Time Fractional Quantum Mechanics

TL;DR

This paper introduces a novel time fractional quantum mechanics framework, extending the Schr"odinger equation with fractional derivatives, impacting fundamental quantum concepts and providing new solutions and properties for free particles.

Contribution

It presents a new space-time fractional Schr"odinger equation with a fractional Planck's constant and explores its implications, including the absence of stationary states and new solution functions.

Findings

No stationary states in time fractional quantum systems

Eigenvalues of pseudo-Hamiltonian differ from energy levels

Derived free particle solutions and kernels in fractional quantum mechanics

Abstract

A time fractional quantum framework has been introduced into quantum mechanics. A new version of the space-time fractional Schr\"odinger equation has been launched. The introduced space-time fractional Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger 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.

Equations

$$i^{\beta }{\hslash }_{\beta }{\partial }_t^{\beta }\psi (x,t)=D_{\alpha ,\beta }(-{\hslash }_{\beta }^2\Delta {)}^{\alpha /2}\psi (x,t)+V(x,t)\psi (x,t),$$

$$1<\alpha \le 2,0<\beta \le 1,$$

$${\partial }_t^{\beta }f(t)=\frac{1}{\Gamma (1-\beta )}\underset{0}{\overset{t}{\int }}d\tau \frac{f^{{}^{\prime }}(\tau )}{(t-\tau {)}^{\beta }},0<\beta \le 1,$$

$$\Gamma (z)=\underset{0}{\overset{\infty }{\int }}dt{e}^{-t}{t}^{z-1}.$$

$$(-{\hslash }_{\beta }^2\Delta {)}^{\alpha /2}\psi (x,t)=\frac{1}{2\pi {\hslash }_{\beta }}\int d{p}_{\beta }\exp \{i\frac{p_{\beta }x}{{\hslash }_{\beta }}\}|p_{\beta }{|}^{\alpha }\phi (p_{\beta },t),$$

$$\psi (x,t)=\frac{1}{2\pi {\hslash }_{\beta }}\int d{p}_{\beta }\exp \{i\frac{p_{\beta }x}{{\hslash }_{\beta }}\}\phi (p_{\beta },t),$$

$$\phi (p_{\beta },t)=\int dr\exp \{-i\frac{p_{\beta }x}{{\hslash }_{\beta }}\}\psi (x,t).$$

$$[{\hslash }_{\beta }]=\mathrm{erg}\cdot {\sec }^{\beta },$$

$$[D_{\alpha ,\beta }]={\mathrm{erg}}^{1-\alpha }\cdot {\mathrm{cm}}^{\alpha }\cdot {\sec }^{-\alpha \beta }.$$

$${\hat{p}}_{\beta }=-i{\hslash }_{\beta }\frac{\partial }{\partial x},$$

$$\hat{x}=x.$$

$$[\hat{x},{\hat{p}}_{\beta }]=i{\hslash }_{\beta },$$

$$[\hat{x},{\hat{p}}_{\beta }]=\hat{x}{\hat{p}}_{\beta }-{\hat{p}}_{\beta }\hat{x}.$$

$${\hat{H}}_{\alpha ,\beta }({\hat{p}}_{\beta },\hat{x})=D_{\alpha ,\beta }|{\hat{p}}_{\beta }{|}^{\alpha }+V(\hat{x},t).$$

$${\hslash }_{\beta }{i}^{\beta }{\partial }_t^{\beta }\psi (x,t)={\hat{H}}_{\alpha ,\beta }({\hat{p}}_{\beta },\hat{x})\psi (x,t)$$

$$=\left(D_{\alpha ,\beta }|{\hat{p}}_{\beta }{|}^{\alpha }+V(\hat{x},t)\right)\psi (x,t),1<\alpha \le 2,0<\beta \le 1,$$

$$(\varphi ,\chi )=\underset{-\infty }{\overset{\infty }{\int }}dx{\varphi }^{\ast }(x,t)\chi (x,t).$$

$$(\varphi ,(-{\hslash }_{\beta }^2\Delta {)}^{\alpha /2}\chi )=((-{\hslash }_{\beta }^2\Delta {)}^{\alpha /2}\varphi ,\chi ).$$

$$i^{\beta }{\hslash }_{\beta }{\partial }_t^{\beta }\psi (\mathbf{r},t)=D_{\alpha ,\beta }(-{\hslash }_{\beta }^2\Delta {)}^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t),$$

$$1<\alpha \le 2,0<\beta \le 1,$$

$$(-{\hslash }_{\beta }^2\Delta {)}^{\alpha /2}\psi (\mathbf{r},t)=\frac{1}{(2\pi {\hslash }_{\beta }{)}^3}\int d^{3}r\exp \{i\frac{{\mathbf{p}}_{\beta }\mathbf{r}}{{\hslash }_{\beta }}\}|{\mathbf{p}}_{\beta }{|}^{\alpha }\phi ({\mathbf{p}}_{\beta },t),$$

$$\psi (\mathbf{r},t)=\frac{1}{(2\pi {\hslash }_{\beta }{)}^3}\int d^{3}p\exp \{i\frac{{\mathbf{p}}_{\beta }\mathbf{r}}{{\hslash }_{\beta }}\}\phi ({\mathbf{p}}_{\beta },t),$$

$$\phi ({\mathbf{p}}_{\beta },t)=\int d^{3}r\exp \{-i\frac{{\mathbf{p}}_{\beta }\mathbf{r}}{{\hslash }_{\beta }}\}\psi (\mathbf{r},t).$$

$${\hat{H}}_{\alpha ,\beta }({\hat{\mathbf{p}}}_{\beta },\hat{\mathbf{r}})=D_{\alpha ,\beta }|{\hat{\mathbf{p}}}_{\beta }{|}^{\alpha }+V(\hat{\mathbf{r}},t),$$

$$\hat{\mathbf{r}}=\mathbf{r},$$

$${\hat{\mathbf{p}}}_{\beta }=-i{\hslash }_{\beta }\frac{\partial }{\partial \mathbf{r}},$$

$$[{\hat{r}}_k,{\hat{p}}_{\beta j}]=i{\hslash }_{\beta }{\delta }_{kj},[{\hat{r}}_k,{\hat{r}}_j]=0,[{\hat{p}}_{\beta k},{\hat{p}}_{\beta j}]=0,k,j=1,2,3,$$

$${\delta }_{kj},=\{\text{QATOP}1k=j,0k\ne j.$$

$${\hslash }_{\beta }{i}^{\beta }{\partial }_t^{\beta }\psi (\mathbf{r},t)={\hat{H}}_{\alpha ,\beta }({\hat{\mathbf{p}}}_{\beta },\hat{\mathbf{r}})\psi (\mathbf{r},t)$$

$$=\left(D_{\alpha ,\beta }|{\hat{\mathbf{p}}}_{\beta }{|}^{\alpha }+V(\hat{\mathbf{r}},t)\right)\psi (\mathbf{r},t),1<\alpha \le 2,0<\beta \le 1,$$

Full text

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, 05.30.-d, 03.65.Sq

1. 1 Introduction

2. 1.1 Shortcomings of time fractional quantum mechanics

3. 1.2 Benefits of time fractional quantum mechanics

4. 1.3 The paper outline

5. 2 Space-time fractional Schrödinger equation

6. 2.1 Pseudo-Hamilton operator in time fractional quantum mechanics

7. 2.2 Hermiticity of pseudo-Hamilton operator

8. 2.3 3D generalization of space-time fractional Schrödinger equation

9. 2.3.1 Space-time fractional Schrödinger equation in momentum representation

10. 2.3.2 Hermiticity of pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$

11. 2.3.3 The parity conservation law

12. 3 Solution to space-time fractional Schrödinger equation

13. 4 Energy in the framework of time fractional quantum mechanics

14. 5 Applications of time fractional quantum mechanics

15. 5.1 A free particle wave function

16. 5.2 A free particle space-time fractional quantum mechanical kernel

17. 5.2.1 Renormalization properties of the space-time fractional quantum mechanical kernel

