In physics, **fractional quantum mechanics** is a generalization of standard quantum mechanics, which naturally comes out when the Brownian-like quantum paths substitute with the Lévy-like ones in the Feynman path integral. This concept was discovered by Nick Laskin who coined the term *fractional quantum mechanics*.^{ [1] }

Standard quantum mechanics can be approached in three different ways: the matrix mechanics, the Schrödinger equation and the Feynman path integral.

The Feynman path integral ^{ [2] } is the path integral over Brownian-like quantum-mechanical paths. Fractional quantum mechanics has been discovered by Nick Laskin (1999) as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. A path integral over the Lévy-like quantum-mechanical paths results in a generalization of quantum mechanics.^{ [3] } If the Feynman path integral leads to the well known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.^{ [4] } The Lévy process is characterized by the Lévy index *α*, 0 < *α* ≤ 2. At the special case when *α* = 2 the Lévy process becomes the process of Brownian motion. The fractional Schrödinger equation includes a space derivative of fractional order *α* instead of the second order (*α* = 2) space derivative in the standard Schrödinger equation. Thus, the fractional Schrödinger equation is a fractional differential equation in accordance with modern terminology.^{ [5] } This is the key point to launch the term fractional Schrödinger equation and more general term *fractional quantum mechanics*. As mentioned above, at *α* = 2 the Lévy motion becomes Brownian motion. Thus, fractional quantum mechanics includes standard quantum mechanics as a particular case at *α* = 2. The quantum-mechanical path integral over the Lévy paths at *α* = 2 becomes the well-known Feynman path integral and the fractional Schrödinger equation becomes the well-known Schrödinger equation.

The fractional Schrödinger equation discovered by Nick Laskin has the following form (see, Refs.[1,3,4])

using the standard definitions:

**r**is the 3-dimensional position vector,*ħ*is the reduced Planck constant,*ψ*(**r**,*t*) is the wavefunction, which is the quantum mechanical function that determines the probability amplitude for the particle to have a given position**r**at any given time*t*,*V*(**r**,*t*) is a potential energy,- Δ = ∂
^{2}/∂**r**^{2}is the Laplace operator.

Further,

*D*is a scale constant with physical dimension [D_{α}_{α}] = [energy]^{1 − α}·[length]^{α}[time]^{−α}, at*α*= 2,*D*_{2}=1/2*m*, where*m*is a particle mass,- the operator (−
*ħ*^{2}Δ)^{α/2}is the 3-dimensional fractional quantum Riesz derivative defined by (see, Refs.[3, 4]);

Here, the wave functions in the position and momentum spaces; and are related each other by the 3-dimensional Fourier transforms:

The index *α* in the fractional Schrödinger equation is the Lévy index, 1 < *α* ≤ 2.

The effective mass of states in solid state systems can depend on the wave vector k, i.e. formally one considers m=m(k). Polariton Bose-Einstein condensate modes are examples of states in solid state systems with mass sensitive to variations and locally in k fractional quantum mechanics is experimentally feasible.

The **de Broglie–Bohm theory**, also known as the *pilot wave theory*, **Bohmian mechanics**, **Bohm's interpretation**, and the **causal interpretation**, is an interpretation of quantum mechanics. In addition to a wavefunction on the space of all possible configurations, it also postulates an actual configuration that exists even when unobserved. The evolution over time of the configuration is defined by a guiding equation that is the nonlocal part of the wave function. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

The **Schrödinger equation** is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

A **wave function** in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters *ψ* and Ψ.

**Fractional calculus** is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D

**Functional integration** is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields.

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, **relativistic wave equations** predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as ψ or Ψ, are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations.

In theoretical physics, the **pilot wave theory**, also known as **Bohmian mechanics**, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat. To solve these problems, the theory is inherently nonlocal.

**Zitterbewegung** ("jittery motion" in German) is a predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of 2*mc*^{2}/*ℏ*, or approximately 1.6×10^{21} radians per second. For the hydrogen atom, zitterbewegung can be invoked as a heuristic way to derive the Darwin term, a small correction of the energy level of the s-orbitals.

The **Schrödinger–Newton equation**, sometimes referred to as the **Newton–Schrödinger** or **Schrödinger–Poisson equation**, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form.

In quantum mechanics, the **Hellmann–Feynman theorem** relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In quantum mechanics, the **momentum operator** is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

The **quantum potential** or **quantum potentiality** is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952.

The **Feynman checkerboard**, or **relativistic chessboard** model, was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin-½ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime as discrete sums.

The **Madelung equations**, or the **equations of quantum hydrodynamics**, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation.

In physics, **relativistic quantum mechanics (RQM)** is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light *c*, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. *Non-relativistic quantum mechanics* refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. *Relativistic quantum mechanics* (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

The **fractional Schrödinger equation** is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term *fractional Schrödinger equation* was coined by Nick Laskin.

In quantum mechanics, energy is defined in terms of the **energy operator**, acting on the wave function of the system as a consequence of time translation symmetry.

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena

- ↑ Laskin, Nikolai (2000). "Fractional quantum mechanics and Lévy path integrals".
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