Canonical transformation

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

Exactly solvable schrödinger equation for a class of multiparameter exponential-type potentials

Abstract: The solution to a spectral problem involving the Schrodinger equation for a particular class of multiparameter exponential-type potentials is presented. The proposal is based on the canonical transformation method applied to a general second-order differential equation, multiplied by a function g(x), to convert it into a Schrodinger-like equation. The treatment of multiparameter exponential-type potentials comes from the application of the transformed results to the hypergeometric equation under the assumption of a specific g(x). Besides presenting the explicit solutions and their spectral values, it is shown that the problem considered in this article unifies and generalizes several former studies. That is, the proposed exactly solvable multiparameter exponential-type potential can be straightforwardly applied to particular exponential potentials depending on the choice of the involved parameters as exemplified for the Hulthen potential and their isospectral partner. Moreover, depending on the function g(x), the proposal can be extended to find different exactly solvable potentials as well as to generate new potentials that could be useful in quantum chemical calculations. © 2011

Exactly solvable effective mass D-dimensional Schrodinger equation for pseudoharmonic and modified Kratzer problems

Abstract: We employ the point canonical transformation (PCT) to solve the D-dimensional Schr\"{o}dinger equation with position-dependent effective mass (PDEM) function for two molecular pseudoharmonic and modified Kratzer (Mie-type) potentials. In mapping the transformed exactly solvable D-dimensional ($D\geq 2$) Schr\"{o}dinger equation with constant mass into the effective mass equation by employing a proper transformation, the exact bound state solutions including the energy eigenvalues and corresponding wave functions are derived. The well-known pseudoharmonic and modified Kratzer exact eigenstates of various dimensionality is manifested.

Path-integral solution for pseudoharmonic potential.

Abstract: We obtained the path-integral solution of a three-dimensional pseudoharmonic potential by applying a suitably chosen point canonical transformation.

Nonbijective canonical transformations and their representation in quantum mechanics

Abstract: In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one‐to‐one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one‐dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x, p) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group Cκ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group Cκ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single compo...

On the one-dimensional compressible Ising model

Abstract: A study of the one-dimensional compressible Ising model, in zero field, with quadratic potential energy and linear exchange parameter, is presented, using a canonical transformation of the hamiltonian. Earlier solutions are corrected and generalized, the role of the four-spin interaction in different ensembles being made explicit. The results agree with the solutions of the Baker-Essam model.