Integration of the Schrödinger equation by canonical transformations

The computation of the one-loop effective action in a radially symmetric background can be reduced to a sum over partial-wave contributions, each of which is the logarithm of an appropriate one-dimensional radial determinant. While these individual radial determinants can be evaluated simply and efficiently using the Gel'fand-Yaglom method, the sum over all partial-wave contributions diverges. A renormalization procedure is needed to unambiguously define the finite renormalized effective action. Here we use a combination of the Schwinger proper-time method, and a resummed uniform DeWitt expansion. This provides a more elegant technique for extracting the large partial-wave contribution, compared to the higher-order radial WKB approach which had been used in previous work. We illustrate the general method with a complete analysis of the scalar one-loop effective action in a class of radially separable SU(2) Yang-Mills background fields. We also show that this method can be applied to the case where the background gauge fields have asymptotic limits appropriate to uniform field strengths, such as, for example, in the Minkowski solution, which describes an instanton immersed in a constant background. Detailed numerical results will be presented in a sequel.

show abstract

“…A particularly elegant method is the one utilizing quantum canonical transformations, as detailed in Refs. [40,41] …”

Section: Discussionmentioning

confidence: 99%

Renormalized effective actions in radially symmetric backgrounds: Partial wave cutoff method

2006

View full text Add to dashboard Cite

show abstract

“…The second is the method of quantum canonical transformation in which the Hamiltonian is reduced successively by a sequence of quantum canonical transformations to a one-variable function. From the simple solution of the one-variable Hamiltonian, the original solutions are obtained by the corresponding inverse sequence of the quantum canonical transformation for the wavefunction [4][5][6][7]. Both of these methods identify the same set of solvable models.…”

Section: Introductionmentioning

confidence: 99%

A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models

Tsaur

Wang

2013

Eur. J. Phys.

Self Cite

View full text Add to dashboard Cite

Solvable models of the Schrödinger equation are important models of quantum systems because they are idealistic approximations of real quantum systems and much insight into real quantum systems can be gained from the exact solutions of the solvable models. In this paper we show that a universal Laplace transform scheme can be used to solve the Schrödinger equations in closed form for all known solvable models. The work demonstrates how to apply the Laplace transform to differential equations with non-constant coefficients, which is useful in many branches of physics in addition to quantum mechanics. The advantages of the Laplace transform over the power expansion method and its connection with the methods of supersymmetry shape-invariant potentials and quantum canonical transformation, which also give closed-form solutions for solvable models, are elucidated.

show abstract

“…9 However they have not been often used in other time-dependent problems than a harmonic oscillator with time-dependent angular frequency. 3 The time-dependent harmonic oscillators have been widely studied by a variety of methods including the invariant operator, 10 the propagator, 11 and the time-space transformation.…”

Section: Application: Time-dependent Harmonic Oscillatormentioning

confidence: 99%

“…The quantum canonical transformations provide a unified approach to the integrability of many time-independent systems and exact solutions of the Schrödinger equation are obtained for systems including the harmonic oscillator, 4 the Morse potential, and the Pöschl-Teller potential, etc. 9 For time-dependent problems, however, a harmonic oscillator with time-dependent angular frequency is probably the only system solved by these transformations. 3 In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.…”

Section: Introductionmentioning

confidence: 99%

Canonical Transformations for Time-Dependent Harmonic Oscillators

2004

Bulletin of the Korean Chemical Society

View full text Add to dashboard Cite

A canonical transformation changes variables such as coordinates and momenta to new variables preserving either the Poisson bracket or the commutation relations depending on whether the problem is classical or quantal respectively. Classically canonical transformations are well established as a powerful tool for solving differential equations. Quantum canonical transformations have been defined and used relatively recently because of the non-commutativeness of the quantum variables. Three elementary canonical transformations and their composite transformations have quantum implementations. Quantum canonical transformations have been mostly used in time-independent Schrödinger equations and a harmonic oscillator with time-dependent angular frequency is probably the only time-dependent problem solved by these transformations. In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.

show abstract

Integration of the Schrödinger equation by canonical transformations

Cited by 9 publications

References 8 publications

Renormalized effective actions in radially symmetric backgrounds: Partial wave cutoff method

Renormalized effective actions in radially symmetric backgrounds: Partial wave cutoff method

A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models

Canonical Transformations for Time-Dependent Harmonic Oscillators

Contact Info

Product

Resources

About