Fractional Schrödinger equation - AbsoluteAstronomy.com

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics

Fractional quantum mechanics

In physics, fractional quantum mechanics is a generalization of standard quantum mechanics. The term fractional quantum mechanics was coined by Nick Laskin...

. 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. The fractional Schrödinger equation has the following form :

Here

is a 3-dimensional vector,

is the Planck constant

Planck constant

The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

is the wavefunction

Wavefunction

Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

, which is the quantum mechanical probability amplitude for the particle to have a given position

at any given time t,

is a potential energy, and

is the Laplace operator

Laplace operator

In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

. Further,

is a scale constant with physical dimension

, (at α = 2, D₂ =1/2m, where m is a particle mass), and the operator

is the 3-dimensional fractional quantum Riesz derivative defined by

Here the wave functions in the space

and momentum

representations 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. Thus, the fractional Schrödinger equation includes a space derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

of fractional order α instead of the second order (α = 2) space derivative in the standard Schrödinger equation

Schrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

. 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

Fractional quantum mechanics

In physics, fractional quantum mechanics is a generalization of standard quantum mechanics. The term fractional quantum mechanics was coined by Nick Laskin...

. At α = 2 fractional Schrödinger equation becomes the well-known Schrödinger equation

Schrödinger equation

.

The fractional Schrödinger equation has the following operator form

where the fractional Hamilton operator

is given by

The Hamilton operator,

corresponds to classical mechanics Hamiltonian function

where

and

are the momentum and the coordinate respectively.

Time-independent fractional Schrödinger equation

The special case when the Hamiltonian

is independent of time

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

where

satisfies

This is the time-independent fractional Schrödinger equation.

Thus, we see that the wave function

oscillates with a definite frequency. In classical physics

Classical physics

What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

the frequency corresponds to the energy. Therefore, the quantum mechanical state has a definite energy E.
The probability to find a particle at

is the absolute square of the wave function

Because of time-independent fractional Schrödinger equation this is equal to

and does not depend upon the time.
That is, the probability of finding the particle at

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.

Time-independent fractional Schrödinger equation

See also