By Grosche C.

During this lecture a quick creation is given into the idea of the Feynman course essential in quantum mechanics. the overall 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 changes and separation of variables should be defined. As straightforward examples I speak about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb strength.

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5. The Coulomb Potential The hydrogen atom is, of course, one of the most interesting subjects in quantum mechanics. In the beginnings of quantum mechanics and with the atom model of Rutherford it was a riddle how to tract a system which is from the classical physics point of view unstable and doomed to vanish into pure radiation. Its was Bohr’s genius, postulating the famous rules that only a countable number of orbits are allowed satisfying the quantum condition pdq = nh (n ∈ N). g. in the case of the helium-atom, and it was Pauli who solved the hydrogen problem in the terms of the “new” quantum mechanics developed by Schr¨odinger and Heisenberg.

0 0 . . . (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.

7) Np(k1 ,k2 ) = Γ(k1 + k2 − κ)Γ(−k1 + k2 + κ) Γ(2k2 ) 2π 2 1 2 × Γ(k1 + k2 + κ − 1)Γ(−k1 + k2 − κ + 1) , [κ = 21 (1 + ip)]. It is possible to state closed expressions for the (energy dependent) Green functions for the P¨oschl-Teller and modified P¨oschl-Teller potential, respectively. 8) √ with m1/2 = 12 (λ ± κ), LE = − 12 + 12 2mE /¯h and x′′ ≥ x′ . A similar expression is valid for the modified P¨oschl-Teller-potential Green function (see [65]) which is, however, omitted here.