C.s.  with

  where H can be t-dep    k

Density Operator Density operator : pk = weighting (or probability) factor with Matrix elements :  nor  d ~ quantum averaging  ens ~ k~ ensemble averaging

  where H can be t-dep

Equilibrium Ensemble System in equilibrium  ensemble stationary :  and i.e. Energy representation :   System in equilibrium  In a general basis ,  is hermitian  detailed balance

Expectation Values Expectation value of a physical quantity G : ( Quantum + ensemble av. ) Assuming knormalized, i.e.,    i.e. knormalized :

5.2. Statistics of the Various Ensembles Microcanonical ensemble : Fixed N, V, E or ( quantum statistics: no Gibbs’ paradox ) ( N, V, E;  ) = # of accessible microstates Equal a priori probabilities postulate  Energy representation:     i.e.

Pure State Only 1 state  p is accessible  3rd law     Energy representation : Thus i.e. idempotent ( is a projector ) In another representation with basis { m } so that ,  normalized  

Mixed State Multiple states are accessible, i.e.  > 1. Any representation : • = set of accessible • state indices Let K be the subspace spanned by the accessible  k’s. Consider any orthonormal basis {n } such that Since { k } is a basis of K, its completeness means  ( is diagonal w.r.t. {n } )

Let k = ensemble member index   So that Postulate of a priori random phases

Canonical Ensemble i.e. E-representation : Canonical ensemble : Fixed N, V, T.    By definition

Grand Canonical Ensemble Grand canonical ensemble : Fixed , V, T Er, s = Er(Ns ) = E of r th state of Ns p’cle sys 

5.3. Examples An Electron in a Magnetic Field signed Single e with spin & magnetic moment Pauli matrices :  A diagonal       agrees with § 3.9-10

A Free Particle in a Box Free particle of mass m in a cubical box of sides L.  with Periodic B.C :  with

( r - representation )  with  ( see next page )

 

 is symmetric  Location uncertainty : Particle density at r :

Alternatively Uising & integrate by parts twice : 

A Simple Harmonic Oscillator  n = 0,1,2,... Hermite polynomials : Rodrigues’ formula

is real Kubo, “Stat Mech.”, p.175 Mathematica 

Probability density :  qis a Gaussian with dispersion ( r.m.s. deviation ) :

Classical limit : (purely thermal)   Quantum limit : (non-thermal)   = Probability density of ground state

 

5.4. Systems Composed of Indistinguishable Particles N non-interacting particles subject to the same 1-particle hamiltonian h.  i= label of the eigenstate assumed by the i th particle. Letn= # of particles occupying the  th eigenstate.  L( , j ) = label of the j th particle that occupies the  th eigenstate.

Note: [ ... ] = 1 if n = 0. Let P denote a permutation of the particle labels :

Distinguishable particles : permutations within the same  counted as the same. permutations across different ’s counted as distinct.  # of distinct microstates is Indistinguishable particles : Boltzmannian ( distinguishable p’cles)

Indistinguishable Particles Particles indistinguishable  Physical properties unchanged under particle exchange i.e.  

Anti-symmetric : Pauli’s exclusiion principle  i.e.  Fermi-Dirac statistics Symmetric :  Bose-Einstein statistics

5.5. The Density Matrix & the Partition Function of a System of Free Particles N non-interacting, indistinguishable particles : Let i stands for ri , & i for ri. e.g., Goal: To write or

Non-interacting particles  Periodic B.C.  Bosons Fermions Mathematica

Consider the N ! permutations among { ki } associated with a given K.  E is unchanged  nk > 1 cases neglected (measure 0)

arbitrary P  P  = I 2-p'cle

from § 5.3 = thermal ( de Broglie ) wavelength

Let with   Mathematica mean inter-particle distance = n = particle density  

Resolution of problems in classical statistics: • Gibbs correction factor ( 1 / N! ). • Phase space volume per state Classical limit : Non-classical systems are said to be degenerate.  n3 = degeneracy discriminant ( no spatial correlation ) Classical limit

Exchange Correlation Let N = 2 : 

Classical limit

Statistical Potential  Mathematica

5. Formulation of Quantum Statistics