18. 5.2.2 Wright function in time fractional quantum mechanics

19. 5.2.3 Fox H-function representation for a free particle space-time fractional quantum kernel

20. 6 Special cases  of time fractional quantum mechanics

21. 6.1 The Schrödinger equation - the case when $\alpha=2$ and $\beta=1$

22. 6.2 The fractional Schrödinger equation - the case when $1<\alpha\leq 2$ and $\beta=1$

23. 6.3 The time Schrödinger equation - the case when $\alpha=2$ and $0<\beta\leq 1$

24. 7 Conclusion

25. 8 Acknowledgments

26. 9 Appendix A

27. 9.1 Fractional generalization of trigonometric functions $\cos(z)$ and $\sin(z)$

28. 10 Appendix B

29. 10.1 Fox’s $H$-function

30. 10.1.1 Some properties of the $H$-function

31. 10.2 Cosine transform of the $H$-function

1 Introduction

In recent years the application of fractional calculus in quantum theory became a rapidly growing area. It was initiated by the discovery of fractional quantum mechanics [1]-[4]. The crucial manifestation of fractional quantum mechanics is fractional Schrödinger equation. The fractional Schrödinger equation includes a spatial derivative of fractional order instead of the second order spatial derivative in the well-known Schrödinger equation. Thus, only the spatial derivative becomes fractional in the fractional Schrödinger equation, while the time derivative is the first order time derivative. Due to the presence of the first order time derivative in the fractional Schrödinger equation, fractional quantum mechanics supports all quantum mechanics fundamentals.

Inspired by the work of Laskin [1]-[2], Naber invented time fractional Schrödinger equation [5]. The time fractional Schrödinger equation involves the time derivative of fractional order instead of the first-order time derivative, while the spatial derivative is the second-order spatial derivative as it is in the well-known Schrödinger equation. To obtain the time fractional Schrödinger equation, Naber mapped the time fractional diffusion equation into the time fractional Schrödinger equation, similarly to the map between the well-known diffusion equation and the standard Schrödinger equation. The mapping implemented by Naber can be considered as a ”fractional” generalization of the Wick rotation [6]. To get the time fractional Schrödinger equation, Naber implemented the Wick rotation in complex $t$-plane by rising the imaginary unit $i$ to the same fractional power as the fractional order of the time derivative in the time fractional diffusion equation. The time fractional derivative in the time fractional Schrödinger equation is the Caputo fractional derivative [7]. Naber has found the exact solutions to the time fractional Schrödinger equation for a free particle and a particle in a potential well [5].

Later on, Wang and Xu [8], and then Dong and Xu [9], combined both Laskin’s equation and Naber’s equation and came up with space-time fractional Schrödinger equation. The space-time fractional Schrödinger equation includes both spatial and temporal fractional derivatives. Wang and Xu found exact solutions to the space-time fractional Schrödinger equation for a free particle and for an infinite square potential well. Dong and Xu found the exact solution to the space-time fractional Schrödinger equation for a quantum particle in $\delta$-potential well.

Here we introduce time fractional quantum mechanics and develop its fundamentals. The wording ”time fractional quantum mechanics” means that the time derivative in the fundamental quantum mechanical equations - Schrödinger equation and fractional Schrödinger equation, is substituted with a fractional time derivative. The time fractional derivative in our approach is the Caputo fractional derivative.

To introduce and develop time fractional quantum mechanics we begin with our own version of the space-time fractional Schrödinger equation. Our space-time fractional Schrödinger equation involves two scale dimensional parameters, one of which can be considered as a time fractional generalization of the famous Planck’s constant, while the other one can be interpreted as a time fractional generalization of the scale parameter emerging in fractional quantum mechanics [1]-[4]. The fractional generalization of Planck’s constant is a fundamental dimensional parameter of time fractional quantum mechanics, while the fractional generalization of Laskin’s scale parameter [1]-[4] plays a fundamental role in both time fractional quantum mechanics and time fractional classical mechanics.

In addition to the above mentioned dimensional parameters, time fractional quantum  mechanics involves two dimensionless fractality parameters $\alpha$, $1<\alpha\leq 2$ and $\beta$, $0<\beta\leq 1$. Parameter $\alpha$ is the order of the spatial fractional quantum Riesz derivative [1] and $\beta$ is the order of the time fractional derivative. In other words, $\alpha$ is responsible for modelling spatial fractality, while parameter $\beta$, which is the order of Caputo fractional derivative, is responsible for modeling temporal fractality.

Time fractional quantum mechanical operators of coordinate, momentum and angular momentum have been introduced and their commutation relationship has been established. The pseudo-Hamilton quantum mechanical operator has been introduced and its Hermiticity has been proven in the framework of time fractional quantum mechanics. The general solution to the space-time fractional Schrödinger equation was found in the case when the pseudo-Hamilton operator does not depend on time. Energy of a quantum system in the framework of time fractional quantum mechanics was defined and calculated in terms of the Mittag-Leffler function. Two new functions associated with the Mittag-Leffler function have been launched and elaborated. These two new functions can be considered as a natural fractional generalization of the well-known trigonometric functions sine and cosine. A fractional generalization of the celebrated Euler equation was discovered. A free particle time fractional quantum kernel was calculated in terms of Fox’s $H$-functions.

In the framework of time fractional quantum mechanics at particular choices of fractality parameters $\alpha$ and $\beta$, we rediscovered the following fundamental quantum equations:

1. Schrödinger equation (Schrödinger equation [10]), $\alpha=2$ and $\beta=1$;

2. Fractional Schrödinger equation (Laskin equation [4]), $1<\alpha\leq 2$ and $\beta=1$;

3. Time fractional Schrödinger equation (Naber equation [5]),  $\alpha=2$ and $0<\beta\leq 1$;

4. Space-time fractional Schrödinger equation (Wang and Xu [8] and Dong and Xu [9] equation), $1<\alpha\leq 2$ and $0<\beta\leq 1$.

1.1 Shortcomings of time fractional quantum mechanics

While fractional quantum mechanics [1]-[4] supports all quantum mechanics fundamentals, time fractional quantum mechanics violates the following fundamental physical laws of Quantum Mechanics:

a. Quantum superposition law;

b. Unitarity of evolution operator;

c. Probability conservation law;

d. Existence of stationary energy levels of quantum system.

Fractional quantum dynamics is governed by a pseudo-Hamilton operator instead of the Hamilton operator in standard quantum mechanics. Eigenvalues of quantum pseudo-Hamilton operator are not the energy levels of a time fractional quantum system.

1.2 Benefits of time fractional quantum mechanics

What benefits does time fractional quantum mechanics bring into quantum theory and its applications?

Despite of the above listed shortcomings, the developments in time fractional quantum mechanics can be considered a newly emerged and attractive application of fractional calculus to quantum theory. Time fractional quantum mechanics helps to understand the significance and importance of the fundamentals of quantum mechanics such as Hamilton operator, unitarity of evolution operator, existence of stationary energy levels of quantum mechanical system, quantum superposition law, conservation of quantum probability, etc.

Besides, time fractional quantum mechanics invokes new mathematical tools, which have never been used in quantum theory before.

From a stand point of quantum mechanical fundamentals, time fractional quantum mechanics seems not as a quantum physical theory but rather as an adequate, convenient mathematical framework, well adjusted to study dissipative quantum systems interacting with environment [11].

1.3 The paper outline

The introduction presents a brief overview of benefits and shortcomings of time fractional quantum mechanics.

