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TURKISH JOURNAL OF PHYSICS

Path Integral Over Single-Sheeted Hyperboloid and Related Potentials

H. Ahmedov

Published Jan 1, 1997• 0 citations
Publisher

Abstract

Path integral over single-sheeted hyperboloid is evaluated. To relate the obtained propagator to quantum system we choose the part of it which is invariant with respect to reflection. This part corresponds to the propagator of the free particle motion on the invariant subspace of M- Imaginary Lobachevsky space. The wave functions and spectra for the quantum systems with potentials -\frac{\nu^{2}+\frac{1}{4}}{\sin h^{2}x} and -\frac{\nu^{2}+\frac{1}{4}}{\sin^{2}x} is given. Note that these potentials are not the special case of Rosen-Morse or the modified P\"{o}schl-Teller ones

DOI
10.55730/1300-0101.2357

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Path integral over single-sheeted hyperboloid is evaluated. To relate the obtained propagator to quantum system we choose the part of it which is invariant with respect to reflection. This part corresponds to the propagator of the free particle motion on the invariant subspace of M- Imaginary Lobachevsky space. The wave functions and spectra for the quantum systems with potentials -\frac{\nu^{2}+\frac{1}{4}}{\sin h^{2}x} and -\frac{\nu^{2}+\frac{1}{4}}{\sin^{2}x} is given. Note that these potentials are not the special case of Rosen-Morse or the modified P"{o}schl-Teller ones
In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.