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

By Grosche C.

During this lecture a quick advent is given into the idea of the Feynman direction indispensable in quantum mechanics. the overall formula in Riemann areas may be given in accordance with the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the speculation of space-time ameliorations and separation of variables could be defined. As common examples I talk about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb capability.

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B= .  0 −1 2 − ǫ2 ω (2) 2 .. ... .. 0 0 0 ... 0 0 .. 2  0 0 .. (N−2) 2 2−ǫ ω −1 −1 2 − ǫ ω (N−1) 2 2   . 8) Thus we get FN m = 2π i ǫ¯h N 2 d N−1 z exp m T z Bz − 2 i ǫ¯h = m 2π i ǫ¯h det B 1 2 . 9) Therefore ′′ ′ F (t , t ) = m 2π i h ¯ f (t′′ , t′ ) 1 2 , where f (t′′ , t′ ) = lim ǫ det B. N→∞ Our final task is to determine det B. Let us consider the j × j matrix   2 − ǫ2 ω (1) 2 −1 ... 0 0   −1 2 − ǫ2 ω (2) 2 . . 0 0   . . . (j)  .. .. .. B =     0 0 . . 2 − ǫ2 ω (j−1) 2 −1 0 0 ...

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. 52) where Vc has to be determined and Ω denotes the D-dimensional unit vector on the S D−1 -sphere.

15) This two-dimensional transformation can be interpreted as a two-dimensional Kustaanheimo-Stiefel transformation [66]. It has the general feature that it performs a change of variables to “square-root” coordinates, note r = ξ 2 + η 2 . g. 16) where the zi are homogeneous bilinear forms in x and y. Following a theorem of Hurwitz and Lam this type of transformation can only be realized in the spacedimensions D = 1, 2, 4, 8. 17) (B k )li xk yl = zi = k,l B(x)li yi . 19) j i and the matrix B satisfies the condition t B(x)B(x) = |x|2 .

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An introduction into the Feynman path integral by Grosche C.


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