In Sec.2 we launched a new version of the space-time fractional Schrödinger equation. Our space-time fractional Schrödinger equation involves two scale dimensional parameters, one of which can be considered as a time fractional generalization of the famous Planck’s constant, while the other one can be interpreted as a time fractional generalization of the scale parameter emerging in fractional quantum mechanics. A 3D generalization of the space-time fractional Schrödinger equation has been developed. We also found the space-time fractional Schrödinger equation in momentum representation. The pseudo-Hamilton operator was introduced and its Hermiticity has been proven. The parity conservation law has been proven in the framework of time fractional quantum mechanics.

In Sec.3 the solution to the space-time fractional Schrödinger equation was obtained in the case when the pseudo-Hamilton operator does not depend on time. It was found that time fractional quantum mechanics does not support a fundamental property of quantum mechanics - conservation of quantum mechanical probability.

The concept of energy of a time fractional quantum mechanical system was introduced in Sec.4. It has been shown that in time fractional quantum mechanics a quantum system does not have stationary states, and eigenvalues of the pseudo-Hamilton operator are not energy levels.

A free particle wave function and a space-time fractional quantum mechanical kernel were obtained as applications of time fractional quantum mechanics in Sec.5. The renormalization properties of a space-time fractional quantum mechanical  kernel were established.  The involvement of the Wright function in time fractional quantum mechanical framework was discussed. Fox’s $H$-function representation for a free particle space-time fractional quantum kernel was found.

Sec.6 deals with special cases of time fractional quantum mechanics. It has been shown that with a particular choice of fractality parameters $\alpha$ and $\beta$, the space-time fractional Schrödinger equation covers the following three special cases: the Schrödinger equation ($\alpha=2$ and $\beta=1$), the fractional Schrödinger equation ($1<\alpha\leq 2$ and $\beta=1$) and the time fractional Schrödinger equation ($\alpha=2$ and $0<\beta\leq 1$).

The conclusion summarizes the major findings of the paper.

In Appendix A we introduce and discuss two new functions closely related to the Mittag-Leffler function. It has been observed that these two new functions can be considered as a natural fractional generalization of the well-known trigonometric functions $\cos(z)$, and $\sin(z).$ Fractional generalization of the celebrated Euler equation was discovered in terms of these two newly introduced functions.

Appendix B presents the definition and some fundamental properties of Fox’s $H$-function.

2 Space-time fractional Schrödinger equation

To introduce time fractional quantum mechanics we begin with our own version of the space-time fractional Schrödinger equation originally introduced by Wang and Xu [8] and by Dong and Xu [9]. The term space-time fractional Schrödinger equation was coined by Dong and Xu [9].

Considering a 1D spatial dimension, we launch the space-time fractional Schrödinger equation in the following form,

[TABLE]

[TABLE]

here $\psi(x,t)$ is the wave function, $i$ is imanaginery unit, $i=\sqrt{-1}$, $V(x,t)$ is potential energy, $\hbar_{\beta}$ and $D_{\alpha,\beta}$ are two scale coefficients, which we introduce into the framework of time fractional quantum mechanics, $\Delta$ is the 1D Laplace operator, $\Delta=\partial^{2}/\partial x^{2}$, and, finally, $\partial_{t}^{\beta}$ is the left Caputo fractional derivative [7] of order $\beta$ defined by

[TABLE]

where $f^{{}^{\prime}}(\tau)$ is first order time derivative, $\ f^{{}^{\prime}}(\tau)=df(\tau)/d\tau$, $\Gamma(1-\beta)$ is the Gamma function111The Gamma function has familiar representaion

$\Gamma(z)=\int\limits_{0}^{\infty}dte^{-t}t^{z-1}.$

(3) , and the operator $(-\hbar_{\beta}^{2}\Delta)^{\alpha/2}$ is a time fractional quantum Riesz derivative222Quantum Riesz fractional derivative was originally introduced in [1].

[TABLE]

where the wave functions in space and momentum representations, $\psi(x,t)$ and $\varphi(p,t)$, are related to each other by the Fourier transforms

[TABLE]

[TABLE]

Quantum scale coefficient $\hbar_{\beta}$ has physical dimension

[TABLE]

and scale coefficient $D_{\alpha,\beta}$ has physical dimension

[TABLE]

The introduction of the scale coefficient $D_{\alpha,\beta}$ was inspired by Bayin [12].

2.1 Pseudo-Hamilton operator in time fractional quantum mechanics

Aiming to obtain the operator form of space-time fractional Schrödinger equation Eq.(1), let us define time fractional quantum momentum operator $\widehat{p}_{\beta}$

[TABLE]

and quantum operator of coordinate $\widehat{x}$,

[TABLE]

Hence, the commutation relation in the framework of time fractional quantum mechanics has the form

[TABLE]

where $[\widehat{x},\widehat{p}_{\beta}]$ is the commutator of two quantum operators $\widehat{x}$ and $\widehat{p}_{\beta}$,

[TABLE]

Using quantum operators of momentum Eq.(9) and coordinate Eq.(10), we introduce a new time fractional quantum mechanical operator $\widehat{H}_{\alpha,\beta}(\widehat{p}_{\beta},\widehat{x})$,

[TABLE]

The operator $\widehat{H}_{\alpha,\beta}(\widehat{p}_{\beta},\widehat{x})$ is not the Hamilton operator of the quantum mechanical system under consideration333In time fractional quantum mechanics the eigenvalues of operator $\widehat{H}_{\alpha,\beta}(\widehat{p}_{\beta},\widehat{x})$ are not energies of a quantum system. In classical time fractional mechanics the principle of least action with classical pseudo-Hamilton function $H_{\alpha,\beta}(p_{\beta},x)$ does not result in Hamilton equations of motion.. Following [9], we will call this operator pseudo-Hamilton operator.

Having the pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}$, let us rewrite Eq.(1) as

[TABLE]

[TABLE]

which is operator form of the space-time fractional Schrödinger equation.

2.2 Hermiticity of pseudo-Hamilton operator

The pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{p}_{\beta},\widehat{x})$ introduced by Eq.(13) is the Hermitian operator in space with scalar product

[TABLE]

In this space, operators $\widehat{p}_{\beta}$ and $\widehat{x}$ defined by Eqs.(9) and (10) are Hermitian operators. The proof can be found in any textbook on quantum mechanics, (see, for example, [13]).

To prove the Hermiticity of quantum mechanical operator $\widehat{H}_{\alpha,\beta}$ let us note that in accordance with the definition of the time fractional quantum Riesz derivative given by Eq.(4) there exists the integration-by parts formula

[TABLE]

Therefore, using this integration-by parts formula we prove straightforwardly Hermiticity of the term $D_{\alpha,\beta}|\widehat{p}_{\beta}|^{\alpha}=D_{\alpha,\beta}(-\hbar_{\beta}^{2}\Delta)^{\alpha/2}$. Next, potential energy operator $V(\widehat{x},t)$ in Eq.(13) is Hermitian operator by virtue of being a function of Hermitian operator $\widehat{x}$.

Thus, we completed the proof of Hermiticity of the pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}$ in the space with scalar product defined by Eq.(15).

2.3 3D generalization of space-time fractional Schrödinger

equation

Considering 3D spatial dimension, we launch the space-time fractional Schrödinger equation of the following form,

[TABLE]

[TABLE]

where $\psi(\mathbf{r},t)$ is the wave function, $\mathbf{r}$ is 3D space vector, $\Delta$ is the Laplacian, $\Delta=(\partial/\partial\mathbf{r)}^{2},$ all other notations are the same as for Eq.(1), and 3D time fractional quantum Riesz derivative $(-\hbar_{\beta}^{2}\Delta)^{\alpha/2}$ is defined by

[TABLE]

where the wave functions in space representation $\psi(\mathbf{r},t)$ and momentum representation $\varphi(\mathbf{p},t)$ are related to each other by the 3D Fourier transforms

[TABLE]

[TABLE]

Further, 3D generalization of the pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}$ in the framework of time fractional quantum mechanics is

[TABLE]

where $\widehat{\mathbf{r}}$ is 3D quantum operator of coordinate

[TABLE]

and $\widehat{\mathbf{p}}_{\beta}$ is 3D time fractional quantum momentum operator introduced by

[TABLE]

with $\hbar_{\beta}$ being the scale coefficient appearing for the first time in Eqs.(1) and (17).

