Fractional Schrödinger equation
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. 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.
- 1 Fundamentals
- 2 Physical applications
- 3 Two critiques and two crises
- 4 See also
- 5 References
- 6 Further reading
- r is the 3-dimensional position vector,
- ħ is the reduced 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,
- Δ = ∂2/∂r2 is the Laplace operator.
- Dα is a scale constant with physical dimension [Dα] = [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, Ref.);
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
where the fractional Hamilton operator is given by
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 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
This is the time-independent fractional Schrödinger equation (see, Ref.).
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. Lysogorski 
where is the quantum mechanical probability density and the vector can be called by the fractional probability current density vector
here we use the notation (see also matrix calculus): .
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 (see, Ref.):
Introducing the momentum operator we can write the vector in the form (see, Ref.)
This is fractional generalization of the well-known equation for probability current density vector of standard quantum mechanics (see, Ref.).
Though the extra term appeared in the paper by Tayurskii and Lysogorski before the Comment by Wei, there were several serious fundamental confusions in this paper.
1) It is well known that He atoms does not have fractional kinetic energy, but the authors claimed fractional Schrodinger equation could be applied to superfluid when they are in fractal space. In fact fractional quantum mechanics has nothing to do with fractional space unless some one can find solid evidence first.
2) The authors said the extra term K is 'the additional source in fractal space' rather than the common space. The authors failed to pointed out that missing term K was Laskin's mistake so that many authors have been calculating scattering problem in their papers using method and definitions similar to standard quantum mechanics.
3) The authors assumed the term K be zero to reach the continuity equation about superfluid, but there were not the needed conditions in superfluid at all.
In one word, the authors made the term K very confusing. Laskin's widely spread mistake was not corrected until Wei's Comment.
The quantum mechanical velocity operator is defined as follows:
Straightforward calculation results in (see, Ref.)
To get the probability current density equal to 1 (the current when one particle passes through unit area per unit time) 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.
Fractional Bohr atom
When is the potential energy of hydrogenlike atom,
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 (the radius of the lowest, n = 1, Bohr orbit) 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 (α − 1)E0 divided by ħc, (α − 1)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 infinite potential well
A particle in a one-dimensional well moves in a potential field , which is zero for and 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 (i) and (iii), the fractional Schrödinger equation can be satisfied only if we take . In the middle region (ii), the time-independent fractional Schrödinger equation is (see, Ref.).
This equation defines the wave functions and the energy spectrum within region (ii), while outside of the region (ii), 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 (see, Ref.). Then the solution in region (ii) can be written as
To satisfy the boundary conditions we have to choose
It follows from the last equation that
Then the even ( under reflection ) solution of the time-independent fractional Schrödinger equation in the infinite potential well is
The odd ( under reflection ) 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 (see, Ref.)
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
Critique: However, there was a repeated argument about Laskin's solution on the infinite square well. Jeng, Hawkins and Schwarz  criticized that Laskin's piece-wise solution was completely wrong since fractional Schrödinger equation is non-local. Bayin supported Laskin, but Yury repeated that Laskin's solution was wrong. Wei had the last words .
Wei pointed out that Laskin's solution is not the solution to the fractional Schrödinger equation when α=1 and α=2, and that there is no mathematical solution to the fractional Schrödinger equation with an infinite square well potential for 0<α<2. Though Laskin's solution is wrong from the mathematical viewpoint, it may be a solution in the sense of the limit, that is, it may be the limit of the solution for a finite square well as the depth of the well increases to infinity.
Fractional quantum oscillator
where 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
We set the total energy equal to E, so that
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,
The above equation gives the energy levels of stationary states for the 1D fractional quantum oscillator (see, Ref.),
This equation is generalization of the well-known energy levels equation of the standard quantum harmonic oscillator (see, Ref.) 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.
Fractional quantum mechanics in solid state systems
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 .
Two critiques and two crises
As we mentioned above, there were two academic critiques on fractional quantum mechanics, which resulted in two crises.
On piece-wise solutions
In 2010, Zeng et al criticized that fractional Schrodinger equation is non-local and Laskin's piece-wise solution for infinite square well was wrong. According to Zeng, all the solutions of fractional Schrodinger equation except one were based on piece-wise method. This became the first crisis in fractional quantum mechanics.
In order to overcome this crisis, Wei provided an elegant reasoning to show that Laskin's solution may be a limit of the solution for the finite square well. Meanwhile, some non-piecewise solutions were reported in
On unusual probability continuity equation
In 2016, Wei pointed out a term in probability continuity equation was missing in Laskin's highly cited paper. Without this term, the scattering problem in fractional quantum mechanics is almost the same as the standard quantum mechanics; with this term, the scattering problem becomes conceptually different. However，all the papers calculated scattering problem in traditional way except one paper by Tayurskii and Lysogorskiy. This becomes the second crisis and researchers are revising their previous papers.
There are two additional points in Wei's comment on fractional quantum mechanics.
1) Schrodinger equation was independent of uncertainty relation. Fractional Schrodinger equation did not bring us new certainty relation.
2) There are no particles with fractional kinetic energy and its almost impossible to find the application of fractional quantum mechanics.
Laskin had a reply, insisting that fractional uncertainty relation was correct and that people have found and will find the applications of fractional quantum mechanics.
- Schrödinger equation
- Path integral formulation
- Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
- Fractional calculus
- Quantum harmonic oscillator
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