1549380323-Statistical Mechanics Theory and Molecular Simulation

404 Quantum ensembles Q(β) = ∑∞ n=0 e−βEn= ∑∞ n=0 e−β(n+1/2) ̄hω. (10.4.17) Recalling that the sum of a geometric series is give ...

Problems 405 After all, one still needs to solve the eigenvalue problem for the Hamiltonian, which involves solution eqn. (10.1. ...

406 Quantum ensembles d. What are the expectation values of the operatorsSˆx,Sˆy, andSˆzat time t for this case? e. What is the ...

Problems 407 b. A harmonic oscillator of frequencyωinddimensions has energy eigen- values given by En= ( n+ d 2 ) ̄hω, but the e ...

408 Quantum ensembles Hint: You might find the Ritz variational principle of quantum mechanics helpful. The Ritz principle state ...

11 The quantum ideal gases: Fermi–Dirac and Bose–Einstein statistics 11.1 Complexity without interactions In Chapters 3 through ...

410 Quantum ideal gases wherexiis the combined coordinate and spin labelxi= (ri,si). TheN-particle func- tion Φ(x 1 ,....,xN) is ...

General formulation 411 periodicity of the box. These can be collected into a vectorni= (nx,i,ny,i,nz,i) of integers, which lead ...

412 Quantum ideal gases to plus signs.^1 ) In the fermion case, the determinant leads to a wave function that is completely anti ...

Boltzmann statistics 413 E{fnm}= ∑ m ∑ n εnfnm, (11.2.19) which is just a sum over all possible energies multiplied by the numbe ...

414 Quantum ideal gases For example, if there were only two states, then the occupation numbers aref 1 and f 2 wheref 1 +f 2 =N. ...

General formulation 415 11.4 General formulation for fermions and bosons For systems of identical fermions or identical bosons, ...

416 Quantum ideal gases the inner sum. The final sum overNcombined with the restricted sum over occupation numbers is mathematic ...

The ideal fermion gas 417 At this point, let us recall the procedure for calculating the equation of state in the grand canonica ...

418 Quantum ideal gases PV kT =g ∫ dnln ( 1 +ζe−βεn ) =g ∫ dnln ( 1 +ζe−^2 π (^2) β ̄h (^2) |n| (^2) /mL 2 ) = 4πg ∫∞ 0 dn n^2 l ...

The ideal fermion gas 419 Pλ^3 gkT = ∑∞ l=1 (−1)l+1ζl l^5 /^2 ρλ^3 g = ∑∞ l=1 (−1)l+1ζl l^3 /^2 , (11.5.7) whereρ=〈N〉/V is the n ...

420 Quantum ideal gases λ^3 ρ g = λ^3 ρ g +a 2 ρ^2 − 1 23 /^2 λ^6 ρ^2 g^2 , (11.5.13) or a 2 = λ^6 23 /^2 g^2 , (11.5.14) from w ...

The ideal fermion gas 421 11.5.2 The high-density, low-temperature limit The high-density, low-temperature limit exhibits the la ...

422 Quantum ideal gases AsT→0,ζ→∞and only the first term in the above expansion survives: ρλ^3 =ρ ( 2 π ̄h^2 mkT ) 3 / 2 ≈ 4 g 3 ...

The ideal fermion gas 423 e f (e) Fig. 11.1The Fermi–Dirac distribution forT= 0 in eqn. (11.5.32) (solid line) and finite temper ...