The basic canonical commutation relationships in 3D case are

[TABLE]

where $\delta_{kj}$ is the Kronecker symbol,

[TABLE]

Having 3D generalization of pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$, let us present Eq.(17) in the form

[TABLE]

[TABLE]

which is operator form of 3D space-time fractional Schrödinger equation.

Heaving operators (22) and (23) we introduce time fractional angular momentum operator $\widehat{\mathbf{L}}_{\beta}$ as cross-product of two above defined operators $\widehat{\mathbf{r}}$ and $\widehat{\mathbf{p}}_{\beta}$

[TABLE]

This equation can be expressed in component form

[TABLE]

if we use 3D Levi-Civita antisymmetric tensor $\varepsilon_{ijk\text{ }}$ which changes its sign under interchange of any pair of indices $i$, $j$, $k$.

It is obvious that the algebra of time fractional angular momentum operators $\widehat{\mathrm{L}}_{\beta i}$ and its commutation relationships with operators $\widehat{r}_{j}$ and $\widehat{p}_{\beta k}$ are the same as for the angular momentum operator of quantum mechanics [13].

2.3.1 Space-time fractional Schrödinger equation in momentum

representation

To obtain the space-time fractional Schrödinger equation in momentum representation let us substitute the wave function $\psi(\mathbf{r},t)$ from Eq.(19) into Eq.(17),

[TABLE]

[TABLE]

Further, multiplying Eq.(29) by $\exp(-i\mathbf{p}_{\beta}\mathbf{r}/\hbar_{\beta})$ and integrating over $d^{3}r$ yields the equation for the wave function $\varphi(\mathbf{p}_{\beta},t)$ in momentum representation

[TABLE]

[TABLE]

where $U_{\mathbf{p}_{\beta},\mathbf{p}_{\beta}^{\prime}}$ is introduced as

[TABLE]

and we used the following representation for the delta function $\delta(\mathbf{r})$

[TABLE]

Equation (30) is the 3D space-time fractional Schrödinger equation in momentum representation.

Substituting the wave function $\psi(x,t)$ from Eq.(5) into Eq.(1), multiplying by $\exp(-ipx/\hbar)$ and integrating over $dx$ bring us the 1D space-time fractional Schrödinger equation in momentum representation

[TABLE]

[TABLE]

where $U_{p_{\beta},p_{\beta}^{\prime}}$ is given by

[TABLE]

and we used the following representation for the delta function $\delta(x)$

[TABLE]

Equation (30) is the 1D space-time fractional Schrödinger equation in momentum representation for the wave function $\varphi(p_{\beta},t)$.

2.3.2 Hermiticity of pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$

To prove that defined by Eq.(17) $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$ is Hermitian operator we introduce the space with scalar product defined by

[TABLE]

where $\phi^{\ast}(\mathbf{r},t)$ means complex conjugate function.

Then, the proof can be done straightforwardly by 3D generalization of the considered presented in Sec.2.2.

2.3.3 The parity conservation law

Here we study invariance of pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$ under inversion transformation. Inversion, or to be precise, spatial inversion consists of the simultaneous change in the sign of all three spatial coordinates

[TABLE]

It easy to see that

[TABLE]

which means that the function $\exp\{i\mathbf{p}_{\beta}\mathbf{x}/\hbar_{\beta}\}$ is the eigenfunction of the 3D time fractional quantum Riesz operator $(-\hbar_{\beta}^{2}\Delta)^{\alpha/2}$ with eigenvalue $|\mathbf{p}_{\beta}|^{\alpha}$.

Thus, the operator $(-\hbar_{\beta}^{2}\Delta)^{\alpha/2}$ is the symmetrized fractional derivative, that is

[TABLE]

Assuming that the potential energy operator $V(\widehat{\mathbf{r}},t)$ is invariant under spatial inversion $V(\widehat{\mathbf{r}},t)=V(-\widehat{\mathbf{r}},t)$, we conclude that pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$ is invariant under inversion, or, in other words, it supports the parity conservation law. The inverse symmetry results in the fact that inversion operator $\widehat{P}_{\mathrm{inv}}$ and the pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{\mathbf{p}}_{\beta},\widehat{\mathbf{r}})$ commute

[TABLE]

Hence, we can divide the wave functions of time fractional quantum mechanical states with defined eigenvalue of the operator $\widehat{P}_{\mathrm{inv}}$ into two classes; (i) wave functions which are not changed upon the action of the inversion operator, $\widehat{P}_{\mathrm{inv}}\psi_{+}(\mathbf{r})=\psi_{+}(\mathbf{r})$, the corresponding states are called even states; (ii) wave functions which change sign under action of the inversion operator, $\widehat{P}_{\mathrm{inv}}\psi_{-}(\mathbf{r})=-\psi_{-}(\mathbf{r})$, the corresponding states are called odd states. Equation(40) represents the parity conservation law for time fractional quantum mechanics, that is, if the state of a time fractional quantum mechanical system has a given parity (i.e. if it is even, or odd), then this parity is conserved.

Thus, we conclude that time fractional quantum mechanics supports the parity conservation law.

3 Solution to space-time fractional Schrödinger equation

It is well know that if the Hamilton operator of a quantum mechanical system does not depend on time, then we can search for the solution to the Schrödinger equation in separable form. In the case when pseudo-Hamilton operator (21) doesn’t depend on time we search for the solution to Eq.(1) assuming that the solution has form,

[TABLE]

where $\varphi(x)$ and $\chi(t)$ are spatial and temporal components of the wave function $\psi(x,t)$444We consider here 1D space-time fractional Schrödinger equation The generalization to 3D case can be done straightforwardly..

It is assumed as well, that initial wave function $\psi(x,t=0)=\psi(x,0)$ is normalized

[TABLE]

Substituting Eq.(41) into Eq.(1) we obtain two equations

[TABLE]

and

[TABLE]

where $\mathcal{E}$ is the eigenvalue of quantum mechanical pseudo-Hamilton operator $\widehat{H}_{\beta}(\widehat{p}_{\beta},\widehat{x})$,

[TABLE]

The solution to Eq.(43) depending on potential energy term $V(\widehat{x})$ can be obtained by the well-known methods of standard quantum mechanics [13]. To find the solution to time fractional equation (44) we can use the Laplace transform method555The Laplace transform $\widetilde{\chi}(s)$ of a function $\chi(t)$ is defined as

$\widetilde{\chi}(s)=\int\limits_{0}^{\infty}dte^{-st}\chi(t).$

(46) where $\chi(t)$ is defined for $t\geq 0$ if the integral exists. The inverse Laplace transform is defined by

$\chi(t)=\frac{1}{2\pi i}\int\limits_{\gamma i-\infty}^{\gamma+i\infty}dse^{st}\widetilde{\chi}(s),$

(47) where the integration is done along the vertical line $\mathop{\mathrm{R}e}(s)=\gamma$ in the complex plane $s$ such that $\gamma$ is greater than the real part of all singularities in the complex plane of $\widetilde{\chi}(s)$. This requirement on integration path in the complex plane $s$ ensures that the contour path is in the region of convergence. In practice, computing the complex integral can be done by using the Cauchy residue theorem.. In the Laplace transform domain Eq.(44) reads

[TABLE]

where $\widetilde{\chi}(s)$ is defined by Eq.(46) and $\chi(0)=\chi(t=0)$ is the initial condition on time dependent component $\chi(t)$ of the wave function $\psi(x,t)$ given by Eq.(41).

