Fractional Schrödinger equation

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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[edit]

The fractional Schrödinger equation in the form originally obtained by Nick Laskin is:[2]

i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_\alpha (-\hbar
^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)

Further,

  • 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.[2]);

(-\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 position and momentum spaces; \psi(\mathbf{r},t) and  \varphi (\mathbf{p},t) are related each other by the 3-dimensional Fourier transforms:


\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot\mathbf{r}/\hbar}\varphi (\mathbf{p},t),\qquad \varphi (\mathbf{p},t)=\int d^3re^{-i
\mathbf{p}\cdot\mathbf{r}/\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.[3] This is the main point of the term fractional Schrödinger equation or a more general term fractional quantum mechanics.[4] 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 \widehat{H}_\alpha is given by


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

The Hamilton operator, \widehat{H}_\alpha corresponds to the 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 position vectors respectively.

Time-independent fractional Schrödinger equation[edit]

The special case when the Hamiltonian H_\alpha 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 \phi (\mathbf{r}) satisfies


H_\alpha \phi (\mathbf{r}) = E\phi (\mathbf{r}),

or


D_\alpha (-\hbar ^2\Delta )^{\alpha /2}\phi (\mathbf{r})+V(\mathbf{r})\phi (
\mathbf{r})=E\phi (\mathbf{r}).

This is the time-independent fractional Schrödinger equation.

Thus, we see that the wave function \psi (\mathbf{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 \mathbf{r} is the absolute square of the wave function | \psi (\mathbf{r},t) |^2 . Because of time-independent fractional Schrödinger equation this is equal to | \phi (\mathbf{r})|^2 and does not depend upon the time. That is, the probability of finding the particle at \mathbf{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.

Probability current density[edit]

The continuity equation for probability current and density follows from the fractional Schrödinger equation:


\frac{\partial \rho (\mathbf{r},t)}{\partial t}+\nabla \cdot \mathbf{j}(
\mathbf{r},t)=0,

where \rho (\mathbf{r},t)=\psi ^{\ast }(\mathbf{r},t)\psi (\mathbf{r},t) is the quantum mechanical probability density and the vector \mathbf{j}(\mathbf{r},t) can be called by the fractional probability current density vector


\mathbf{j}(\mathbf{r},t)=\frac{D_\alpha \hbar }i\left( \psi ^{*}(\mathbf{r}
,t)(-\hbar ^2\Delta )^{\alpha /2-1}\mathbf{\nabla }\psi (\mathbf{r},t)-\psi (
\mathbf{r},t)(-\hbar ^2\Delta )^{\alpha /2-1}\mathbf{\nabla }\psi ^{*}(
\mathbf{r},t)\right) ,

where we use the notation (see also matrix calculus): 
\mathbf{\nabla =\partial /\partial r}
.

Introducing the momentum operator \widehat{\mathbf{p}}=\frac{\hbar }{i}
\frac{\partial }{\partial \mathbf{r}} we can write the vector \mathbf{j} in the form (see, Ref.[2])


\mathbf{j=}D_{\alpha }\left( \psi (\widehat{\mathbf{p}}^{2})^{\alpha /2-1}
\widehat{\mathbf{p}}\psi ^{\ast }+\psi ^{\ast }(\widehat{\mathbf{p}}^{\ast
2})^{\alpha /2-1}\widehat{\mathbf{p}}^{\ast }\psi \right).

This is fractional generalization of the well-known equation for probability current density vector of standard quantum mechanics (see, Ref.[7]).

Velocity operator[edit]

The quantum mechanical velocity operator \widehat{\mathbf{v}} is defined as follows:


\widehat{\mathbf{v}}=\frac{i}{\hbar }(H_{\alpha }\widehat{\mathbf{r}}\mathbf{
-}\widehat{\mathbf{r}}H_{\alpha }),

Straightforward calculation results in (see, Ref.[2])


\widehat{\mathbf{v}}=\alpha D_{\alpha }|\widehat{\mathbf{p}}^{2}|^{\alpha
/2-1}\widehat{\mathbf{p}}\,.

