Models of damped oscillators in quantum mechanics

Abstract

Abstract. We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. An IntroductionWe continue an investigation of the one-dimensional Schrödinger equations with variable quadratic Hamiltonians of the form where3) 5) and the function µ (t) satisfies the characteristic equation µ ′′ − τ (t) µ ′ + 4σ (t) µ = 0 (1.6)

Time reversal for modified oscillators

“…But d(0) = −λ = 0 in this case, and a more detailed analysis of the asymptotic behavior gives an additional antisymmetric term in the above propagator [56]. Hamiltonians (1.2) and (1.7) correspond to the cases a 1 = cos 2 t, b 1 = sin 2 t, c 1 = 2d 1 = sin 2t (3.17) and a 2 = cosh 2 t, b 2 = sinh 2 t, c 2 = 2d 2 = − sinh 2t.…”

Обращение Времени Для Модифицированных Осцилляторов

“…Наше условие (3.4) принимает вид тригонометрического тождества 14) которое подтверждает симметрию пропагатора. Для квантового осциллятора с трением [56] …”

“…Однако при этом d(0) = −λ ̸ = 0, и более подробный анализ асимптотик дает дополнительный антисимметричный член в приведенном выше пропагаторе [56].…”

“…Другим примером является осциллятор с трением и с переменными коэффици-ентами a = (ω 0 /2)e −2λt , b = (ω 0 /2)e 2λt , c = d = 0 [56]. Классическое уравнениё…”

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

“…A goal of this paper is to make a modest step in this direction (see also [32,53] and the references therein). We use explicit solutions from recent papers on variable quadratic Hamiltonians in nonrelativistic quantum mechanics [49,[54][55][56][57][58][59][60][61] to describe steady-state and transient solutions to linear cable equations derived for membrane compartments with a non-necessarily constant or monotonically changing radius and propose, en passage, a new hyperbolic representation for the neurite compartments.…”