Path integral of the hydrogen atom, Jacobi's principle of least action and one-dimensional quantum gravity

Fujikawa, Kazuo

doi:10.1016/s0550-3213(96)00584-6

Cited by 11 publications

(15 citation statements)

References 47 publications

(37 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1.1 has been reduced to a Hamiltonian of two 2D harmonic oscillators in parabolic coordinates using some simple trick, mentioned in Ref. [11].…”

Section: Discussionmentioning

confidence: 99%

Path Integral Solution of Noncentral Potential

Mandal

2000

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

We have studied the path integral solution of a system of particle moving in certain class of non-central potential without using Kustannheimo-Stiefel transformation. The Hamiltonian of the system has been converted to a separable Hamiltonian of Liouville type in parabolic coordinates and has further reduced to a Hamiltonian corresponding to two 2-dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. Hartmann ring-shaped potential and compound Coulomb plus Aharanov-Bohm potential have also been studied as special cases.

show abstract

“…1.1 has been reduced to a Hamiltonian of two 2D harmonic oscillators in parabolic coordinates using some simple trick, mentioned in Ref. [11].…”

Section: Discussionmentioning

confidence: 99%

Path Integral Solution of Noncentral Potential

Mandal

2000

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

show abstract

“…This is reduced to a simpler form by using some canonical transformation and redefining U 's and V 's as [12] …”

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

confidence: 99%

“…25, it is nothing but the Green's functions of the operator 1 H−E as discussed at beginning of this section. And the integration in the RHS can be done in a straightforward manner [12]. Thus we obtain the explicit Green's functions for the system of non central non-Hermitian potential.…”

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

confidence: 99%

“…In this article we consider a general and very important non-central PT symmetric non-Hermitian system in the PI formulation. First we show that how the Hamiltonian corresponding to this potential is reduced to a separable Hamiltonian of Liouville type [12] in a different coordinate system. This further enables us to map the system into two non-interacting 2d harmonic oscillators with the appropriate choice of coordinates.…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Mourya

Mandal

2016

Springer Proceedings in Physics

View full text Add to dashboard Cite

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 .

show abstract

“…Details of the Coulomb path integral and applications of the Duru-Kleinert(DK) method can be found in the comprehensive textbook 13 by Kleinert. Consideration from gauge invariance viewpoint on the DK method was first given by Fujikawa to make it clear that the essence of the DK transformation can be viewed as the special choice of the gauge fixing condition for a system with invariance under reparametrization of time 14,15 . An application of Fujikawa's approach, which is based on the view point of the Jacobi's principle of least action, was found by the present author in formulating an exactly solvable path integral of one-dimensional Coulomb system 16 which does not posses any oscillator coordinates to be transformed to.…”

Section: Introductionmentioning

confidence: 99%

On the effective potential of Duru-Kleinert path integrals

Sakoda

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

We propose a new method to evaluate the effective potential in the path integral for the fixed-energy amplitude as well as for the pseudotime evolution kernel in the formalism by Duru and Kleinert. Restriction to the postpoint or the prepoint prescriptions in formulating time sliced path integrals is avoided by leaving off the use of expectation values for correction terms. This enables us to consider an arbitrary ordering prescription and to examine the ordering dependence of the effective potential. To investigate parameter dependences, we introduce the ordering parameter $\alpha$ in addition to the splitting parameter $\lambda$ in the formulation of the time sliced path integral. The resulting path integrals are found to be independent of the ordering parameter although the explicit dependence, given by a contribution proportional to $(1-2\lambda)^{2}$, on the splitting parameter remains. As an application, we check the relationship between path integrals for the radial oscillator and the radial Coulomb system in arbitrary dimensions

show abstract

Path integral of the hydrogen atom, Jacobi's principle of least action and one-dimensional quantum gravity

Cited by 11 publications

References 47 publications

Path Integral Solution of Noncentral Potential

Path Integral Solution of Noncentral Potential

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

On the effective potential of Duru-Kleinert path integrals

Contact Info

Product

Resources

About