Hence,


\mathbf{j=}\frac{1}{\alpha }\left( \psi \widehat{\mathbf{v}}\psi ^{\ast
}+\psi ^{\ast }\widehat{\mathbf{v}}\psi \right) ,\qquad 1<\alpha \leq 2.

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


\psi (\mathbf{r},t)=\sqrt{\frac{\alpha }{2\mathrm{v}}}\exp \left[\frac{i}{\hbar }(
\mathbf{p}\cdot\mathbf{r}-Et)\right],\qquad E=D_{\alpha }|\mathbf{p}|^{\alpha
},\qquad 1<\alpha \leq 2,

where \mathrm{v} is the particle velocity, \mathrm{v}=\alpha D_{\alpha
}p^{\alpha -1}.

Then we have


\mathbf{j=}\frac{\mathbf{v}}{\mathrm{v}},\qquad \mathbf{v}=\alpha D_{\alpha
}|\mathbf{p}^{2}|^{\frac{\alpha }{2}-1}\mathbf{p,}

that is, the vector \mathbf{j} is indeed the unit vector.

Physical applications[edit]

Fractional Bohr atom[edit]

Main article: Bohr atom

When V(\mathbf{r}) is the potential energy of hydrogenlike atom,


V(\mathbf{r})=-\frac{Ze^{2}}{|\mathbf{r}|},

where e is the electron charge and Z is the atomic number of the hydrogenlike atom, (so Ze is the nuclear charge of the atom), we come to following fractional eigenvalue problem,


D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\phi (\mathbf{r})-\frac{Ze^{2}}{|
\mathbf{r|}}\phi (\mathbf{r})=E\phi (\mathbf{r}).

This eigenvalue problem has first been solved in.[5]

Using the first Niels Bohr postulate yields


 \alpha
D_{\alpha }\left( \frac{n\hbar }{a_{n}}\right) ^{\alpha }=\frac{Ze^{2}}{a_{n}
},

and it gives us the equation for the Bohr radius of the fractional hydrogenlike atom


a_{n}=a_{0}n^{\alpha /(\alpha -1)}.

Here a0 is the fractional Bohr radius (the radius of the lowest, n = 1, Bohr orbit) defined as,


a_{0}=\left( \frac{\alpha D_{\alpha }\hbar ^{\alpha }}{Ze^{2}}\right) ^{1/(\alpha -1)}.

The energy levels of the fractional hydrogenlike atom are given by


E_{n}=(1-\alpha )E_{0}n^{-\alpha/(\alpha -1)},\qquad 1<\alpha \leq 2,

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 = ∞,


E_{0}=\left( \frac{Ze^{2}}{\alpha D_{\alpha }^{1/\alpha }\hbar }\right) ^{\alpha/(\alpha -1)}.

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 (\alpha -1)E_{0}/\hbar c is transformed into

\mathrm{Ry}=me^{4}/2\hbar ^{3}c,

which is the well-known expression for the Rydberg formula.

According to the second Niels Bohr postulate, the frequency of radiation \omega associated with the transition, say, for example from the orbit m to the orbit n, is,


\omega =\frac{(1-\alpha )E_{0}}{\hbar }\left[ \frac{1}{n^{\frac{\alpha }{
\alpha -1}}}-\frac{1}{m^{\frac{\alpha }{\alpha -1}}}\right]  
.

The above equations are fractional generalization of the Bohr model. In the special Gaussian case, when (α = 2) those equations give us the well-known results of the Bohr model.[6]

The infinite potential well[edit]

A particle in a one-dimensional well moves in a potential field V({x}), which is zero for -a\leq x\leq a and which is infinite elsewhere,


V(x)=\infty ,\qquad x<-a\qquad \qquad (\mathrm{i})

V(x)=0,\quad -a\leq x\leq a\quad \quad \quad \ (\mathrm{ii})

V(x)=\infty ,\qquad \ x>a\qquad \qquad \ (\mathrm{iii})

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 \psi (x), which can be written as


\psi (x,t)=\left(-i\frac{Et}{\hbar }\right)\phi(x)
,

where \phi (x), is now time independent. In regions (i) and (iii), the fractional Schrödinger equation can be satisfied only if we take \phi (x)=0. In the middle region (ii), the time-independent fractional Schrödinger equation is


D_\alpha (\hbar \nabla )^\alpha \phi (x)=E\phi (x).

