Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian

2017

Annals of Physics

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Some general properties of the wave functions of complex-valued potentials with real spectrum are studied. The main results are presented in a series of lemmas, corollaries and theorems that are satisfied by the zeros of the real and imaginary parts of the wave functions on the real line. In particular, it is shown that such zeros interlace so that the corresponding probability densities ρ(x) are never null. We find that the profile of the imaginary part V I (x) of a given complex-valued potential determines the number and distribution of the maxima and minima of the related probability densities. Our conjecture is that V I (x) must be continuous in R, and that its integral over all the real line must be equal to zero in order to get control on the distribution of the maxima and minima of ρ(x). The applicability of these results is shown by solving the eigenvalue equation of different complex potentials, these last being either PT -symmetric or not invariant under the PT -transformation.

“…These last properties facilitate the interpretation of V I (x) as a rightful perturbation (for |λ| << 1) which would be associated with dissipation (see e.g. [39,40]).…”

Section: Pt -Symmetric Potentialsmentioning

confidence: 86%

Section: Pt -Symmetric Potentialsmentioning

confidence: 99%

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Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum

Jaimes-Nájera

2017

Annals of Physics

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“…and x 2 =η(t) the (dimensionless) classical momentum [167][168][169][170]. The time-dependent coefficient of the quadratic term obeys the Riccati equatioṅ…”

Section: Time-dependent Oscillator Wave Packetsmentioning

confidence: 99%

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

Integrability, Supersymmetry and Coherent States

2019

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A short review of the main properties of coherent and squeezed states is given in introductory form. The efforts are addressed to clarify concepts and notions, including some passages of the history of science, with the aim of facilitating the subject for nonspecialists. In this sense, the present work is intended to be complementary to other papers of the same nature and subject in current circulation.

“…We look for a real-valued function e − 1 2 χ 2 F (χ) that is a solution of the eigenvalue equation (36), with no zeros in R × [t 0 , ∞). Let us consider the general solution…”

Section: Nonstationary Oscillatorsmentioning

confidence: 99%

Coherent states for exactly solvable time-dependent oscillators generated by Darboux transformations

Cruz¹,

Razo²,

et al. 2020

Phys. Scr.

The Darboux method is commonly used in the coordinate variable to produce new exactly solvable (stationary) potentials in quantum mechanics. In this work we follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to include the time-variable as a parameter of the transformation. The new potentials are nonstationary and define Hamiltonians which are not integrals of motion for the system under study. We take the stationary oscillator of constant frequency to produce nonstationary oscillators, and also provide an invariant that serves to define uniquely the state of the system. In this sense our approach completes the program of the BSS method since the eigenfunctions of the invariant are an orthonormal basis for the space of solutions of the related Schrödinger equation. The orthonormality holds when the involved functions are evaluated at the same time. The dynamical algebra of the nonstationary oscillators is generated by properly chosen ladder operators and coincides with the Heisenberg algebra. We also construct the related coherent states and show that they form an overcomplete set that minimizes the quadratures defined by the ladder operators. These states are not invariant under time-evolution since their time-dependence relies on the basis of states and not on the complex eigenvalue that labels them. Some concrete examples are provided.