By K. K. Thornber (auth.), George J. Papadopoulos, J. T. Devreese (eds.)

ISBN-10: 1468491407

ISBN-13: 9781468491401

ISBN-10: 1468491423

ISBN-13: 9781468491425

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**Additional info for Path Integrals: And Their Applications in Quantum, Statistical and Solid State Physics**

**Example text**

R. ])]dt/h} x OR. (t). (t)] plays the role of the "classical" action functional in this path integral. STATIONARY-PHASE APPROXIMATIONS Unfortunately, very few path integrals can be evaluated exactly so various approximate methods are invoked. One such approach uses the stationary-phase approximation also called the steepest-descent method. ll In this approach an integral (with sufficiently analytic integrand) of the form A = Jeif(x)dx , and integrated along the real line, is first transformed into an integral along a contour in the complexified x-plane that is arranged to cross over a saddle point defined by a solution of f'(x ) = 0 and oriented so the integrand falls off as rapidly as poss~ble on either side.

26a). Then the point transformation (28) followed by a gauge transformation will produce the associated self-adjoint problem Hln> = Aln>. We note that (28) is just the transformation which is also employed in the work of Horsthemke and Bach 16/. Clearly. the orthogonality relation (15) and the symmetry of the propagator (20). proven for constant diffusion. remain true also for the general case. Jw _1 JU non > (31) 2 with wo(x) given by (4). As before we wish to explore the "classical mechanics" behind the Fokker-Planck equation, now for arbitrary diffusion.

Let ~(s) be the measure on infinite sequences {sk} of real variables for which 2 2 Jexp(a~~sk)d~(S) = exp(~ ~hk) for any set {~} such that ~~ < 00. e. On the other hand, representation through a sequence of finitedimensional integrals and stationary-phase approximations leads to the extremal conditions sk = a~ for all k, and thus to a squaresummable sequence {sk} that lies outside the class on which ~ is concentrated. Continuous-Representation Path Integrals In the last section both discrete and formally continuous path integral expressions were developed for the propagator in phase-space continuous representations.

### Path Integrals: And Their Applications in Quantum, Statistical and Solid State Physics by K. K. Thornber (auth.), George J. Papadopoulos, J. T. Devreese (eds.)

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