Further, from Eq.(48) we have

[TABLE]

which can be presented as geometric series (the range of convergence is given by the criteria $|\mathcal{E}s^{-\beta}/i^{\beta}\hbar_{\beta}|<1$, see, [12]),

[TABLE]

Then the inverse Laplace transform yields

[TABLE]

where $E_{\beta}(z)$ is the Mittag-Leffler function [14] defined by the series

[TABLE]

with $\Gamma(x)$ being the Gamma function given by Eq.(3).

In the limit case, when $\beta=1$, the Mittag-Leffler function $E_{\beta}(z)$ becomes the exponential function

[TABLE]

and $\chi(t)$ goes into

[TABLE]

here $\hbar$ is Planck’s constant and $\mathcal{E}$ is an eigenvalue of quantum mechanical Hamilton operator $\widehat{H}_{\alpha}(\widehat{p},\widehat{x}),$

[TABLE]

where the scale coefficient $D_{\alpha}=D_{\alpha,\beta}|_{\beta=1}$ was introduced originally in [1].

It is easy to see, that the time dependent component $\chi(t)$ (51) of the wave function $\psi(x,t)$ can be written as

[TABLE]

if we use the following new expression for the Mittag-Leffler function $E_{\beta}(i^{\beta}z)$,

[TABLE]

in  terms of two functions $Ec_{\beta}(z)$ and $Es_{\beta}(z)$ introduced by Eqs.(130) and (131), see Appendix A.

It has been shown in Appendix A that Eq.(57) can be considered as a fractional generalization of the celebrated Euler equation, which is recovered from Eq.(57) in the limit case $\beta=1.$

Finally, we have the solution to the time fractional Schrödinger equation (1) given by

[TABLE]

We see from Eq.(58) that the time fractional quantum mechanics does not support normalization condition for the wave function. If normalization condition (42) holds at the initial time moment $t=0$, then at any time moment $t>0$ it becomes time dependent

[TABLE]

[TABLE]

here normalization condition Eq.(42) was used.

Therefore, we come to the conclusion that in the framework of time fractional quantum mechanics total quantum mechanical probability $\int\limits_{-\infty}^{\infty}dx|\psi(x,t)|^{2}$ is time dependent. In other words,  time fractional quantum mechanics does not support a fundamental property of the quantum mechanics - conservation of quantum mechanical probability.

4 Energy in the framework of time fractional quantum mechanics

It is well known that the energy levels of a quantum mechanical system in stationary states are defined by Hamilton operator eigenvalues. In time fractional quantum mechanics a quantum system doesn’t have stationary states and an eigenvalues of pseudo-Hamilton operator are not energy levels.

To define and calculate what we call as energy of time fractional quantum system, let us introduce time fractional quantum mechanical operator of energy follow to Bayin [12],

[TABLE]

When $\beta=1$, the Caputo fractional derivative Eq.(2) becomes the ordinary first order time derivative, the quantum scale coefficient $\hbar_{\beta}$ becomes the well-known fundamental Planck’s constant, $\hbar_{\beta}|_{\beta=1}=\hbar$. Thus, in the case $\beta=1$ time fractional quantum mechanical operator of energy (60) becomes what is called sometimes as operator of energy,

[TABLE]

However, it has to be clear that time is not quantum mechanical operator in the framework of time fractional quantum mechanics as well as in the framework of standard quantum mechanics.

Having energy operator $\widehat{\mathrm{E}}_{\beta}$ (60) and wave function given by Eq.(58) we can calculate the energy of time fractional quantum mechanical system.

[TABLE]

where $\partial_{t}^{\beta}$ is the Caputo fractional time derivative (2) and $\psi^{\ast}(x,t)$ stands for complex conjugated wave function.

Substituting $\psi(x,t)$ given by Eq.(41) with $\chi(t)$ defined by Eq.(56) yields for $\mathrm{E}_{\beta}$

[TABLE]

With help of Eq.(42) the last equation can be rewritten as

[TABLE]

This equation defines the energy $\mathrm{E}_{\beta}$ of time fractional quantum system with the pseudo-Hamilton operator $\widehat{H}_{\alpha,\beta}(\widehat{p}_{\beta},\widehat{x})$ introduced by Eq.(13). The energy $\mathrm{E}_{\beta}$ defined by Eq.(62) is real due to Hermiticity of the pseudo-Hamilton operator Eq.(13). We see, that the energy $\mathrm{E}_{\beta}$ of time fractional quantum system depends on time $t$, eigenvalue $\mathcal{E}$ of pseudo-Hamilton operator, and fractality parameter $\beta$. Thus, we come to the conclusion 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 time fractional quantum system.

Let us note, that in the limit case $\beta=1$ we have (see, Eqs.(134) and (135) in Appendix A),

[TABLE]

Then, it follows from Eqs.(64) and (65) that

[TABLE]

here $\mathrm{E}_{1}$ is the energy of physical quantum system and $\mathcal{E}$ is the eigenvalue of quantum mechanical Hamilton operator $\widehat{H}_{\alpha}(\widehat{p},\widehat{x})$ given by Eq.(55).

Therefore, in the limit case $\beta=1$ we recover the well-known statement of standard quantum mechanics and fractional quantum mechanics [1]-[4] that the energy spectrum of a quantum system is a set of eigenvalues of Hamilton operator.

5 Applications of time fractional quantum mechanics

5.1 A free particle wave function

For a free particle when $V(x,t)=0$, the 1D space-time fractional Schrödinger equation (1) reads

[TABLE]

here $\psi(x,t)$ is the wave function, $i$ is imanaginery unit, $i=\sqrt{-1}$, $\hbar_{\beta}$ is scale coefficient, $\Delta$ is 1D Laplace operator, $\Delta=\partial^{2}/\partial x^{2}$, and, finally, $\partial_{t}^{\beta}$ is the left Caputo fractional derivative of order $\beta$ defined by Eq.(2).

We are searching for solution to Eq.(67) with the initial condition $\psi_{0}(x)$,

[TABLE]

By applying the Fourier transform to the wave function $\psi(x,t)$

[TABLE]

with $\varphi(p_{\beta},t)$ defined by

[TABLE]

we obtain from Eq.(67)

[TABLE]

with the initial condition $\varphi_{0}(p_{\beta})$ given by

[TABLE]

The equation (71) is the 1D space-time fractional Schrödinger equation in momentum representation.

The solution to the problem introduced by Eqs.(71) and (72) is

[TABLE]

where $E_{\beta}$ is the Mittag-Leffler function given by Eq.(52).

Hence, the solution to the 1D space-time fractional Schrödinger equation Eq.(67) with initial condition given by Eq.(68) can be presented as

[TABLE]

5.2 A free particle space-time fractional quantum mechanical kernel

The solution (74) to space-time fractional Schrödinger equation can be expressed in the form

[TABLE]

if we introduce into consideration a free particle space-time fractional quantum mechanical kernel $K_{\alpha,\beta}^{(0)}(x,t)$ defined by

[TABLE]

[TABLE]

The space-time fractional quantum mechanical kernel $K_{\alpha,\beta}^{(0)}(x,t)$ satisfies

[TABLE]

where $\delta(x)$ is delta function.

It follows immediately from Eq.(76) that the Fourier transform $K_{\alpha,\beta}^{(0)}(p_{\beta},t)$ of space-time fractional quantum mechanical kernel is

[TABLE]

[TABLE]

This is space-time fractional quantum mechanical kernel in momentum representation.

