Fractional Schrödinger equation

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by N. Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The fractional Schrödinger equation has the following form :

$i\backslash hbar\; \backslash frac\{\backslash partial\; \backslash psi\; (\backslash mathbf\{r\},t)\}\{\backslash partial\; t\}=D\_\backslash alpha\; (-\backslash hbar$

^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t).

Here r is a 3-dimensional vector, ℏ is the Planck constant, ψ(r,t) is the wavefunction, which is the quantum mechanical probability amplitude for the particle to have a given position r at any given time t, V(r,t) is a potential energy, and Δ = ∂²/∂r² is the Laplace operator. Further, D_α is a scale constant with physical dimension [D_α] = erg^1 − α ⋅ cm^α ⋅ sec^−α, (at α = 2, D₂ =1/2m, where m is a particle mass), and the operator (−ℏ²Δ)^α/2 is the 3-dimensional fractional quantum Riesz derivative defined by

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

Here the wave functions in the space ψ(r,t) and momentum φ(p,t) representations are related each other by the 3-dimensional Fourier transforms

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

The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2. Thus, 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. This is the main point of the term fractional Schrödinger equation or a more general term fractional quantum mechanics. At α = 2 fractional Schrödinger equation becomes the well-known Schrödinger equation.

The fractional Schrödinger equation has the following operator form

i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=\widehat{H}_\alpha \psi (\mathbf{r},t),

where the fractional Hamilton operator Ĥ_α is given by

\widehat{H}_\alpha =D_\alpha (-\hbar ^2\Delta )^{\alpha /2}+V(\mathbf{r},t).

The Hamilton operator, Ĥ_α corresponds to classical mechanics Hamiltonian function

H_\alpha (\mathbf{p},\mathbf{r})=D_\alpha |\mathbf{p}|^\alpha +V(\mathbf{r},t),

where p and r are the momentum and the coordinate respectively.

Time-independent fractional Schrödinger equation

The special case when the Hamiltonian H_α is independent of time

H_\alpha =D_\alpha (-\hbar ^2\Delta )^{\alpha /2}+V(\mathbf{r}), is of great importance for physical applications. It is easy to see that in this case there exist the special solution of the fractional Schrödinger equation

\psi (\mathbf{r},t)=e^{-(i/\hbar )Et}\phi (\mathbf{r}), where ϕ(r) satisfies

H_αϕ(r) = Eϕ(r),

or

D_α(−ℏ²Δ)^α/2ϕ(r) + V(r)ϕ(r) = Eϕ(r).

This is the time-independent fractional Schrödinger equation.

Thus, we see that the wave function ψ(r,t) oscillates with a definite frequency. In classical physics the frequency corresponds to the energy. Therefore, the quantum mechanical state has a definite energy E. The probability to find a particle at r is the absolute square of the wave function |ψ(r,t)|². Because of time-independent fractional Schrödinger equation this is equal to |ϕ(r)|² and does not depend upon the time. That is, the probability of finding the particle at r is independent of the time. One can say that the system is in a stationary state. In other words, there is no variation in the probabilities as a function of time.

See also

• Fractional quantum mechanics
• Schrödinger equation
• Path integral formulation
• Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
• Fractional calculus