Time fractional quantum mechanics

Abstract

A time fractional quantum framework has been introduced into quantum mechanics. A new version of the space-time fractional Schrödinger equation has been launched. The introduced space-time fractional Schrödinger equation has a new scale parameter, which is a fractional generalization of Planck's constant in quantum physics.It has been shown that the presence of a time fractional time derivative in the space-time fractional Schrödinger equation significantly impacts quantum mechanical fundamentals.Time fractional quantum mechanical operators of coordinate, momentum and angular momentum were defined and their commutation relationships were established. The pseudo-Hamilton operator was introduced and its Hermicity has been proven.Two new functions related to the Mittag-Leffler function have been introduced to solve the space-time fractional Schrödinger equation. Energy of a time fractional quantum system has been defined and calculated in terms of the newly introduced functions. It has been found that in the framework of time fractional quantum mechanics there are no stationary states, and the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time fractional quantum system.A free particle solution to the space-time fractional Schrödinger equation was found. A free particle space-time fractional quantum mechanical kernel has been found and expressed in terms of the H-function. Renormalization properties of a free particle solution and the space-time fractional quantum kernel were established.Some particular cases of time fractional quantum mechanics have been analyzed and discussed. PACS number(s): 05.40.Fb, 03.65.Sq

Comb Model: Non-Markovian versus Markovian

“…Therefore, this comb dynamics determines a quantum friction, which leads to the FTSE. This issue has been the focus of extensive studies, reflected in recent reviews [16,33,34].…”

Dispersive estimates for time and space fractional Schrödinger equations

“…4 In physics, fractional calculus are introduced to study the quantum phenomena, see other studies. [5][6][7][8] It is well known that the Schrödinger equation is the most important equations in the quantum mechanics, and the well-posedness of this equation has attracted the attention of thousands of researches, see previous studies [9][10][11] for more information. There have been three main fractional generalizations of the Schrödiger equation: (1) space fractional Schrödinger equation (SFSE), see other works 5,6,12 ; (2) time fractional Schrödinger equation (TFSE), see previous studies [13][14][15] ; (3) and both time and space fractional Schrödinger equation (TSFSE), see previous works.…”

Two‐sided and regularised Riesz‐Feller derivatives

“…The Riesz-Feller potentials are very useful in several physical applications as it is the case of diffusion [8][9][10][11][12] and quantum mechanics. [13][14][15][16][17][18][19][20] In these fields, partial differential equations are used to describe the underlying phenomena by using Riemann-Liouville (RL) and mainly Caputo (C) derivatives for the time derivative and the Riesz-Feller potential for space derivative. In the general fractional time-space derivatives case, the solution is expresed in terms of Mellin-Barnes integrals.…”