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A. K. Kapoor

Publications and source records attributed to A. K. Kapoor.

29 records · Page 2Linked to original sources
arXiv2004

Periodic Quasi - Exactly Solvable Models

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lam{é} potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the Riccati type quantum Hamilton-Jacobi equation is primarily responsible for the surprisingly large number of allowed solvability conditions in the associated Lam{é} case. We also study the singularity structure of the quantum momentum function, which yields the band edge eigenvalues and eigenfunctions.

quant-ph↗
arXiv2004

Quantum Hamilton-Jacobi analysis of PT symmetric Hamiltonians

We apply the quantum Hamilton-Jacobi formalism, naturally defined in the complex domain, to a number of complex Hamiltonians, characterized by discrete parity and time reversal (PT) symmetries and obtain their eigenvalues and eigenfunctions. Examples of both quasi-exactly and exactly solvable potentials are analyzed and the subtle differences, in the singularity structures of their quantum momentum functions, are pointed out. The role of the PT symmetry in the complex domain is also illustrated.

quant-ph↗
arXiv2003

Calculation of Band Edge Eigenfunctions and Eigenvalues of Periodic Potentials through the Quantum Hamilton - Jacobi Formalism

We obtain the band edge eigenfunctions and the eigenvalues of solvable periodic potentials using the quantum Hamilton - Jacobi formalism. The potentials studied here are the Lam{é} and the associated Lam{é} which belong to the class of elliptic potentials. The formalism requires an assumption about the singularity structure of the quantum momentum function $p$, which satisfies the Riccati type quantum Hamilton - Jacobi equation, $ p^{2} -i \hbar \frac{d}{dx}p = 2m(E- V(x))$ in the complex $x$ plane. Essential use is made of suitable conformal transformations, which leads to the eigenvalues and the eigenfunctions corresponding to the band edges in a simple and straightforward manner. Our study reveals interesting features about the singularity structure of $p$, responsible in yielding the band edge eigenfunctions and eigenvalues.

quant-ph↗
arXiv1998

Hamiltonian path integral quantization in polar coordinates

Using a scheme proposed earlier we set up Hamiltonian path integral quantization for a particle in two dimensions in plane polar coordinates.This scheme uses the classical Hamiltonian, without any $O(\hbar^2)$ terms, in the polar varivables. We show that the propagator satisfies the correct Schrödinger equation.

quant-ph↗
arXiv1998

A new orthogonalization procedure with an extremal property

Various methods of constructing an orthonomal set out of a given set of linearly independent vectors are discussed. Particular attention is paid to the Gram-Schmidt and the Schweinler-Wigner orthogonalization procedures. A new orthogonalization procedure which, like the Schweinler- Wigner procedure, is democratic and is endowed with an extremal property is suggested.

quant-ph↗
arXiv1996

Quantum Hamilton-Jacobi formalism and the bound state spectra

It is well known in classical mechanics that, the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. The equivalent quantum action variable appearing in the quantum Hamilton-Jacobi formalism, can, analogously provide the energy eigenvalues of a bound state problem, without having to solve the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well known solvable potentials. It is also shown, how this method provides an understanding, as to why approximate quantization schemes such as ordinary and supersymmetric WKB, can give exact answers for certain potentials.

quant-ph↗
arXiv1995

On Exactness Of The Supersymmetric WKB Approximation Scheme

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-ω^2(x)} dx = n \hbar π$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2πn \hbar$, originating from the quantum Hamilton-Jacobi formalism reveals that, the locations and the residues of the poles that contribute to these integrals match identically, for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactness of the SWKB for these potentials is accounted for. Three non-exact cases are also analysed; the origin of this non-exactness is shown to be due the presence of additional singularities in $\sqrt{E-ω^2(x)}$, like branch cuts in the $x-$plane.

quant-ph↗
arXiv1995

HAMILTONIAN PATH INTEGRAL QUANTIZATION IN ARBITRARY CO-ORDINATES AND EXACT PATH INTEGRATION

We briefly review a hamiltonian path integral formalism developed earlier by one of us. An important feature of this formalism is that the path integral quantization in arbitrary co-ordinates is set up making use of only classical hamiltonian without addition of adhoc $\hbar^2$ terms. In this paper we use this hamiltonian formalism and show how exact path integration may be done for several potentials.

hep-th↗
arXiv1995

Local Scaling of Time in Hamiltonian Path Integration

Inspired by the usefulness of local scaling of time in the path integral formalism, we introduce a new kind of hamiltonian path integral in this paper. A special case of this new type of path integral has been earlier found useful in formulating a scheme of hamiltonian path integral quantization in arbitrary coordinates. This scheme has the unique feature that quantization in arbitrary co-ordinates requires hamiltonian path integral to be set up in terms of the classical hamiltonian only, without addition of any adhoc $ O(\hbar ^2) $terms. In this paper we further study the properties of hamiltonian path integrals in arbitrary co-ordinates with and without local scaling of time and obtain the Schrodinger equation implied by the hamiltonian path integrals. As a simple illustrative example of quantization in arbitrary coordinates and of exact path integration we apply the results obtained to the case of Coulomb problem in two dimensions.

hep-th↗