Solution of the multidimensional quantum harmonic oscillator with time-dependent frequencies through Fourier, Hermite and Wigner transforms

Burgan, J. R.; Feix, Marc; Fijalkow, E.; Munier, A.

doi:10.1016/0375-9601(79)90567-x

Cited by 49 publications

(16 citation statements)

References 6 publications

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“…This idea of time-scaling the coordinates is not new and has been widely exploited in many different fields of physics. In 1979, Burgan et al [15] studied the Schrödinger equation for a multidimensional quantum harmonic oscillator with time-dependent frequencies. By introducing an appropriate time-dependent scaling of the spatial coordinates, they were able to transform the problem to a free-particle motion and to derive an exact analytical solution.…”

Section: Introductionmentioning

confidence: 99%

Time scaling with efficient time-propagation techniques for atoms and molecules in pulsed radiation fields

et al. 2011

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We present an ab initio approach to solving the time-dependent Schrödinger equation to treat electron-and photon-impact multiple ionization of atoms or molecules. It combines the already known time-scaled coordinate method with a high-order time propagator based on a predictor-corrector scheme. In order to exploit in an optimal way the main advantage of the time-scaled coordinate method, namely, that the scaled wave packet stays confined and evolves smoothly toward a stationary state, of which the squared modulus is directly proportional to the electron energy spectra in each ionization channel, we show that the scaled bound states should be subtracted from the total scaled wave packet. In addition, our detailed investigations suggest that multiresolution techniques like, for instance, wavelets are the most appropriate ones to represent the scaled wave packet spatially. The approach is illustrated in the case of the interaction of a one-dimensional model atom as well as atomic hydrogen with a strong oscillating field.

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Section: Introductionmentioning

confidence: 99%

Time scaling with efficient time-propagation techniques for atoms and molecules in pulsed radiation fields

et al. 2011

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“…The problem of the harmonic oscillator with time-dependent frequency has been studied extensively in Refs. [41][42][43], which is reviewed in Appendix D. We consider a UV theory obtained by adding the single-trace and double-trace couplings to the reference theory S 0 in a translationally invariant way. In this case, the initial wavefunction is a Gaussian product state in the K-space.…”

Section: The Rg Hamiltonianmentioning

confidence: 99%

Constraints on beta functions in field theories

Lee

2022

SciPost Phys.

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The \betaβ-functions describe how couplings run under the renormalization group flow in field theories. In general, all couplings that respect the symmetry and locality are generated under the renormalization group flow, and the exact renormalization group flow is characterized by the \betaβ-functions defined in the infinite dimensional space of couplings. In this paper, we show that the renormalization group flow is highly constrained so that the \betaβ-functions defined in a measure zero subspace of couplings completely determine the \betaβ-functions in the entire space of couplings. We provide a quantum renormalization group-based algorithm for reconstructing the full \betaβ-functions from the \betaβ-functions defined in the subspace. As examples, we derive the full \betaβ-functions for the O(N)O(N) vector model and the O_L(N) \times O_R(N)OL(N)×OR(N) matrix model entirely from the \betaβ-functions defined in the subspace of single-trace couplings.

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“…The transformation method has been extensively used in order to obtain exact solutions of the Schrödinger equation [18]. This method involves an appropriate rescaling of the space and time variables in the Schrödinger equation as well as a unitary transformation of the wavefunction [8].…”

Section: Ermakov-lewis Invariantmentioning

confidence: 99%

Solution of the Schr dinger equation for time-dependent 1D harmonic oscillators using the orthogonal functions invariant

Fernández‐Guasti¹,

Moya‐Cessa²

2003

J. Phys. A: Math. Gen.

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An extension of the classical orthogonal functions invariant to the quantum domain is presented. This invariant is expressed in terms of the Hamiltonian. Unitary transformations which involve the auxiliary function of this quantum invariant are used to solve the time-dependent Schrödinger equation for a harmonic oscillator with time-dependent parameter. The solution thus obtained is in agreement with the results derived using other methods which invoke the Lewis invariant in their procedures.

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Solution of the multidimensional quantum harmonic oscillator with time-dependent frequencies through Fourier, Hermite and Wigner transforms

Cited by 49 publications

References 6 publications

Time scaling with efficient time-propagation techniques for atoms and molecules in pulsed radiation fields

Time scaling with efficient time-propagation techniques for atoms and molecules in pulsed radiation fields

Constraints on beta functions in field theories

Solution of the Schr dinger equation for time-dependent 1D harmonic oscillators using the orthogonal functions invariant

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