Solution of the path integral for the H-atom

Duru, Ì. H.; Kleinert, H.

doi:10.1016/0370-2693(79)90280-6

Cited by 278 publications

(142 citation statements)

References 4 publications

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“…and perform a canonical transformation ξp ξ =p u ; √ ηp η =p v (17) to simplify the kinetic term in H in Eq. 15 as…”

Section: Green's Functions For the Pt-symmetric Non-central Potentialmentioning

confidence: 99%

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Green’s Function of a General PT-Symmetric Non-Hermitian Non-central Potential

Mourya

Mandal

2016

Springer Proceedings in Physics

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We study the path integral solution of a system of particle moving in certain class of PT symmetric non-Hermitian and non-central potential. The Hamiltonian of the system is converted to a separable Hamiltonian of Liouville type in parabolic coordinates and is further mapped into a Hamiltonian corresponding to two 2-dimensional simple harmonic oscillators (SHOs). Thus the explicit Green's functions for a general non-central PT symmetric non hermitian potential are calculated in terms of that of 2d SHOs. The entire spectrum for this three dimensional system is shown to be always real leading to the fact that the system remains in unbroken PT phase all the time .

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“…and perform a canonical transformation ξp ξ =p u ; √ ηp η =p v (17) to simplify the kinetic term in H in Eq. 15 as…”

Section: Green's Functions For the Pt-symmetric Non-central Potentialmentioning

confidence: 99%

“…[15,16] where energy spectrum has been calculated by solving Schroedinger equation using complicated KS transformation [17,18].…”

Section: Reality Of the Spectrummentioning

confidence: 99%

Green’s Function of a General PT-Symmetric Non-Hermitian Non-central Potential

Mourya

Mandal

2016

Springer Proceedings in Physics

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“…and let us apply the procedure of path reparametrization [29,30], performing the time transformation → defined by = 2 ( ) (16) or, in discrete form…”

Section: The Greenmentioning

confidence: 99%

“…The normalized wave functions are given by (29) for the azimuthal part, and can be deduced from expressions (30) and (42) for the polar and radial parts, , the potential (1) is reduced to the form…”

Section: First Case the Ring-shaped Oscillatormentioning

confidence: 99%

Path integral treatment of a noncentral electric potential

et al. 2013

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Abstract:We present a rigorous path integral treatment of a dynamical system in the axially symmetric potentialIt is shown that the Green's function can be calculated in spherical coordinate system for V (θ) = . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number ≥ 0, are recovered.

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“…The most important analytical methods that have been used in literature to solve these problems are supersymmetry (SUSY) [3,4], Nikiforov-Uvarov (NU) method [5], Pekeris approximation [6], variational method [7], hypervirial perturbation method [8], shifted 1/N expansion (SE) and the modified shifted 1 /N expansion (MSE) [9], exact quantization rule (EQR) [10], perturbative formalism [11][12][13], polynomial solution [14,15], wave function ansatz method [16,17], path integral method [18,19], Lie algebraic method [20][21][22][23], Fourier Grid Hamiltonian method [24], and asymptotic iteration method (AIM) [25,26] to solve the radial Schrödinger equation exactly.…”

Section: Introductionmentioning

confidence: 99%