Applying the Laplace transform with respect to the time variable $t$ we obtain the Fourier-Laplace transform $K_{\alpha,\beta}^{(0)}(p_{\beta},u)$ of space-time fractional quantum mechanical kernel

[TABLE]

[TABLE]

Taking into account Eq.(77) we have

[TABLE]

5.2.1 Renormalization properties of the space-time fractional

quantum mechanical kernel

Aiming* *to study renormalization properties of the space-time fractional quantum mechanical kernel, let us re-write $K_{\alpha,\beta}^{(0)}(p_{\beta},s)$  given by Eq.(80) in integral form

[TABLE]

[TABLE]

where we introduced two functions $N_{\alpha,\beta}(p_{\beta},u)$ and $L_{\beta}(u,s)$ defined by

[TABLE]

and

[TABLE]

Having new function $N_{\alpha,\beta}(p_{\beta},u)$ we obtain

[TABLE]

[TABLE]

It is easy to see, that the scaling

[TABLE]

holds for function $N_{\alpha,\beta}(x,u)$ given by Eq.(84).

Having new function $L_{\beta}(u,s)$ we obtain

[TABLE]

[TABLE]

where $\sigma$ is greater than the real part of all singularities in the complex plane of $L_{\beta}(u,s)$.

It is easy to see that the scaling

[TABLE]

holds for function $L_{\beta}(u,t)$ given by Eq.(86).

In terms of above introduced functions $N_{\alpha,\beta}(x,u)$ and $L_{\beta}(u,t)$ Eq.(76) reads

[TABLE]

where functions $N_{\alpha,\beta}(x,u)$ and $L_{\beta}(u,t)$ are defined by Eqs.(84) and (86).

The equivalent representation for$K_{\alpha,\beta}^{(0)}(x,t)$ is

[TABLE]

[TABLE]

Hence, one can consider Eq.(89) as an alternative representation for space-time fractional quantum mechanical kernel $K_{\alpha,\beta}^{(0)}(x,t)$. The interesting feature of this representation is that it separates space and time variables for space-time fractional quantum mechanical kernel $K_{\alpha,\beta}^{(0)}(x,t)$. It follows from Eqs.(84) and (88) that

[TABLE]

With help of Eqs.(85) and (87) we conclude that the scaling

[TABLE]

holds for space-time fractional quantum mechanical kernel $K_{\alpha,\beta}^{(0)}(x,t)$ given by Eq.(76).

Defined by Eqs.(87) and (86) function $L_{\beta}(u/t^{\beta},1)$ can be re-written as

[TABLE]

where we introduced a new variable $z=u/t^{\beta}$.

5.2.2 Wright function in time fractional quantum mechanics

Our intent now is to show that the function $L_{\beta}(z,1)$ defined by Eq.(92) can be presented in terms of the Wright function (see, for example, [15]). By deforming the integration path in the right hand side of Eq.(92) into the Hankel contour $H_{a}$ we have

[TABLE]

A series expansion for $\exp\{-zs^{\beta}\}$ yields

[TABLE]

[TABLE]

[TABLE]

Using the well-known Hankel representation of the Gamma function (see, for example, [16]),

[TABLE]

we obtain for $L_{\beta}(z,1)$

[TABLE]

In the notations of Ref.[15], the Wright function $\phi(a,\beta;z)$ is expressed as

[TABLE]

Therefore, we come to following representation of our function $L_{\beta}(z,1)$ in terms of the Wright function $\phi$

[TABLE]

Knowledge of particular cases for the Wright function let us obtain the particular cases for the function $L_{\beta}(z,1)$.

1. When $\beta=1/2$ we have

[TABLE]

2. When $\beta=1/2$ we have

[TABLE]

where $\delta(z)$ is delta function. This property is useful to recover fractional quantum mechanics and standard quantum mechanics formulas from the general equation (88).

Thus, function $L_{\beta}(z,1)$ involved into representation (89) of space-time fractional quantum kernel, is expressed by Eq.(98) as the Wright function. It introduces into the time fractional quantum mechanics a well developed mathematical tool - Wright function. In other words, many well-known results related to the Wright function can be applied to study fundamental properties of the space-time fractional quantum kernel and develop a variety of new applications of time fractional quantum mechanics.

5.2.3 Fox H-function representation for a free particle

space-time fractional quantum kernel

The space-time fractional quantum kernel given by Eq.(76) can be expressed in terms of the Fox $H$-function. The definition of the $H$-function and its fundamental properties can be found in [17]. For paper integrity and reader’s audience convenience we presented the definition of the Fox $H$-function and some its properties in Appendix B.

In terms of the Fox $H$-function the Mittag-Leffler function $E_{\beta}(z)$ is presented by Eq.(154), see, Appendix B. Then, $K_{\alpha,\beta}^{(0)}(x,t)$ given by Eq.(76) reads

[TABLE]

With help of the cosine transform of the $H$-function defined by of Eq.(159), see Appendix B, we obtain

[TABLE]

[TABLE]

which is the expression for the space-time fractional quantum kernel $K_{\alpha,\beta}^{(0)}(x,t)$ in terms of $H_{3,3}^{2,1}$-function.

Alternatively, using the Property 4 given by Eq.(157), see Appendix B, $K_{\alpha,\beta}^{(0)}(x,t)$ can be presented as

[TABLE]

[TABLE]

Thus, the equations (102) and  (103) introduce a new family of a free particle space-time fractional quantum mechanical kernels parametrized by two fractality parameters $\alpha$ and $\beta$. Newly introduced kernel $K_{\alpha,\beta}^{(0)}(x,\tau)$ does not satisfy quantum superposition law defined by Eq.(24) in [2], due to the fractional time derivative of order $\beta$ in Eq.(67).

6 Special cases  of time fractional quantum mechanics

With particular choice of fractality parameters $\alpha$ and $\beta$ the space-time fractional Schrödinger equation Eq.(1) covers the following three special cases: the Schrödinger equation ($\alpha=2$ and $\beta=1$), the fractional Schrödinger equation ($1<\alpha\leq 2$ and $\beta=1$) and the time fractional Schrödinger equation ($\alpha=2$ and $0<\beta\leq 1$).

6.1 The Schrödinger equation - the case when $\alpha=2$

and $\beta=1$

When $\alpha=2$ and $\beta=1$ we have

[TABLE]

and

[TABLE]

where $m$ is mass of a quantum particle, $p$ is momentum of a quantum particle and $\hbar$ is the well-known Planck’s constant.

In this case space-time fractional Schrödinger equation Eq.(1) goes into the celebrated Schrödinger equation [10]

[TABLE]

For the quantum mechanical kernel $K_{2,1}^{(0)}(x,t)=$ $K_{\alpha,\beta}^{(0)}(x,t)|_{\alpha=2,\beta=1},$ where $K_{\alpha,\beta}^{(0)}(x,t)$ is defined by Eq.(88) we obtain

[TABLE]

here

[TABLE]

[TABLE]

and

[TABLE]

Substitution of Eqs.(108) and (109) into Eq.(107) yields

[TABLE]

The intergral in Eq.(110) can be evaluated analytically and the result is

[TABLE]

which is Feynman’s free particle quantum kernel [18].

6.2 The fractional Schrödinger equation - the case when $1<\alpha\leq 2$ and $\beta=1$

In the case when $\beta=1$, $D_{\alpha,\beta}$ goes into the scale coefficient $D_{\alpha}$ emerging in fractional quantum mechanics [1], [2],

[TABLE]

which has physical dimension,

[TABLE]

In this case the space-time fractional Schrödinger equation Eq.(1) goes into Laskin’s fractional Schrödinger equation [4]

[TABLE]

with $\hbar$ being fundamental Planck’s constant.

