Quantization method and Schrödinger equation of fractional time and their weak effects on Hamiltonian: Phase transitions of energy and wave functions

Highlights

• A modeling method to describe quantum systems with fractional time is consistent with fundamental quantum physics laws.

• The fractional time quantization methods and fractional time Schrödinger equation are derived.

• The distinct nonlinear phenomena are found in fractional time quantum system: the phase transitions of energy and wave functions.

• Bound states in continuum (BICs) phenomena firstly found in fractional time quantum mechanics.

Abstract

Fractional time quantum mechanics (FTQM) is a method of describing the time evolution of quantum dynamics based on fractional derivatives. For any potential, we obtain the modeling method to describe quantum systems with fractional time consistent with fundamental quantum physics laws, which makes fractional time effects naturally enter quantum mechanics. The method is only using the start states, not based on usually directly replacing the integer derivatives by fractional ones or parallel introductions of standard models. In the process, we solve three open problems perplexing the studies on FTQM: What is the essential quantization method? How does one represent the fractional time Hamiltonian while retaining physical significance? How does one avoid the violations of current models of FTQM for many fundamental quantum physics laws? Then, a FTQM framework is built by amalgamating two quantization methods under a unified foundation of fractional time. The framework contains the quantization method, the Hamiltonian, the Hamilton operator, the Schrödinger equation, the energy correspondence relation, the Bohr correspondence principle and the time-energy uncertainty relation. And the effects of fractional time are revealed: containing historical information of particle’s motions; representing weak actions of Hamilton operator. An example is provided, and analytic expressions of the energy and wave functions are obtained. These account for the distinct nonlinear phenomena: the phase transition of energy and wave functions, which can not be revealed in the previous methods. The phase transitions cause some classical physical effects and phenomena: (1) energy gaps filled by energy levels; (2) increase in particle orbits; (3) a famous bound states in continuum (BICs) firstly found in FTQM; (4) a new explanation of the discrete energy levels from the perspective of energy level filling.

Introduction

The concepts and methods of fractional time, based on the theories of mathematical evolution operators and physical time evolution, were introduced and developed to describe the evolution of actions over time, therein considering historical information by comparing them with standard methods of point time affecting future behavior and states that are infinitely close to the current moment [1]. From an intuitive view, fractional time can be regarded as holistically treating time as a series of variable intervals rather than points at different positions constituting segments [2]. Therefore, the fractional time methodology aims to describe the evolution of quantum systems by involving historical information based on changes in the time interval instead of describing the evolution of physical systems based on changes in point time.

The study of fractional time has been expanded to various fields in mathematics to science and technology, such as quantum mechanics, electrodynamics, statistical mechanics and the mechanics of materials. In Ref [3], the relations between dissipative constant of damped oscillations and the order of fractional derivative were revealed. In Refs [4], [5], [6], [7], fractal time random walks, continuous time random walks were described, and relevant experimental results and experimental designs were reported. In Refs [8], a time-fractional Allen-Cahn equation with volume constraint was proposed. In Refs [9], [10], a contrary to the spontaneous emission phenomenon of the isotropic photonic crystal was found, and fractional time is proposed as a better mathematical method to study spontaneous emission dynamics from the optical system. The result was consistent with the experimental results in Ref [11]. In Ref [12], the physically-based macroscale models based on time-fractional diffusion for the simulation of the mass transport process through the solid porous media were considered. In Refs [13], [14], [15], [16], some experimental results and experimental designs of relaxation processes described by fractional time were mentioned. In particular, a summary of fractional differential dynamics can be seen in Ref [17].

The principles of fractional quantum mechanics were presented by Laskin in 2000 [18], [19], [20]. Then, a fractional derivative was introduced to describe fractional time in quantum dynamics and formed the corresponding model by Naber [21]. Hundreds of papers and monographs have mentioned, used and/or developed the model [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]. For example, in Ref [49], the fractional time Schrödinger equation was isospectral to a comb model when the parameter $\alpha =0.5$. In Ref [50], the micro-seismic power-laws pectra of sand with a fractional time Schrödinger equation was obtained. In Ref [51], a full spectra collocation approximation of using a shifted Legendre collocation method of fractional time Schrödinger equation was shown. In Ref [22], the influence of time discreteness on the transition coefficients was provided within the FTQM. In Ref [23], the fractional Schrödinger equation was considered with a cusp interaction. In Ref [24], the free particle’s fractional time kernel expressed it by H-function was found.

However, for research on the basic framework of fractional time quantum mechanics (FTQM), researchers have only directly replaced the integer time derivatives by fractional ones but with few mathematical derivations or physical analysis. This represents only one type of parallel introductions of the standard models, and the derivations were skipped [25], [26], [27], [28], [49]. Thus, a mathematical framework or method to describe FTQM remains lacks [24], [25], [26], [27], [29], [49]. Moreover, various authors [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48] (see App. Appendix A for their detailed views) noted that the previous models violated the following fundamental quantum physics laws, which were summarized in the review of FTQM [24]:

• Existence of stationary energy levels in quantum systems [21], [22], [24], [29], [33], [34], [35], [43];

• Unitarity of the evolution operator [21], [22], [23], [24], [26], [28], [29], [30], [33], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48];

• Quantum superposition law [24];

• Probability conservation law [21], [22], [24], [25], [27], [31], [32], [34], [35], [36], [43], [44].