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 \phi (x) has to be continuous everywhere, thus we impose the boundary conditions \phi (-a)=\phi (a)=0 for the solutions of the time-independent fractional Schrödinger equation (see, Ref.[5]). Then the solution in region (ii) can be written as


\phi (x)=A\exp(ikx) +B\exp(-ikx).

To satisfy the boundary conditions we have to choose


A = -B\exp(-i2ka),

and


\sin(2ka) =0.

It follows from the last equation that


2ka = n\pi.

Then the even (\phi _n^{\mathrm{even}}(-x) = \phi _n^{\mathrm{even}}(x) under reflection x\rightarrow -x) solution of the time-independent fractional Schrödinger equation \phi ^{\mathrm{even}}(x) in the infinite potential well is


\phi _n^{\mathrm{even}}(x)=\frac 1{\sqrt{a}}\cos \left[ \frac{
n\pi x}{2a}\right] , \quad n = 1, 3, 5, ....

The odd (\phi _n^{\mathrm{odd}}(-x) = -\phi _n^{\mathrm{odd}}(x) under reflection x\rightarrow -x) solution of the time-independent fractional Schrödinger equation \phi ^{\mathrm{even}}(x) in the infinite potential well is


\phi _n^{\mathrm{odd}}(x)=\frac 1{\sqrt{a}}\sin \left[ \frac{
n\pi x}{2a}\right] , \quad n = 2, 4, 6, ....

The solutions \phi ^{\mathrm{even}}(x) and \phi ^{\mathrm{odd}}(x) have the property that


\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{even}}(x)\phi _{n}^{\mathrm{even}
}(x)=\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{odd}}(x)\phi _{n}^{\mathrm{odd}
}(x)=\delta _{mn},

where \delta _{mn} is the Kronecker symbol and


\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{even}}(x)\phi _{n}^{\mathrm{odd}
}(x)=0.

The eigenvalues of the particle in an infinite potential well are (see, Ref.[5])


E_n=D_\alpha \left( \frac{\pi \hbar }{2a}\right) ^\alpha n^\alpha ,\qquad
\qquad n=1,2,3....,\qquad 1<\alpha \leq 2.

It is obvious that in the Gaussian case (α = 2) above equations are transformed into the standard quantum mechanical equations for a particle in a box (for example, see Eq.(20.7) in [7])

The state of the lowest energy, the ground state, in the infinite potential well is represented by the \phi _n^{\mathrm{even}}(x) at n=1,


\phi _{\mathrm{ground}}(x)\equiv \phi _1^{\mathrm{even}}(x)=\frac 1{\sqrt{a}
}\cos \left(\frac{\pi x}{2a}\right),

and its energy is


E_{\mathrm{ground}}=D_{\alpha }\left( \frac{\pi \hbar }{2a}\right) ^{\alpha
}.

Fractional quantum oscillator[edit]

Fractional quantum oscillator introduced by Nick Laskin (see, Ref.[2]) is the fractional quantum mechanical model with the Hamiltonian operator H_{\alpha ,\beta } defined as


H_{\alpha,\beta}=D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}+q^{2}|\mathbf{
r}|^{\beta },\quad 1<\alpha \leq 2,\quad 1<\beta \leq 2,  
,

where q is interaction constant.

The fractional Schrödinger equation for the wave function \psi (\mathbf{r},t) of the fractional quantum oscillator is,


i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_{\alpha }(-\hbar
^{2}\Delta )^{\alpha /2}\psi (\mathbf{r},t)+q^{2}|\mathbf{r}|^{\beta }\psi (
\mathbf{r},t)

Aiming to search for solution in form


\psi (\mathbf{r},t)=e^{-iEt/\hbar }\phi (\mathbf{r}),

we come to the time-independent fractional Schrödinger equation,


D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\phi (\mathbf{r},t)+q^{2}|\mathbf{
r}|^{\beta }\phi (\mathbf{r},t)=E\phi (\mathbf{r},t).

