Exact quantum theory of a time-dependent bound quadratic Hamiltonian system

Yeon, Kyu-Hwang; Lee, Kang Ku; Um, Chung-In; George, Thomas F.; Pandey, Lakshmi N.

doi:10.1103/physreva.48.2716

Cited by 64 publications

(76 citation statements)

References 18 publications

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“…The exact quantization of TDHS become possible using several techniques such as propagator method Yeon et al, 1993Yeon et al, , 1996, invariant operator method (Choi, 2003(Choi, , 2004Choi and Gweon, 2003;Um et al, 1997) and unitary(or canonical) transformation method (Choi et al, 2002;Ji and Kim, 1996;Landovitz et al, 1979;Zhang et al, 2001Zhang et al, , 2002a. We will use unitary transformation method in order to derive quantum mechanical solution of mesoscopic capacitance coupled circuit with a timedependent power source.…”

Section: Introductionmentioning

confidence: 99%

Thermal State for the Capacitance Coupled Mesoscopic Circuit with a Power Source

Choi

2007

Int J Theor Phys

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The Schrödinger equation of the mesoscopic capacitance coupled circuit with an arbitrary power source is solved by means of two step unitary transformation. The original Hamiltonian transformed to a very simple form by unitary operators so that it can be easily treated. We derived the exact full wave functions in Fock state. By making use of these wave functions and introducing the Lewis-Riesenfeld invariant operator, the thermal state have been constructed. The fluctuations of charges and currents are evaluated in thermal state. For T → 0, the uncertainty products between charges and currents in thermal state recovers exactly to that of Fock state with n, m = 0.

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Section: Introductionmentioning

confidence: 99%

Thermal State for the Capacitance Coupled Mesoscopic Circuit with a Power Source

Choi

2007

Int J Theor Phys

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“…The invariant operator can then be used to find exact wave functions of quantum states. During the past several years, this LR method has been widely used for the study of a general quadratic hamiltonian systems [7,8,9,10,11]. For the harmonic oscillator with time-dependent mass and frequency, the wave functions and the kernel (propagator) have been found [7,8,11].…”

Section: Introductionmentioning

confidence: 99%

Exact quantum states of a general time-dependent quadratic system from classical action

Song

1999

Phys. Rev. A

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A generalization of driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered.The generalization which gives a general quadratic Hamiltonian system does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous and one particular solutions. The kernels are then used to find wave functions which satisfy the Schrödinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization which gives a rather complicated quadratic Hamiltonian is simply interpreted as acting an unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.

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“…Quantum systems with quadratic Hamiltonians (see, for example, [2], [5], [6], [10], [14], [15], [18], [20], [26], [46], [64], [65], [66], [68] and the references therein) have attracted substantial attention over the years because of their great importance in many advanced quantum problems. Examples are coherent and squeezed states, uncertainty relations, Berry's phase, quantization of mechanical systems and Hamiltonian cosmology.…”

Section: Discussionmentioning

confidence: 99%

On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

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Abstract. We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.The purpose of this Letter is to elaborate on a "missing" class of solutions to the time-dependent Schrödinger equation for the simple harmonic oscillator in one dimension. We also provide an interesting computer-animated feature of these solutions -the phase space oscillations of the electron density and the corresponding probability distribution of the particle linear momentum. As a result, a dynamic visualization of the fundamental Heisenberg Uncertainty Principle [27] is given [45], [62]. Symmetry and Hidden SolutionsThe time-dependent Schrödinger equation for the simple harmonic oscillator,has the following six-parameter family of square integrable solutionswhere H n (x) are the Hermite polynomials [51] and On the other hand, the "dynamic harmonic oscillator states" (1.2)-(1.9) are eigenfunctions,of the time-dependent quadratic invariant,with the required operator identity [15], [54]:(1.12)Here, the time-dependent annihilation a (t) and creation a † (t) operators are explicitly given bywith p = i −1 ∂/∂x in terms of our solutions (1.4)-(1.9). These operators satisfy the canonical commutation relation, 14) and the oscillator-type spectrum (1.10) of the dynamic invariant E can be obtained in a standard way by using the Heisenberg-Weyl algebra of the rasing and lowering operators (a "second quantization" [1], [42], the Fock states): Here, ψ n (x, t) = e i(2n+1)γ(t) Ψ n (x, t) (1.16) is the relation to the wave functions (1.2) with ϕ n (t) = − (2n + 1) [68] and the references therein) have attracted substantial attention over the years because of their great importance in many advanced quantum problems. Examples are coherent and squeezed states, uncertainty relations, Berry's phase, quantization of mechanical systems and Hamiltonian cosmology. More applications include, but are not limited to charged particle traps and motion in uniform magnetic fields, molecular spectroscopy and polyatomic molecules in varying external fields, crystals through which an electron is passing and exciting the oscillator modes, and other mode interactions with external fields. Quadratic Hamiltonians have particular applications in quantum electrodynamics because the electromagnetic field can be represented as a set of generalized driven harmonic oscillators [13], [20].The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresp...

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Exact quantum theory of a time-dependent bound quadratic Hamiltonian system

Cited by 64 publications

References 18 publications

Thermal State for the Capacitance Coupled Mesoscopic Circuit with a Power Source

Thermal State for the Capacitance Coupled Mesoscopic Circuit with a Power Source

Exact quantum states of a general time-dependent quadratic system from classical action

On a hidden symmetry of quantum harmonic oscillators

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