By Grosche C.

During this lecture a brief creation is given into the speculation of the Feynman direction crucial in quantum mechanics. the overall formula in Riemann areas could be given in response to the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the idea of space-time alterations and separation of variables might be defined. As simple examples I talk about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb strength.

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**Extra resources for An introduction into the Feynman path integral**

**Example text**

J) .. .. .. B = 0 0 . . 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.

This we have already discussed in chapter II. The ordering ambiguity appears in the “b(t)”-term, where we have in the corresponding Hamiltonian a px-term. e. 3) as synonymous. : x(t) = xCl (t) + y(t), where y(t) denotes a fluctuating path about the classical one. The classical path obeys, of course, the Euler Lagrange equations: δL[xCl , x˙ Cl ] d ∂L[xCl , x˙ Cl ] ∂L[xCl , x˙ Cl ] = − = 0, δxCl dt ∂ x˙ Cl ∂xCl xCl (t′ ) = x′ , xCl (t′′ ) = x′′Cl . 4) Expanding we obtain for the action 1 δ 2 S[x] S[x] = S[xCl ] + 2 δx2 2 ′′ ′ t′′ y = S[xCl (t ), xCl (t )] + t′ x=xCl 35 m 2 c 2 y˙ − y + by y˙ dt.

29) Let us note that the most general solution for the general quadratic Lagrangian is due to Grosjean and Goovaerts [52, 53]. 39 Important Examples 3. 1. The General Radial Path Integral Radial path integrals have been first discussed by Edwards and Gulyaev [30] and Arthurs [4]. Edwards and Gulyaev discussed the two- and three-dimensional cases, Arthurs concentrated on D = 2. Further progress have been made by Peak and Inomata [84] who calculated the path integral for the radial harmonic oscillator including some simple applications.

### An introduction into the Feynman path integral by Grosche C.

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