The Hamiltonian H_{\alpha,\beta} 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[edit]

The energy levels of 1D fractional quantum oscillator with the Hamiltonian function 
H_{\alpha}=D_{\alpha }|p|^{\alpha }+q^{2}|x|^{\beta } can be found in semiclassical approximation.

We set the total energy equal to E, so that


E=D_{\alpha }|p|^{\alpha }+q^{2}|x|^{\beta },

whence


|p|=\left( \frac{1}{D_{\alpha }}(E-q^{2}|x|^{\beta })\right)
^{1/\alpha }
.

At the turning points p=0. Hence, the classical motion is possible in the range |x|\leq (E/q^{2})^{1/\beta }.

A routine use of the Bohr-Sommerfeld quantization rule yields


2\pi \hbar (n+\frac{1}{2})=\oint
pdx=4\int\limits_{0}^{x_{m}}pdx=4\int\limits_{0}^{x_{m}}D_{\alpha
}^{-1/\alpha }(E-q^{2}|x|^{\beta })^{1/\alpha }dx,

where the notation \oint  means the integral over one complete period of the classical motion and x_{m}=(E/q^{2})^{1/\beta } is the turning point of classical motion.

To evaluate the integral in the right hand we introduce a new variable y=x(E/q^{2})^{-1/\beta }. Then we have


\int\limits_0^{x_m}D_\alpha ^{-1/\alpha }(E-q^2|x|^\beta )^{1/\alpha }dx=\frac 1{D_\alpha ^{1/\alpha }q^{2/\beta }}E^{\frac 1\alpha +\frac 1\beta
}\int\limits_0^1dy(1-y^\beta )^{1/\alpha }.

The integral over dy can be expressed in terms of the Beta-function,


\int\limits_{0}^{1}dy(1-y^{\beta })^{1/\alpha }=\frac{1}{\beta }
\int\limits_{0}^{1}dzz^{\frac{1}{\beta }-1}(1-z)^{\frac{1}{\alpha }}=\frac{1
}{\beta }\Beta \left(\frac{1}{\beta },\frac{1}{\alpha }+1\right).

Therefore


2\pi \hbar (n+\frac 12)=\frac 4{D_\alpha ^{1/\alpha }q^{2/\beta }}E^{\frac
1\alpha +\frac 1\beta }\frac 1\beta \Beta\left(\frac 1\beta ,\frac 1\alpha +1\right).

The above equation gives the energy levels of stationary states for the 1D fractional quantum oscillator (see, Ref.[2]),


E_{n}=\left( \frac{\pi \hbar \beta D_{\alpha }^{1/\alpha }q^{2/\beta }}{2\Beta(
\frac{1}{\beta },\frac{1}{\alpha }+1)}\right) ^{\frac{\alpha \beta }{\alpha
+\beta }}\left(n+\frac{1}{2}\right)^{\frac{\alpha \beta }{\alpha +\beta }}.

This equation is generalization of the well-known energy levels equation of the standard quantum harmonic oscillator (see, Ref.[7]) and is transformed into it at α = 2 and β = 2. It follows from this equation that at \frac{1}{\alpha }+\frac{1}{\beta }=1 the energy levels are equidistant. When 1<\alpha \leq 2 and 1<\beta \leq 2 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.

See also[edit]

References[edit]

  1. ^ N. Laskin, (2000), Fractional Quantum Mechanics and Lévy Path Integrals. Physics Letters 268A, 298-304.
  2. ^ N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: http://arxiv.org/abs/quant-ph/0206098)
  3. ^ S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993
  4. ^ N. Laskin, (2000), Fractional Quantum Mechanics, Physical Review E62, 3135-3145. (also available online: http://arxiv.org/abs/0811.1769)
  5. ^ N. Laskin, (2000), Fractals and quantum mechanics. Chaos 10, 780-790
  6. ^ N. Bohr, (1913), Phil. Mag. 26, 1, 476, 857
  7. ^ L.D. Landau and E.M. Lifshitz, Quantum mechanics (Non-relativistic Theory), Vol.3, Third Edition, Course of Theoretical Physics, Butterworth-Heinemann, Oxford, 2003

Further reading[edit]