Form-Preserving Transformations for Hamiltonians with Linear Terms in the Momentum

Schulze-Halberg, Axel

doi:10.1007/s10702-005-1127-2

Cited by 2 publications

(1 citation statement)

References 23 publications

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“…The linear potential with a time-dependent coefficient has been extensively discussed, see for example Myers [115] (1960) and [116], also more recently [117][118][119][120][121][122][123]. In this case the time-dependent Hamiltonian (for d = 1) is…”

Section: Translationsmentioning

confidence: 99%

Exact time dependence of solutions to the time-dependent Schrödinger equation

Lohe¹

2008

J. Phys. A: Math. Theor.

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Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

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Section: Translationsmentioning

confidence: 99%

Exact time dependence of solutions to the time-dependent Schrödinger equation

Lohe¹

2008

J. Phys. A: Math. Theor.

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Effective Mass Hamiltonians With Linear Terms in the Momentum: Darboux Transformations and Form-Preserving Transformations

Schulze-Halberg

2007

Int. J. Mod. Phys. A

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We define form-preserving transformations and Darboux transformations for time-dependent, effective mass Hamiltonians with additional linear terms. We give reality conditions for both transformations, guaranteeing the transformed potential to be real-valued. We further show that our form-preserving transformation preserves normalizability of the Schrödinger wave function. Our results generalize all former results on form-preserving transformations and Darboux transformations for the time-dependent Schrödinger equation. This paper is a sequel of Refs. 16–18.

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Form-Preserving Transformations for Hamiltonians with Linear Terms in the Momentum

Cited by 2 publications

References 23 publications

Exact time dependence of solutions to the time-dependent Schrödinger equation

Exact time dependence of solutions to the time-dependent Schrödinger equation

Effective Mass Hamiltonians With Linear Terms in the Momentum: Darboux Transformations and Form-Preserving Transformations

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