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 See also
- 4 References
- 5 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.
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 continuity equation for probability current and density follows from the fractional Schrödinger equation:
where is the quantum mechanical probability density and the vector can be called by the fractional probability current density vector
where we use the notation (see also matrix calculus): .
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.).
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,
This eigenvalue problem has first been solved 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 (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
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
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.
- 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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