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Canonical transformation

About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.


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TL;DR: In this article, the authors consider the problem of removing ordering ambiguity in position dependent mass quantum systems characterized by a generalized position-dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian.
Abstract: We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schr$o$dinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering we show that the class of systems can also be exactly solvable and are also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Li$e$nard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

20 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduce a canonical transformation of the fundamental single-mode field operators a and b that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states, and examine the structure and properties of the quantum states defined as eigenvectors of the transformed annihilation operator b.
Abstract: We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators ${X}_{1}$ and ${X}_{2}$ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function ${g}^{(2)}(0)$ and of the fourth-order correlation function ${g}^{(4)}(0)$ are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.

20 citations

Journal ArticleDOI
TL;DR: In this article, two types of canonical forms for certain classes of functional differential equations are introduced, which preserve oscillatory or non-oscillatory behaviour of solutions. And they are also suitable for studying both-side solutions of equivalent functional-differential equations.
Abstract: Functional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.

20 citations

Journal ArticleDOI
TL;DR: In this article, the bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse and Coulomb Schrodinger equations are reviewed in a unified way using the point canonical transformation method.
Abstract: The bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse, and the D-dimensional radial Coulomb Schrodinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrodinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a d-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.

20 citations

Journal ArticleDOI
TL;DR: In this paper, the authors examined the possibilities of improving the Wigner method by propagating the Weyl-Wigner function in an interaction picture, and showed that the classical interaction picture in a natural way comes into play.
Abstract: published in Advance ACS Abstracts, March 15, 1994. 0022-3654/94/2098-3212~04.50~0 successive propagations of the state vector within the Gaussian wavepacket method using the full Hamiltonian or a zeroth-order channel Hamiltonian, respectively. In the second approach introduced by Skodje'4 a canonical transformation of the classical coordinates into a set of classical interaction coordinates is performed, and the interaction Hamiltonian to be used in the Gaussian wavepacket method is found by applying the quanti- zation procedure for generalized canonical coordinates proposed by Heller.lz In this paper we examine the possibilities of improving the Wigner method by propagating the Wigner function in an interaction picture. It is organized in the following way: We begin by recalling the main features of the Weyl-Wigner representation of quantum mechanics. Then we give a short description of the quantum and the classical interaction pictures. We discuss subsequently the Wigner method in the interaction picture and show that the classical interaction picture, in a natural way, comes into play. Finally we present a numerical application and discuss the results. For simplicity all derivations are kept in one dimension, the extension to several dimensions being straightforward. To avoid confusion, quantum mechanical operators are written in capital letters (with the exception of the position and momentum operators) supplied with a hat, the corresponding phase space functions denoted by the same letter without the hat and classical functions are written in script.

20 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202218
202158
202042
201932
201829