Fractional quantum mechanics

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]

Fundamentals

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.

Fractional Schrödinger equation

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

[math]\displaystyle{ i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_\alpha (-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)\, }[/math]

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/∂r² is the Laplace operator.

Further,

D_α is a scale constant with physical dimension [D_α] = [energy]^{1 − α}·[length]^α[time]^−α, at α = 2, D₂ =1/2m, where m is a particle mass,
the operator (−ħ²Δ)^α/2 is the 3-dimensional fractional quantum Riesz derivative defined by (see, Refs.[3, 4]);

[math]\displaystyle{ (-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot \mathbf{r}/\hbar}|\mathbf{p}|^\alpha \varphi ( \mathbf{p},t), }[/math]

Here, the wave functions in the position and momentum spaces; [math]\displaystyle{ \psi(\mathbf{r},t) }[/math] and [math]\displaystyle{ \varphi (\mathbf{p},t) }[/math] are related each other by the 3-dimensional Fourier transforms:

[math]\displaystyle{ \psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot\mathbf{r}/\hbar}\varphi (\mathbf{p},t),\qquad \varphi (\mathbf{p},t)=\int d^3re^{-i \mathbf{p}\cdot\mathbf{r}/\hbar }\psi (\mathbf{r},t). }[/math]

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