Download PDF by Grosche C.: An introduction into the Feynman path integral

By Grosche C.

During this lecture a quick creation is given into the speculation of the Feynman course vital in quantum mechanics. the overall formula in Riemann areas should be given in accordance with the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the speculation of space-time variations and separation of variables can be defined. As ordinary examples I speak about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb strength.

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20) which is the formula for the propagator with the Maslov correction (note sin ωτ = | sin ωT |). e. we consider the propagator at caustics. 21) = exp − πn δ(x′ − (−1)n x′′ ). 2 Let us finally determine the energy dependent Green function. 729], a1 > a2 , ℜ( 21 + µ − ν) > 0): ∞ coth2ν 0 √ a1 + a2 x exp − t cosh x I2µ (t a1 a2 sinh x)dx 2 2 Γ( 1 + µ − ν) = √ 2 Wν,µ (a1 t)Mν,µ (a2 t). 22) Here Wν,µ (z) and Mν,µ (z) denote Whittaker-functions. 22) for ν = E/2¯ Whittaker- and parabolic cylinder-functions to obtain G(x′′ , x′ ; E) = − ×D− 1 + E 2 h ¯ω 1 2 m E 1 Γ − π¯hω 2 h ¯ω 2mω ′ (x + x′′ + |x′′ − x′ |) D− 1 + E − 2 h ¯ω ¯h 2mω ′ (x + x′′ − |x′′ − x′ |) .

The path integral is given by x(t′′ )=x′′ ′′ i Dx(t) exp ¯h ′ K(x , x ; T ) = x(t′ )=x′ 61 t′′ t′ m 2 Z e2 dt . 6) Important Examples Discussion of this path integral are due to Duru and Kleinert [27,28] and Inomata [58]. However, the lattice-formulation is not trivial for the 1/r-term. In fact, it is too singular for a path integral, respectively, a stochastic process and some regularization must be found. This is known for some time, and it turns out that the KustaanheimoStiefel transformation does the job.

2. 17), namely the harmonic oscillator with V (r) = 21 mω 2 r 2 . The calculation has first been performed by Peak and Inomata [84]. However, we present the more general case with timedependent coefficients following Goovaerts [39]. This example will be of great virtue in the solution of various path integral problems. 2 The Radial Harmonic Oscillator We have to study Kl (r ′′ , r ′ ; τ ) = (r ′ r ′′ ) 1−D 2 lim N→∞ (D) µl j=1 = (r ′ r ′′ ) 2−D 2 N→∞ N × N/2 ∞ ∞ r(1) r(1) · · · 0 r(N−1) dr(N−1) 0 iǫ im 2 2 2 · Il+ D−2 (r(j) + r(j−1) )− mω 2 (t(j) )r(j) 2 2ǫ¯h 2¯ h exp j=1 = (r ′ r ′′ ) m i ǫ¯h dr(N−1) 0 iǫ im 2 (r(j) − r(j−1) )2 − mω 2 (t(j) )r(j) 2ǫ¯h 2¯ h [r(j) r(j−1) ] · exp lim ∞ dr(1) · · · 0 N × ∞ N/2 m 2π i ǫ¯h 2−D 2 lim N→∞ N/2 α i ei α(r ′2 +r ′′ 2 ∞ )/2 ∞ r(1) dr(1) · · · 0 m r(j) r(j−1) i ǫ¯h r(N−1) dr(N−1) 0 2 2 2 × exp i(β(1) r(1) + β(2) r(2) + · · · + β(N−1) r(N−1) ) × Il+ D−2 (− i αr(0) r(1) ) .

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An introduction into the Feynman path integral by Grosche C.


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