Download e-book for iPad: An introduction into the Feynman path integral by Grosche C.

By Grosche C.

During this lecture a brief advent is given into the speculation of the Feynman course critical in quantum mechanics. the overall formula in Riemann areas could be given in line with the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the speculation of space-time ameliorations and separation of variables could be defined. As common examples I speak about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb capability.

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1. The 1/r-Potential in R2 [28, 57] We consider the Euclidean two-dimensional space with the singular potential V (r) = −Z e2 /r (r = |x|, x ∈ R2 ). Here, e2 denotes the square of an electric charged Z its multiplicity (including sign). However, as already noted, this potential is not the potential of a point charge in R2 . The classical Lagrangian now has the form L(x, x) ˙ = m 2 Z e2 x˙ + . 5) Of course, bound and continuous states can exist, depending on the sign of Z. The path integral is given by x(t′′ )=x′′ ′′ i Dx(t) exp ¯h ′ K(x , x ; T ) = x(t′ )=x′ 61 t′′ t′ m 2 Z e2 dt .

The indices i and k will be omitted in the following. 3) where the lattice interpretation reads ǫ/[f (z (j−1) )f (z (j) )] = δ (j) ≡ δ. Of course, we identify z(t) ≡ z[s(t)] and x(t) ≡ x[s(t)]. 6) with the remaining path integration z(s′′ )=z ′′ G(z) ˆ ′′ , z ′ ; s′′ ) = K(z f 2−d (z) Dz(s) z(0)=z ′ × exp i ¯h s′′ 0 m g 2 (z) 2 z˙ − f 2 (z) W (z) + ∆W (z) + f 2 (z)E ds . 2). Note the difference in comparison with a combined space-time transformation where a factor [f (z ′ )f (z ′′ )]1/4 would instead appear.

That such a procedure is actually legitimate is beyond the scope of these notes. It was justified by Fischer, Leschke and M¨ uller [35] for the radial path integral and for the P¨oschl-Teller path integral as well (see below), where one also has to be careful with the appropriate Besselian functional measure. 83) can be found in references [49] and [92]. 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 .

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An introduction into the Feynman path integral by Grosche C.


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