By Grosche C.
During this lecture a quick advent is given into the speculation of the Feynman course quintessential in quantum mechanics. the overall formula in Riemann areas may be given according to the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the speculation of space-time differences and separation of variables can be defined. As easy examples I talk about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb power.
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Extra info for An introduction into the Feynman path integral
The metric tensor in polar coordinates is (gab ) = diag(1, r 2 , r 2 sin2 θ1 , . . , r2 sin2 θ1 . . sin2 θD−2 ). 3) reduces to: cos ψ (1,2) = cos θ (1) cos θ (2) + sin θ (1) sin θ (2) cos(φ(1) − φ(2) ). The D-dimensional measure dx expressed in polar coordinates is D−1 dx = r D−1 drdΩ = r D−1 (sin θk )D−1−k drdθk k=1 D−1 (sin θk )D−1−k dθk . 5) dΩ denotes the (D − 1)-dimensional surface element on the unit sphere S D−1 and Ω(D) = 2π D/2 /Γ(D/2) is the volume of the D-dimensional unit (S D−1 -) sphere.
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) ) .
34) 2 (3) From now on we will denote kl (T ) = kl (T ). 2) The boundary conditions we have already discussed; we have for the radial wave (D) (D) functions ul and Rl respectively: (D) ul (r) → r l+ D−1 2 (D) , Rl (r) → r l (r → 0). 36) with (1) kl (r ′′ , r ′ ; T ) = lim N→∞ m 2π i ǫ¯ h −∞ N × (D) µl ∞ ∞ N 2 [r(j) r(j−1) ] exp j=1 i dr(1) · · · ¯h N j=1 dr(N−1) −∞ m 2 ∆ r(j) − ǫV (|r(j) |) . 38) the second term is the so called “mirror” term. It is not allowed to drop it, even for very small T → 0, since this violates the boundary condition which demands that k0V =0 vanishes for r ′ , r ′′ → 0.