For the quantum mechanical kernel $K_{\alpha,1}^{(0)}(x,t)=$ $K_{\alpha,\beta}^{(0)}(x,t)|_{\beta=1},$ where $K_{\alpha,\beta}^{(0)}(x,t)$ is defined by Eq.(88) we obtain

[TABLE]

here

[TABLE]

and

[TABLE]

Substitution of Eqs.(125) and (117) into Eq.(124) yields

[TABLE]

It was shown in [2], [19] that the integral in Eq.(118) can be expressed in terms of the $H_{2,2}^{1,1}$-function,

[TABLE]

The expression (119) for a free particle quantum kernel $K_{\alpha,1}^{(0)}(x,t)$ can be obtained directly from Eq.(103). It is easy to see, that when $\beta=1$ Eq.(103) reads

[TABLE]

[TABLE]

[TABLE]

By using Property 1 (see, Appendix B) we re-write the above expression as

[TABLE]

The next step is to use Property 2 (see, Appendix B) to come to Eq.(118) for a free particle fractional quantum kernel $K_{\alpha,1}^{(0)}(x,t)$. It has been shown in [19] that at $\alpha=2$ Eq.(119) goes into Eq.(111).

6.3 The time Schrödinger equation - the case when $\alpha=2$ and $0<\beta\leq 1$

In the case when $\alpha=2$ we have

[TABLE]

where $D_{2,\beta}$ is the scale coefficient with physical dimension $[D_{2,\beta}]=\mathrm{g}^{-1}\mathrm{\cdot sec}^{2-2\beta}$.

In this case the space-time fractional Schrödinger equation Eq.(1) goes into the time fractional Schrödinger equation of the form

[TABLE]

This equation can be considered as an alternative to Naber’s time fractional Schrödinger equation [5].

For the quantum mechanical kernel $K_{2,\beta}^{(0)}(x,t)=$ $K_{\alpha,\beta}^{(0)}(x,t)|_{\alpha=2},$ where $K_{\alpha,\beta}^{(0)}(x,t)$ is defined by Eq.(88) we obtain

[TABLE]

here

[TABLE]

and $L_{\beta}(u,t)$ is given by Eq.(86).

[TABLE]

[TABLE]

To express a free particle time fractional quantum kernel $K_{2,\beta}^{(0)}(x,t)$ in terms of the $H$-function let’s put $\alpha=2$ in Eq.(103)

[TABLE]

[TABLE]

[TABLE]

Using the Property 2 yields

[TABLE]

Next, by using the Property 1 and then the Property 2 we find

[TABLE]

Thus, we found a new expression of a free particle time fractional quantum kernel $K_{2,\beta}^{(0)}(x,\tau)$ in terms of $H_{1,1}^{1,0}$-finction. It can be shown that at $\beta=1$ Eq.(129) goes into Eq.(111).

7 Conclusion

Fractional quantum mechanics has emerged as a field over 15 years ago, attracting the attention of many researchers. The original idea behind fractional quantum mechanics was to develop a path integral over Lévy-like quantum paths instead of the well-known Feynman path integral over Brownian-like quantum paths. The basic outcome of implementing this idea is an alternative path integral approach, which results in a new fundamental equation – the fractional Schrödinger equation. In other words, if the Feynman path integral approach to quantum mechanics allows one to reproduce the Schrödinger equation, then the path integral over Levy-like paths leads one to the fractional Schrödinger equation. This is a manifestation of a new non-Gaussian paradigm, based on deep relationships between the structure of fundamental physics equations and fractal dimensions of “underlying” quantum paths. The fractional Schrödinger equation includes the spatial derivative of fractional order $\alpha$ ($\alpha$ is the Lévy index), instead of the second order spatial derivative in the well-known Schrödinger equation. Thus, the fractional Schrödinger equation is the fractional differential equation in accordance with modern terminology. This is the main point of the term, fractional Schrödinger equation, and the more general term, fractional quantum mechanics.

In this paper we introduce the concept of time fractional quantum mechanics. The wording ”time fractional quantum mechanics” means that the time derivative in the fundamental quantum mechanical equations - Schrödinger equation and fractional Schrödinger equation, is substituted with a fractional time derivative. The time fractional derivative in our approach is the Caputo fractional derivative.

It has to be noted, that despite some shortcomings from the stand point of quantum mechanical fundamentals, time fractional quantum mechanics attracts the attention of researchers as an interesting new application of fractional calculus to quantum mechanics. Many preliminary developments were done in the field of time fractional quantum mechanics. Naber invented time fractional Schrödinger equation [5]. The time fractional Schrödinger equation involves a time derivative of fractional order instead of first-order time derivative, while the spatial derivative is the second-order spatial derivative as it is in the well-known Schrödinger equation. Naber has found the exact solutions to the time fractional Schrödinger equation for a free particle and a particle in a potential well [5].

Later on, Wang and Xu [8], and then Dong and Xu [9], combined both Laskin’s equation and Naber’s equation and came up with space-time fractional Schrödinger equation. The space-time fractional Schrödinger equation includes both spatial and temporal fractional derivatives. Wang and Xu found exact solutions to space-time fractional Schrödinger equation for a free particle and for an infinite square potential well [8]. Dong and Xu found the exact solution to space-time fractional Schrödinger equation for a quantum particle in $\delta$-potential well [9].

To introduce and develop a time fractional quantum mechanics we begin with our own version of the space-time fractional Schrödinger equation. Our space-time fractional Schrödinger equation involves two scale dimensional parameters, one of which can be considered as time fractional generalization of the famous Planck’s constant, while the other one can be interpreted as a time fractional generalization of the scale parameter introduced by Laskin in fractional quantum mechanics [1]-[4]. The fractional generalization of Planck’s constant is a fundamental dimensional parameter of time fractional quantum mechanics, while the time fractional generalization of Laskin’s scale parameter [1]-[4] plays a fundamental role in both time fractional quantum mechanics and time fractional classical mechanics.

In addition to the above mentioned dimensional parameters, time fractional quantum  mechanics involves two dimensionless fractality parameters $\alpha$, $1<\alpha\leq 2$ and.$\beta$, $0<\beta\leq 1$. Parameter $\alpha$ is the order of spatial fractional quantum Riesz derivative [1], [2] and $\beta$ is the order of time fractional derivative Caputo derivative. In other words, $\alpha$ is responsible for modelling spatial fractality, while the parameter $\beta$, is responsible for modeling temporal fractality.

Time fractional quantum mechanical operators of coordinate, momentum and angular momentum have been introduced and their commutation relationship has been established. The pseudo-Hamilton  quantum mechanical operator has been introduced and its Hermiticity has been proven in the framework of time fractional quantum mechanics. The general solution to the space-time fractional Schrödinger equation was found in the case when the pseudo-Hamilton operator does not depend on time. Energy of a quantum system in the framework of time fractional quantum mechanics was defined and calculated in terms of the Mittag-Leffler function. Two new functions associated with the Mittag-Leffler function have been launched and elaborated. These two new functions can be considered as a natural fractional generalization of the well-known trigonometric functions sine and cosine. A fractional generalization of the celebrated Euler equation was discovered. A free particle space-time fractional quantum kernel was calculated in terms of the Fox’s $H$-functions. It has been shown that a free particle space-time fractional quantum kernel can be alternatively expressed with help of the Wright function.

The framework of time fractional quantum mechanics, depending on choices of fractality parameters $\alpha$ and $\beta$, covers the following fundamental quantum equations:

1. The Schrödinger equation (Schrödinger equation [10]), $\alpha=2$ and $\beta=1$;

2. Fractional Schrödinger equation (Laskin equation [4]), $1<\alpha\leq 2$ and $\beta=1$;

3. Time fractional Schrödinger equation (Naber equation [5]),  $\alpha=2$ and $0<\beta\leq 1$;

4. Space-time fractional Schrödinger equation (Wang and Xu [8] and Dong and Xu [9] equation), $1<\alpha\leq 2$ and $0<\beta\leq 1$.

8 Acknowledgments

I am grateful to Prof. Mark M. Meerschaert, for inviting me to give a talk at A Workshop on Future Directions in Fractional Calculus Research and Applications, which took place on 17 - 21 October 2016, at Michigan State University, East Lansing, MI 48823.

