Solvable Time-Dependent Models in Quantum Mechanics

Chuyi Chen

A nonperturbative procedure for solving the time-dependent Schrödinger equation, called the multi-projection approach or phase dynamics of quantum mechanics, is derived and illustrated. With the help of this new procedure, time-dependent quantum systems become generally solvable (if corresponding stationary systems are solvable) and the phase character of quantum mechanics reveals itself very clearly.

Sergio A Albeverio, Fritz Gesztesy, Raphael J. Høegh-Krohn +1 more

A. A. Suzko, Grigori Karlovich Giorgadze

Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space-time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed.

Sergio A Albeverio, Friedrich Gesztesy, Raphael J. Høegh-Krohn +1 more

A. Dhahbi, YASSINE CHARGUI, Adel Trablesi

The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.

A. A. Suzko