Fractional Schrödinger equation
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin 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.
Fundamentals
The fractional Schrödinger equation in the form originally obtained by Nick Laskin is:- r is the 3-dimensional position vector,
- ħ is the reduced Planck constant,
- ψ 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 is a potential energy,
- Δ = ∂2/∂r2 is the Laplace operator.
- Dα is a scale constant with physical dimension = 1 − α·α−α, at α = 2, D2 =1/2m, where m is a particle mass,
- the operator α/2 is the 3-dimensional fractional quantum Riesz derivative defined by ;
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 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
where the fractional Hamilton operator is given by
The Hamilton operator, corresponds to the classical mechanics Hamiltonian function introduced by Nick Laskin
where p and r are the momentum and the position vectors respectively.
Time-independent fractional Schrödinger equation
The special case when the Hamiltonian is independent of timeis 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
or
This is the time-independent fractional Schrödinger equation.
Thus, we see that the wave function 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 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.
Probability current density
The conservation law of fractional quantum mechanical probability has been discovered for the first time by D.A.Tayurskii and Yu.V. Lysogorskiwhere
is the quantum mechanical probability density and the vector
can be called by the fractional probability current density vector
and
here we use the notation :.
It has been found in Ref. that there are quantum physical conditions when the new term is negligible and we come to the continuity equation for quantum probability current and quantum density :
Introducing the momentum operator we can write the vector
in the form
This is fractional generalization of the well-known equation for probability current density
vector of standard quantum mechanics.
Velocity operator
The quantum mechanical velocity operator is defined as follows:Straightforward calculation results in
Hence,
To get the probability current density equal to 1 the wave function of a free
particle has to be normalized as
where is the particle velocity,.
Then we have
that is, the vector is indeed the unit vector.
Physical applications
Fractional Bohr atom
When is the potential energy of hydrogenlike atom,where e is the electron charge and Z is the atomic number of the hydrogenlike atom,, we come to following fractional eigenvalue problem,
This eigenvalue problem has first been introduced and solved by Nick Laskin in.
Using the first Niels Bohr postulate yields
and it gives us the equation for the Bohr radius of the fractional hydrogenlike atom
Here a0 is the fractional Bohr radius defined as,
The energy levels of the fractional hydrogenlike atom are given by
where E0 is the binding energy of the electron in the lowest Bohr orbit
that is, the energy required to put it in a state with E = 0 corresponding to n = ∞,
The energy E0 divided by ħc, E0/ħc, can be considered as fractional generalization of the
Rydberg constant of standard quantum mechanics. For α = 2 and Z = 1 the formula
is transformed into
which is the well-known expression for the Rydberg formula.
According to the second Niels Bohr postulate, the frequency of radiation associated with the transition, say, for example from the orbit m to the orbit n, is,
The above equations are fractional
generalization of the Bohr model. In the special Gaussian case, when those equations give us the well-known results of the Bohr model.
The infinite potential well
A particle in a one-dimensional well moves in a potential field, which is zero forand which is infinite elsewhere,
It is evident a priori that the energy spectrum will be discrete. The solution of the fractional Schrödinger equation for the stationary state with well-defined energy E is described by a wave function, which can be written as
where, is now time independent.
In regions and,
the fractional Schrödinger
equation can be satisfied only if we take. In the middle region
, the time-independent fractional Schrödinger equation is.
This equation defines the wave functions and the energy spectrum within region, while outside
of the region, x<-a and x>a, the wave functions are zero. The wave function has to be continuous everywhere, thus we impose the boundary conditions for the solutions of the time-independent fractional Schrödinger equation. Then the solution in region can be written as
To satisfy the boundary conditions we have to choose
and
It follows from the last equation that
Then the even solution of the time-independent fractional Schrödinger equation in the infinite potential well is
The odd solution of the time-independent fractional Schrödinger equation in the infinite potential well is
The solutions and have the
property that
where is the Kronecker symbol and
The eigenvalues of the particle in an infinite potential well are
It is obvious that in the Gaussian case above equations are ö
transformed into the standard quantum mechanical equations for a particle in a box
The state of the lowest energy, the ground state, in the infinite potential
well is represented by the at n=1,
and its energy is
Fractional quantum oscillator
Fractional quantum oscillator introduced by Nick Laskin is the fractional quantum mechanical model with the Hamiltonian operator defined aswhere q is interaction constant.
The fractional Schrödinger equation for the wave
function of the fractional quantum oscillator is,
Aiming to search for solution in form
we come to the time-independent fractional Schrödinger equation,
The Hamiltonian is the fractional
generalization of the 3D quantum harmonic oscillator Hamiltonian of standard quantum
mechanics.
Energy levels of the 1D fractional quantum oscillator in semiclassical approximation
The energy levels of 1D fractional quantum oscillator with the Hamiltonian function were found in semiclassical approximation.We set the total energy equal to E, so that
whence
At the turning points. Hence, the classical motion is
possible in the range.
A routine use of the Bohr-Sommerfeld quantization rule yields
where the notation means the integral over one complete period of
the classical motion and is the turning point
of classical motion.
To evaluate the integral in the right hand we introduce a new variable. Then we have
The integral over dy can be expressed in terms of the Beta-function,
Therefore,
The above equation gives the energy levels of stationary states for the 1D fractional quantum oscillator,
This equation is generalization of the well-known energy levels equation of the
standard quantum harmonic oscillator and is transformed into it at α = 2 and β = 2.
It follows from this equation that at the energy levels are equidistant. When and the equidistant energy levels can be for α = 2 and β = 2 only. It means that the only standard quantum harmonic oscillator has an equidistant energy spectrum.