Additionally, regarding FTQM, another open problem is that the significance of the fractional time Hamiltonian [24]. To date, FTQM has been governed by two forms of the Hamiltonian. One form,

$$H_{\alpha }=\frac{p^2}{2m}+V,$$

has obvious physical significance since it is one type of parallel introductions of the standard Hamiltonian; however, its eigenvalues are not the energies of systems, even in time-independent potentials. Many results show that this Hamiltonian will cause undesired results, such as violations of energy conservation, and the non-unitarity of the evolution operator [21], [24], [33]. Some authors even called it a pseudo-Hamiltonian [24], [49]. The other form,

$$H_{\nu }=-\frac{\beta }{i^{\nu }}{\partial }_x^2{D}_t^{1-\nu }+\frac{\alpha }{i^{\nu }}{D}_t^{1-\nu }+\frac{D^{\nu }{|}_{t=0}}{t^{1-\nu }\Gamma \left(\nu \right)},$$

has complex forms with little physical significance [21], [24]. It can also cause undesired results, such as non-conserved probabilities [21].

In this work, we aim to build a mathematical method to describe FTQM and solve the following three open problems of FTQM. In this method, the effects of fractional time naturally enter quantum mechanics, instead of usually directly replacing the integer derivatives by fractional ones or parallel introductions of the standard models. Then, we build the framework of FTQM which is similar to the standard quantum mechanics established by Schrödinger, Heisenberg and Feynman in different ways [52], [53], namely, obtaining the quantization method, the Hamiltonian, the Hamilton operator, the Schrödinger equation, the energy correspondence relation, the Bohr correspondence principle and the time-energy uncertainty relation. We also verify the framework of FTQM satisfy the above fundamental quantum physics laws (a)-(d), which the previous models violated.

In practical terms, we solve the following three open problems on FTQM:

• What is the essential quantization of fractional time?

  We amalgamate two quantization methods via strict derivations under a unified foundation of fractional time: we utilize Feynman Path Integral to externally describe the behaviors of a particle with the effects experienced in short intervals, and wave mechanics to describe the behaviors of a particle from the energy it possesses to extend the intervals. Then, we present a method to describe the time evolution of particle’s motion involved historical information between the start point and final point. These methods provide a quantization for describing phenomena in fractional time that has a stronger applicability than the Feynman Path Integral approach and the wave mechanics, even including both techniques. The corresponding results of standard quantum mechanics are contained naturally by our method. An example is provided to illustrate these results.

• How does one represent a physically significant fractional time Hamiltonian?

  The fractional time Hamiltonian and a corresponding model (the fractional time Schrödinger equation) with obvious physical significance are obtained; they naturally connect to those in standard quantum mechanics. We compare the Hamiltonian with previous versions in fractional time, and find out the reason why the previous fractional time systems are governed by a pseudo-Hamiltonian, and why their Hamiltonians are always time dependent. We prove that our Hamiltonian satisfies the basic principles of physics. Moreover, we present a kind of weak Hamilton operator: it represents a type of weak effect of the standard Hamilton operator, being essentially different from the single-fold action of the Hamilton operator H in the standard quantum mechanics. Specifically, it decomposes the actions of the standard Hamiltonian operator, as shown in Fig. 1.

• How do ones avoid the violations in current methods and models of FTQM for many fundamental quantum physics laws?

  We prove that the basis of FTQM we obtain consists with all the above fundamental quantum physics laws and explain the reason why the previous models violated those laws (a)-(d).

An example is provided to show the effects of FTQM. We obtain analytic expressions of the energy and wave functions of the fractional Schrödinger equation for an infinite potential well. These results indicate that some distinct nonlinear phenomena: the phase transitions of energy and wave functions. When authors used the previous methods of FTQM, which involve the fundamental quantum physics laws, these distinct nonlinear phenomena are neglected [21], [28], [54].

Moreover, the phase transitions cause some classical physical effects and phenomena: (1) Energy gaps are filled by energy levels. (2) The number of particle orbits increase. (3) A famous BICs have been firstly found in FTQM; its research history is as long as quantum mechanics and are still widely studied [53], [55], [56], [57], [58], [59], [60], [61], [62], [63]. Until recently, they were constructed in three ways [55], [56], [57], [58], [59], [64], [65]. We compare our method with current methods and show that our method is different. The BICs we obtain do not need complex interactions among particles or specific frail potentials, and they are stable. (4) A new explanation of the discrete energy levels from the perspective of energy level filling is provided.