An introduction into the Feynman path integral by Grosche C.

By Grosche C.

During this lecture a brief creation is given into the idea of the Feynman course crucial in quantum mechanics. the final formula in Riemann areas might be given in accordance with the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the idea of space-time modifications and separation of variables can be defined. As effortless examples I speak about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb power.

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2 − ǫ2 ω (j−1) 2 −1 0 0 ... 2 The Harmonic Oscillator One can show that the following recursion relations holds: det B (j+1) = (2 − ǫ2 ω (j+1) 2 ) det B (j) − det B (j−1) with det B (1) = 2 − ǫ2 ω (1) 2 and det B (0) = 1. Let us define g (j) = ǫ det B (j) , then we have g (j+1) − 2g (j) + g (j−1) = −ǫ2 ω (j+1) 2 g (j) . Turning to a continuous notation we find for the function g(t) = f (t, t′ ) a differential equation: g¨(t) + ω 2 (t)g(t) = 0, with g(t′ ) = 0, g(t ˙ ′ ) = 1. 12) The two last equation follow from g(t′ ) = g0 = limǫ→0 ǫ det B (0) and g(t ˙ ′ ) = limǫ→0 g(t′ + ǫ) − g(t′ ) /ǫ = limǫ→0 (det B (1) − det B (0) ) = 1.

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.

5) which is the known result. The corresponding energy-dependent Green function is given by (D = 1): ∞ G(1) (x′′ , x′ ; E) = i K(x′′ , x′ ; T ) ei ET /¯h dT = − 0 m exp 2E i |x′′ − x′ | √ 2mE . 9) π/2z e−z . 2. The Harmonic Oscillator After this easy task we proceed to the path integral calculation of the harmonic oscillator. 2 The Harmonic Oscillator Here we assume that the various coefficients may be time-dependent. 2) x(t′ )=x′ x(t′ )=x′ In this quadratic Lagrangian, of course, an ordering problem appears.

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