Download 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 course crucial 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 idea of space-time alterations and separation of variables can be defined. As common examples I talk about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb strength.

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5) . 4). 6) t′′ im Dy(t) exp 2¯ h ′ 2 2 y˙ − ω (t)y dt . t′ y(t′ )=0 2 ˙ Here we have used the abbreviations mω 2 (t) = c(t) + b(t), y ′ = y(t′ ) and y ′′ = y(t′′ ). 7) Let us introduce a N − 1-dimensional vector z = (x1 , . . , xN−1 )T and the (N − 1) × (N − 1) matrix B:  2 − ǫ2 ω (1) 2  −1  ..  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 .

2c) Here, of course we can analytically continue from integer values of m and n to, say, real numbers α and β, respectively . Similarly we can state a path integral identity for the modified P¨osch-Teller potential which is defined as V (η,ν) ν 2 − 41 ¯ 2 η 2 − 41 h . 3) This can be achieved by means of the path integration of the SU(1, 1) group manifold. One gets NM K (M P T ) ′′ ′ (r ′′ ) exp Φn(η,ν) ∗ (r ′ )Φ(η,ν) n (r , r ; T ) = − n=0 ∞ + 0 dp Φp(η,ν) ∗ (r ′ )Φ(η,ν) (r ′′ ) exp p ih ¯T 2(k1 − k2 − n) − 1 2m − ih ¯T 2 p .

In the literature often use is been made of the asymptotic form of the modified Bessel functions 1 Iν (z) ≃ (2πz)− 2 ez−(ν 2 − 41 )/2z (|z| ≫ 1, ℜ(z) > 0). 20) reads: (D) Kl (r ′′ , r ′ ; T ) × exp ′ ′′ − D−1 2 = (r r )  i  ¯h N j=1 lim N→∞ m 2π i ǫ¯h ∞ N/2 0 ∞ dr(1) · · · 1 D−2 2 m 2 (l + 2 ) − 4 2 (r(j) − r(j−1) ) − ¯h 2ǫ 2mr(j) r(j−1) 43 dr(N−1)   − ǫV (r(j) ) . 21) Important Examples This last equation seems to suggests a Lagrangian formulation of the radial path integral: r(t′′ )=r′′ ?

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