An introduction into the Feynman path integral by Grosche C.

By Grosche C.

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

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6). 7) which casts the original Lagrangian into the form Z e2 . 9) with determinant g = det(gab ) = 16(ξ 2 + η 2 )2 and dV = dx1 dx2 = 4(ξ 2 + η 2 )dξdη. 10) are given by pξ = ¯ ∂ h ξ + 2 , i ∂ξ ξ + η 2 pη = ¯ ∂ h η + 2 . 5 we start by considering the Legendre transformed Hamiltonian HE = − which gives ¯ 2 ∂2 h ∂2 + 2m ∂x21 ∂x22 − ∂2 1 ∂2 ¯h2 ˆ + HE = − 2m 4(ξ 2 + η 2 ) ∂ξ 2 ∂η 2 Z e2 − E. r − Z e2 − E. 1 with the momentum operators and vanishing quantum potential ∆V : pξ = ¯ ∂ h , i ∂ξ pη = ¯ ∂ h , i ∂η ∆V = 0.

91) 4. Other Elementary Path Integrals There are two further path integral solutions based on the SU(2) [5, 25, 60] and SU(1, 1) [5, 61] group path integration, respectively. Γ(α + β + l + 1) = 2(α + β + 2l + 1) Γ(α + l + 1)Γ(β + l + 1) 1 1 2 1 × (sin x)α+ 2 (cos x)β+ 2 Pn(α,β) (cos 2x). Γ(α + β + l + 1) α + β + 2l + 1 = 2α+β+1 Γ(α + l + 1)Γ(β + l + 1) √ β α × 2 sin 2x (1 − cos 2x) 2 (1 + cos 2x) 2 Pn(α,β) (cos 2x). 2c) Here, of course we can analytically continue from integer values of m and n to, say, real numbers α and β, respectively .

83) is very important in numerous applications. Let us note the free particle case. 85) with wave-functions and energy spectrum Ψp (r) = r 2−D 2 √ p Jl+ D−2 (pr), 2 ¯ 2 p2 h . e. motion on the half-line) √ Ψp (r) = r p J 21 (pr) = 2 sin pr, πr Ep = ¯ 2 p2 h . 87) However, there is an ambiguity in the boundary condition for r = 0 for D = 1. e. Ψp (0) = 0, corresponds to a specific self adjoint extension of the Hamiltonian H = −¯h2 d2 /dx2 for functions Ψ ∈ L2 ([0, ∞)) on the half-line. Finally we calculate the energy dependent Green function for the radial harmonic oscillator.

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