Unitary relation for the time-dependent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">SU</mml:mi><mml:mn /><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mn /></mml:math>systems

Song, Donghoon

doi:10.1103/physreva.68.012108

Cited by 9 publications

(3 citation statements)

References 23 publications

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“…The time-dependent harmonic oscillator, for which the dynamical algebra is su(1, 1), is one example of this approach, as followed by Lewis and Riesenfeld [3]. More directly, one can express the time evolution operator U as a general element of the group corresponding to A, with time-dependent parameters, and then use (2) to solve for these parameters, see [22][23][24][25][26][27][28][29][30][31].…”

Section: Dynamical Invariants and Unitary Transformationsmentioning

confidence: 99%

“…As observed in section 4.4 these states are eigenfunctions of K 0 ζ as defined by (61) and can be determined by performing the ζtransformation (28) on the dilatation function ρ which satisfies the Ermakov equation (25). The special solution ρ in (101), which satisfies the Ermakov equation ρ 3 ρ = ω 2 0 with the initial conditions ρ(t 0 ) = 1, ρ(t 0 ) = 0, leads by means of (28) to the following general solution ρ ζ : (31) and may be calculated using (30). An alternative parametrization of the general solution ρ ζ in terms of real parameters a, b is shown in (19) and leads to simpler expressions.…”

Section: Translationsmentioning

confidence: 99%

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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: Dynamical Invariants and Unitary Transformationsmentioning

confidence: 99%

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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“…In recent years, the study of Hamiltonian with explicitly time-dependent coefficients becomes very popular. [1][2][3][4][5][6][7][8][9][10][11] The mathematical challenge and important applications in various areas of physics, such as quantum optics, 12 cosmology, 13 and nanotechnology, 14 are the main reasons for intensive studies. The most common problem in this area is the harmonic oscillator with time-dependent frequency and/or mass.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Pepore¹,

Winotai²,

Osotchan³

et al. 2006

ScienceAsia

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The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.

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Metaplectic operator approach to a time-dependent generalized harmonic oscillator

Lee

2021

J. Korean Phys. Soc.

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The metaplectic operator is a unitary operator corresponding to a time-dependent classical linear canonical transformation. We present the explicit expression for the metaplectic operator. The parameter in the metaplectic operator is expressed in terms of the solution of a classical time-dependent generalized harmonic oscillator. The time evolution operator, Lewis and Riesenfeld invariant, and the wave function of the time-dependent generalized harmonic oscillator are studied using the metaplectic operator, and the results are compared with those from other studies. The metaplectic operator method is a simpler method for studying the time-dependent generalized harmonic oscillator.

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Unitary relation for the time-dependentSU(1,1)systems

Cited by 9 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

Untitled

Metaplectic operator approach to a time-dependent generalized harmonic oscillator

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