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thermodynamic simulations & feynman quantum path integration 40915.6.2 WLS Ising Model AssessmentRepeat the assessment conducted in §15.4.2 for the thermodynamic properties ofthe Ising model but use WLS in place of the Metropolis algorithm.15.7 Unit III. Feynman Path Integrals ⊙Problem: As is well known, a classical particle attached to a linear spring undergoessimple harmonic motion with a position in space as a function of time given byx(t)=A sin(ω 0 t + φ). Your problem is to take this classical space-time trajectoryx(t) and use it to generate the quantum wave function ψ(x, t) for a particle boundin a harmonic oscillator potential.15.8 Feynman’s Space-Time Propagation (Theory)Feynman was looking for a formulation of quantum mechanics that gave a moredirect connection to classical mechanics than does Schrödinger theory and thatmade the statistical nature of quantum mechanics evident from the start. He followeda suggestion by Dirac that Hamilton’s principle of least action, which canbe used to derive classical mechanics, may be the ¯h → 0 limit of a quantum leastactionprinciple. Seeing that Hamilton’s principle deals with the paths of particlesthrough space-time, Feynman posultated that the quantum wave function describingthe propagation of a free particle from the space-time point a =(x a ,t a )tothepoint b =(x b ,t b ) can expressed as [F&H 65]∫ψ(x b ,t b )=dx a G(x b ,t b ; x a ,t a )ψ(x a ,t a ), (15.39)where G is the Green’s function or propagator√[mG(x b ,t b ; x a ,t a ) ≡ G(b, a)=2πi(t b − t a ) expi m(x b − x a ) 22(t b − t a )]. (15.40)Equation (15.39) is a form of Huygens’s wavelet principle in which each point onthe wavefront ψ(x a ,t a ) emits a spherical wavelet G(b; a) that propagates forwardin space and time. It states that a new wavefront ψ(x b ,t b ) is created by summationover and interference with all the other wavelets.Feynman imagined that another way of interpreting (15.39) is as a form ofHamilton’s principle in which the probability amplitude (wave function ψ) fora particle to be at B is equal to the sum over all paths through space-time originatingat time A and ending at B (Figure 15.7). This view incorporates the statisticalnature of quantum mechanics by having different probabilities for travel along−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 409