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

By Grosche C.

During this lecture a brief advent is given into the speculation of the Feynman direction fundamental in quantum mechanics. the final formula in Riemann areas can be given according to the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the speculation of space-time modifications and separation of variables may be defined. As trouble-free examples I talk about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb capability.

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25), the angular dependence comes out wrong. 29) which is also vanishing for r ′ , r ′′ → 0, but not in the right manner. The points r ′ = r ′′ = 0 are essential singularities, where as the correct vanishing is powerlike. 22) seems very suggestive, but it is quit useless in explicit calculations. e. 30) cannot be calculated. The integrals turn out to be repeated integrals over errorfunctions which are not tractable. Also the method of Arthurs [4] fails. In this method one assumes that the integrations in the limit ǫ → 0 are effectively from −∞ to +∞.

24) (“+” for −π/2 < arg(z) < 3π/2, “-” for −3π/2 < arg(z) < π/2). 17) can be justified (adopting an argument due to Nelson [80]), but things are not such as easy. 17). 24) has the right boundary conditions at the origin. We have for z → 0: √ 1 2 2πz e−z Iλ (z) ≃ z λ+ 2 1 + O(z 2 ) (2λ)!! 1 The Radial Path Integral (D) (D) for all j. 28) (D) which is the correct boundary condition for Rn,l . 25), the angular dependence comes out wrong. 29) which is also vanishing for r ′ , r ′′ → 0, but not in the right manner.

47) denotes a “classical Lagrangian” on the lattice. 47), except that one has to take all trigonometrics at mid-points. 3. 1 The Radial Path Integral with the effective Hamiltonian p2r(j) Hef f (pr(j) , r(j) , {pθ(j) , θ(j) }) = 2m   1 1  2 1 pθ(j) + + p2θ2 + · · · + p2φ(j)  + ∆VP rod ({θ(j) }) (j) (j) (j) 2 2 2 2 1 2mr(j) sin θ1 sin θ1 . . 50) with the effective Lagrangian ˙ Lef f (r, r, ˙ {θ, θ}) m = [r˙ 2 + r 2 θ˙12 + r 2 sin2 θ1 θ˙22 + · · · + r 2 (sin2 θ1 . . 47). 47) is too complicated for explicit calculations.

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