In this paper, we construct a fractional time corresponding operator to derive the uniform fractional time Schrödinger equation, which implies the relationship between the fractionaloperatorandtheenergy. Weverifythreebasicprinciplesoftheuniformfractional time Schrödinger equation: (a) Unitarity of the time evolution operator; (b) Existence of stationary energy spectra; (c) Probability conservation law. We apply the uniform equation to study three typical particles: free particle, an infinite potential well and a rectangular barrier. This uniform equation brings some typical physical phenomena, including (1) non-symmetry basic wave functions; (2) continuous energy spectrum for a free particle and an infinite potential well; (3) The barrier penetration in the rectangular barrier. These phenomena were not found in previous researches and the standard quantum mechanics, which will attract further applications in various fields of science.
In this paper, we construct a fractional time corresponding operator to derive the uniform fractional time Schrödinger equation, which implies the relationship between the fractionaloperatorandtheenergy. Weverifythreebasicprinciplesoftheuniformfractional time Schrödinger equation: (a) Unitarity of the time evolution operator; (b) Existence of stationary energy spectra; (c) Probability conservation law. We apply the uniform equation to study three typical particles: free particle, an infinite potential well and a rectangular barrier. This uniform equation brings some typical physical phenomena, including (1) non-symmetry basic wave functions; (2) continuous energy spectrum for a free particle and an infinite potential well; (3) The barrier penetration in the rectangular barrier. These phenomena were not found in previous researches and the standard quantum mechanics, which will attract further applications in various fields of science.
A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schrödinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum mechanics, hermiticity of the Hamilton operator, parity conservation law and the current density. Applications of fractional quantum mechanics cover dynamics of a free particle, new representation for a free particle quantum mechanical kernel, infinite potential well, bound state in δ-potential well, linear potential, fractional Bohr atom and fractional oscillator. We also review fundamentals of the Lévy path integral approach to fractional statistical mechanics.
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.
The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order.The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [𝑀𝐿 2 /𝑇 2 ].The action 𝑆 is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced.The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order.The present work corrects a number of errors in Naber's work.The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained.The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity.A generalization to the case of a particle moving in a potential is also given.
This study investigates the time-fractional Schrödinger equation (TFSE) for a free particle using the Caputo fractional derivative. The model extends the classical Schrödinger equation by incorporating memory effects through a non-integer order time derivative. By applying the Fourier transform method, an explicit solution is derived in terms of the Mittag–Leffler function, which generalizes the classical exponential evolution. The analysis reveals that the resulting dynamics are non-unitary, leading to a time-dependent total probability that deviates from the conservation property of standard quantum mechanics. Numerical illustrations demonstrate the influence of the fractional order ν on wave packet evolution and probability behavior. The results highlight the role of fractional calculus in modeling non-Markovian quantum systems and provide insight into the transition from fractional to classical dynamics as v→1. .
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
Applications of Time-Fractional Schrodinger Equations (TFSEs) to quantum processes are instructive for understanding and describing the time behavior of real physical systems. By applying three popular TFSEs, namely Naber's TFSE I, Naber's TFSE II, and XGF's TFSE, to a basic open system model of a two-level system (qubit) coupled resonantly to a dissipative environment, we solve exactly for Time-Fractional Single Qubit Open Systems (TFSQOSs). However, the three TFSEs perform badly for the following reasons. On the other hand, in the respective frameworks of the three TFSEs, the total probability for obtaining the system in a single-qubit state is not equal to one with time at fractional order, implying that time-fractional quantum mechanics violates quantum mechanical probability conservation. On the other hand, the latter two TFSEs are not capable of describing the non-Markovian dynamics of the system at all fractional order, only at some fractional order. To address this, we introduce a well-performed TFSE by constructing a new analytic continuation of time combined with the conformable fractional derivative, in which for all fractional order, not only does the total probability for the system equal one at all times but also the non-Markovian features can be observed throughout the time evolution of the system. Furthermore, we study the performances of the four TFSEs applying to an open system model of two isolated qubits each locally interacting with its dissipative environment. By deriving the exact solutions for time-fractional two qubits open systems, we show that our TFSE still possesses the above two advantages compared with the other three TFSEs.