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

By Grosche C.

During this lecture a quick creation is given into the speculation of the Feynman direction quintessential in quantum mechanics. the final formula in Riemann areas should be given in accordance with the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the idea of space-time variations and separation of variables could be defined. As straightforward examples I talk about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb power.

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Furthermore we have set α = m/ǫ¯ h 2 2 (j) and β(j) = α[1 − ǫ mω (t )/2]. 59) γ| < π/4 and α, β > 0. 60) is valid for ν > −1 and ℜ(α) > 0. 60) we obtain for KlN (T ): KlN (T ) = α i N 2 exp i β ′2 2 (r + r ′ ) 2 ∞ 0 ∞ r(1) dr(1) · · · 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) ) . . 61) Important Examples where the coefficients αN , pN and qN are given by N−1 αN = α k=1 pN qN α = − 2                       α 2γk N−1 k=1 α2k 4γk α2 α = − 2 4γN−1                      k α1 = α, αk+1 = α γ1 = β1 , α 2γk j=1 γk+1 = βk+1 − (k ≥ 1) α2 .

42) 2 h ¯ Rewriting the Hamiltonian H = − 2m ∆LB yields: p2r 2m 1 1 + p2θ2 + . . 43) 2 2 2 sin θ1 sin θ1 . . sin θD−2 H(pr , r, {pθ , θ}) = + 1 p2 2mr 2 θ1 with ∆VW eyl (r, {θ}) = − 1 ¯2 h 1 +... + . 1+ 2 2 2 8mr sin θ1 sin θ1 . . sin2 θD−2 and the hermitean momenta ¯ ∂ h D−1 pr = + i ∂r 2r D−1−ν ¯h ∂ + cot θν pθν = i ∂θν 2 ¯h ∂ pφ = . e. we have ∆VW eyl = ∆Vprod . 46) where (j) LN , {θ (j) }, r (j−1) , {θ (j−1)}) Cl (r m (j) (j−1) 2 (j) (j) (j−1) 2 ) + sin2 θ1 (θ2 − θ2 ) +... = 2 (r (j) − r (j−1) )2 + r (j) 2 (θ1 − θ1 2ǫ (j) (j−1) .

60) we obtain for KlN (T ): KlN (T ) = α i N 2 exp i β ′2 2 (r + r ′ ) 2 ∞ 0 ∞ r(1) dr(1) · · · 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) ) . . 61) Important Examples where the coefficients αN , pN and qN are given by N−1 αN = α k=1 pN qN α = − 2                       α 2γk N−1 k=1 α2k 4γk α2 α = − 2 4γN−1                      k α1 = α, αk+1 = α γ1 = β1 , α 2γk j=1 γk+1 = βk+1 − (k ≥ 1) α2 .

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