1549380323-Statistical Mechanics Theory and Molecular Simulation

444 Quantum ideal gases 11.6. Consider a system ofNidentical bosons. Each particle can occupy one of two single-particle energy ...

Problems 445 Hˆ|ψ−〉=E−|ψ−〉. Here,E+< E−, and E±=E∓ 1 2 ∆. a. Show that if ∆/kT <<1 for an ensemble of such systems at t ...

12 The Feynman path integral 12.1 Quantum mechanics as a sum over paths The strangeness of the quantum world is evident in syste ...

Sum over paths 447 individual amplitudes. This picture of quantum mechanics is known as thesum over pathsorpath integralformulat ...

448 The Feynman path integral S D Fig. 12.2Interference pattern expected for classical electrons impinging on the double-slit ap ...

Sum over paths 449 Ultimately, an infinite number of amplitudes must be summed in order toobtain the overall probability, which ...

450 The Feynman path integral With this heuristic introduction to the sum over paths in mind, we now proceed to derive the Feynm ...

Path integral derivation 451 can be obtained by evaluating the density matrix at an imaginary inverse temperature β=it/ ̄h. In f ...

452 The Feynman path integral SubstitutingΩ into eqn. (12.2.7) givesˆ ρ(x,x′;β) = lim P→∞ 〈x′|ΩˆP|x〉= lim P→∞ 〈x′|ΩˆΩˆΩˆ···Ωˆ|x〉 ...

Path integral derivation 453 〈xk+1|e−β K/Pˆ |xk〉= ∫ dp〈xk+1|p〉〈p|xk〉e−βp (^2) / 2 mP . (12.2.16) Finally, using eqn. (9.2.43), e ...

454 The Feynman path integral ρ(x,x′;β) = lim P→∞ ( mP 2 πβ ̄h^2 )P/ 2 ∫ dx 2 ···dxP ×exp { − 1 ̄h ∑P k=1 [ mP 2 β ̄h (xk+1−xk)^ ...

Path integral derivation 455 x Imaginary time 0 ℏ/2 ℏ x x’ x Real time 0 t/2 t x x’ β β Fig. 12.6 Representative paths in the pa ...

456 The Feynman path integral x Imaginary time 0 ℏ/2 ℏ x β β Fig. 12.7 Representative paths in the discrete path sum for the can ...

Thermodynamics 457 extra factor ofP in the force constant and the fact that eqn. (12.2.29) hasP− 1 integrations in one spatial d ...

458 The Feynman path integral common case for which we need to evaluate eqn. (12.3.2) is ultimately the simplest. If Aˆis purely ...

Thermodynamics 459 〈Aˆ〉= 1 Q(L,T) lim P→∞ ( mP 2 πβ ̄h^2 )P/ 2 ∫ dx 1 ···dxP [ 1 P ∑P k=1 a(xk) ] ×exp { − 1 ̄h ∑P k=1 [ mP 2 β ...

460 The Feynman path integral Suppose, next, thatAˆis a function of just the momentum operator:Aˆ=Aˆ(ˆp). In this case, it is no ...

Thermodynamics 461 E=− ∂ ∂β lnQ(L,T) = 1 Q(L,T) ∂Q(L,T) ∂β . (12.3.18) SinceQ(L,T) is expressible using only cyclic paths, these ...

462 The Feynman path integral = lim P→∞ 〈PP(x 1 ,...,xP)〉f, (12.3.23) where PP(x 1 ,...,xP) = P βL − 1 L ∑P k=1 [ mP β^2 ̄h^2 (x ...

Functional integrals 463 limit,x 1 ,...,xP+1becomes the complete set of points needed to specify a continuous functionx(s) satis ...