9 Appendix A

9.1 Fractional generalization of trigonometric functions $\cos(z)$

and $\sin(z)$

Here we introduce two new functions $Ec_{\beta}(z)$ and $Es_{\beta}(z)$ defined by the series

[TABLE]

and

[TABLE]

From definitions (130)-(131) we can obtain,

[TABLE]

and

[TABLE]

where $E_{\beta}(z)$ is the Mittag-Leffler function given by Eq.(52) and $i$ is imanaginery unit, $i=\sqrt{-1}$.

Hence, functions $Ec_{\beta}(z)$ and $Es_{\beta}(z)$ can be considered as a natural fractional generalization of the well-known trigonometric functions $\cos(z)$, and $\sin(z)$ respectively. Indeed, when $\beta=1$ $Ec_{\beta}(z)$ and $Es_{\beta}(z)$ become

[TABLE]

and

[TABLE]

The new expression for the Mittag-Leffler function $E_{\beta}(i^{\beta}z)$ given by Eq.(57) can be considered as a fractional generalization of the celebrated Euler equation, which is recovered from Eq.(57) in the limit case $\beta=1$,

[TABLE]

Two new expressions (132) and (133) can be considered as a fractional generalization of the well-known equations

[TABLE]

and

[TABLE]

Let us note, that Eq.(57) can be further generalized to

[TABLE]

[TABLE]

where function $E_{\beta,\gamma}(z)$ is defined by the series

[TABLE]

and two new functions $Ec_{\beta,\gamma}^{\sigma}(z)$ and $Es_{\beta,\gamma}^{\sigma}(z)$ are introduced by

[TABLE]

and

[TABLE]

respectively.

In terms of functions $Ec_{\beta,\gamma}^{\sigma}(z)$ and $Es_{\beta,\gamma}^{\sigma}(z)$, the functions $Ec_{\beta}(z)$ and $Es_{\beta}(z)$ defined by Eqs.(130) and (131) are given by

[TABLE]

10 Appendix B

10.1 Fox’s $H$-function

The $H$-function introduced by Fox [20], possesses many interesting properties which can be used in time fractional quantum mechanics to calculate integrals and perform transformations to study limiting cases at particular choices of fractality parameters.

Fox’s $H$-function is defined by the Mellin-Barnes type integral [20], [22], [17] (we follow the notations of the book [17])

[TABLE]

[TABLE]

where function $\chi(s)$ is given by

[TABLE]

and

[TABLE]

here $m$, $n$, $p$ and $q$ are non negative integers satisfying $0\leq n\leq p$, 1$\leq m\leq q$; and the empty products are interpreted as unity. The parameters $A_{j}$ ($j=1,...,p$) and $B_{j}$ ($j=1,...,q$) are positive numbers; $a_{j}$ ($j=1,...,p$) and $b_{j}$ ($j=1,...,p$) are complex numbers such that

[TABLE]

for $\nu$, $\lambda=0,1,...$; $h=1,...,m$; $j=1,...,n$.

The $C$ is a contour separating the points

[TABLE]

which are the poles of $\Gamma(b_{j}-B_{j}s)$ $(j=1,...,m)$, from the points

[TABLE]

which are the poles of $\Gamma(1-a_{j}+A_{j}s)$ $(j=1,...,n)$. The contour $C$ exists on account of (147). These assumptions will be retained throughout.

In the contracted form the $H$-function in (145) will be denoted by one of the following notations:

[TABLE]

The Fox $H$-function is an analytic function of $z$ which makes sense (i) for every $z\neq 0$ if $\mu>0$ and (ii) for $0<|z|<\beta^{-1}$ if $\mu=0$, where

[TABLE]

and

[TABLE]

Due to the occurrence of the factor $z^{s}$ in (145), the $H$ function is in general multiple-valued, but is one-valued on the Riemann surface of $\log z$.

The Mittag-Leffler $E_{\beta}(z)$ function [14] given by series (52) has the following representation in terms of the $H$-function

[TABLE]

To represent an $H$-function in computable form let us consider the case when the poles $s=(b_{j}+\nu)/B_{j}$ ($j=1,...,m;$ $\nu=0,1,...$) of $\prod\limits_{j=1}^{m}{}^{{}^{\prime}}\Gamma(b_{j}-B_{j}s)$ are simple, that is, where

[TABLE]

[TABLE]

and the prime means the product without the factor $j=h$.

Then we obtain the following expansion for the $H$-function

[TABLE]

[TABLE]

which exists for all $z\neq 0$ if $\mu>0$ and for $0<|z|,\beta^{-1}$ if $\mu=0$, where $\mu$ and $\beta$ are given by the Eqs.(149) and (150). The prime means the product without the factor $j=h$.

The formula (152) can be used for the calculation of special values of the Fox function and to derive the asymptotic behavior for $z\rightarrow 0$.

The asymptotic expansion for $|z|\rightarrow\infty$ are treated in Ref.[22] in the general case. In particular, for $\mu>0$ and $n\neq 0$

[TABLE]

as $|z|\rightarrow\infty$ uniformly on every closed subsector of $|\arg z|\leq\frac{1}{2}\pi\lambda$. The residues have to be taken at the points $s=(a_{j}-1-\nu)/\alpha_{j}$ ($j=1,...,n$; $\nu=0,1,...$) and $\lambda$ is defined by

[TABLE]

Symmetries in the parameters of the $H$ function are detected by regarding the definitions (145) and (146).

10.1.1 Some properties of the $H$-function

Here we present a list of mainly used properties of the $H-$function. The results of this section follow readily from the definition of the $H-$function (145) and hence no proofs are given here.

Property 1 The $H-$function is symmetric in the pairs $(a_{1},A_{1}),...,(a_{n},A_{n})$, likewise $(a_{n+1},A_{n+1}),...,(a_{p},A_{p})$; in $(b_{1},B_{1}),...,(b_{m},B_{m})$ and in $(b_{m+1},B_{m+1}),...,(b_{q},B_{q})$.

Property 2 If one of the $(a_{j},A_{j})$ $(j=1,...,n)$ is equal to one of the $(b_{j},B_{j})$ $(j=m+1,...,q)$ or one of the $(b_{j},B_{j})$ $(j=1,...,m)$ is equal to one of the $(a_{j},A_{j})$ $(j=n+1,...,p)$, then the $H-$function reduces to one of the lower order, and $p,q$ and $n$ (or $m)$ decrease by unity.

Thus, we have the following reduction formulas,

[TABLE]

[TABLE]

provided $n\geq 1$ and $q>m$,  and

[TABLE]

[TABLE]

provided $m\geq 1$ and $p>n$.

Property 3

[TABLE]

[TABLE]

This is an important property of the $H$-function because it enables us to transform an $H-$function with $\mu=\sum\limits_{j=1}^{m}B_{j}-\sum\limits_{j=1}^{n}A_{j}>0$ and $\arg x$ to one with $\mu<0$ and $\arg(1/x)$ and vice versa.

Property 4

[TABLE]

where $k>0$.

Property 5

[TABLE]

[TABLE]

Further interesting and important properties of the Fox $H$-function and expression for elementary and special functions in terms of the $H$-function are listed in Ref.[17].

10.2 Cosine transform of the $H$-function

The cosine transform of $H$-function $H_{p,q}^{m,n}\left[at^{\beta}{\LARGE|}\QATOP{(a_{p},A_{p})}{(b_{q},B_{q})}\right]$ is defined by [21]

[TABLE]

[TABLE]

where Re[$\nu+\alpha\min_{1\leq j\leq m}(\frac{b_{j}}{B_{j}})]>0,$ Re[$\nu+\alpha\max_{1\leq j\leq n}(\frac{a_{j}-1}{A_{j}})]<0$, $|\arg a|<\pi\lambda/2$, with $\lambda$ defined by Eq.(153).