Fractional quantum mechanics

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

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

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]);

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

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

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

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.