Quantum Well in Fractional Quantum Mechanics

TL;DR

This paper derives an exact energy spectrum solution for a particle in a quantum well within fractional quantum mechanics, introduces a graphical method for energy level determination, and discusses implications for emulating fractional quantum effects.

Contribution

It provides the first exact solution for energy levels in a fractional quantum well and a simple graphical algorithm for their calculation.

Findings

Exact energy spectrum derived for fractional quantum well

Graphical algorithm for energy level determination

Potential for emulating fractional quantum mechanics in quantum wells

Abstract

Within the framework of fractional quantum mechanics, an exact solution has been found for the energy spectrum of a quantum particle confined in a quantum well - a symmetric one-dimensional finite potential well. A simple graphical algorithm is described for obtaining the number of discrete levels in a quantum well and their associated energy values. The presented results open up new possibilities for emulating fractional quantum mechanics using quantum wells.

Equations

$$i\hslash \frac{\partial \psi (x,t)}{\partial t}=-D_{\alpha }(\hslash \nabla {)}^{\alpha }\psi (x,t)+V(x)\psi (x,t),1<\alpha \le 2,$$

$$(\hslash \nabla {)}^{\alpha }\psi (x,t)=-\frac{1}{2\pi \hslash }\underset{-\infty }{\overset{\infty }{\int }}dp{e}^{i\frac{px}{\hslash }}|p{|}^{\alpha }\phi (p,t),$$

$$\phi (p,t)=\underset{-\infty }{\overset{\infty }{\int }}dx{e}^{-i\frac{px}{\hslash }}\psi (x,t),$$

$$\psi (x,t)=\frac{1}{2\pi \hslash }\underset{-\infty }{\overset{\infty }{\int }}dp{e}^{i\frac{px}{\hslash }}\phi (p,t).$$

$$V(x)=U,x<-a,(\mathrm{i}),$$

$$V(x)=0,-a\le x\le a,(\mathrm{ii}),$$

$$V(x)=U,x>a.(\mathrm{iii}).$$

$$-D_{\alpha }(\hslash \nabla {)}^{\alpha }\varphi (x)+V(x)\varphi (x)=E\varphi (x).$$

$$-D_{\alpha }(\hslash \nabla {)}^{\alpha }{\varphi }_{\mathrm{i},\mathrm{iii}}(x)+U{\varphi }_{\mathrm{i},\mathrm{iii}}(x)=E{\varphi }_{\mathrm{i},\mathrm{iii}}(x),$$

$$-D_{\alpha }(\hslash \nabla {)}^{\alpha }{\varphi }_{\mathrm{ii}}(x)=E{\varphi }_{\mathrm{ii}}(x),$$

$${\varphi }_{\mathrm{i},\mathrm{iii}}(x)=A{e}^{\kappa x}+B{e}^{-\kappa x},$$

$$\kappa =\frac{1}{\hslash }(\frac{U-E}{D_{\alpha }}{)}^{1/\alpha },E<U,1<\alpha \le 2,$$

$${\varphi }_{\mathrm{ii}}(x)=C\cos \mathrm{k}x+D\sin \mathrm{k}x,$$

$$\mathrm{k}=\frac{1}{\hslash }(\frac{E}{D_{\alpha }}{)}^{1/\alpha },1<\alpha \le 2.$$

$${\varphi }_{\mathrm{i}}(-a)={\varphi }_{\mathrm{ii}}(-a),{\varphi }_{\mathrm{ii}}(a)={\varphi }_{\mathrm{iii}}(a),$$

$${\varphi }_{\mathrm{i}}^{\prime }(-a)={\varphi }_{\mathrm{ii}}^{{}^{\prime }}(-a),{\varphi }_{\mathrm{ii}}^{{}^{\prime }}(a)={\varphi }_{\mathrm{iii}}^{{}^{\prime }}(a),$$

$${\varphi }_{\mathrm{i},\mathrm{ii},\mathrm{iii}}^{\prime }(\pm a)=\frac{d}{dx}{\varphi }_{\mathrm{i},\mathrm{ii},\mathrm{iii}}(x){|}_{x=\pm a}.$$

$${\varphi }_{\mathrm{ii}}^{(\mathrm{even})}(x)=C\cos \mathrm{k}x.$$

$${\varphi }_{\mathrm{i}}^{(\mathrm{even})}(x)=B{e}^{-\kappa x},x>a,$$

$${\varphi }_{\mathrm{iii}}^{(\mathrm{even})}(x)=A{e}^{\kappa x},x<-a.$$

$$C\cos \mathrm{k}a=B{e}^{-\kappa a},$$

$$\mathrm{k}C\sin \mathrm{k}a=\kappa B{e}^{-\kappa a}.$$

$$C\cos \mathrm{k}(-a)=A{e}^{-\kappa a},$$

$$-\mathrm{k}C\sin \mathrm{k}(-a)=\kappa A{e}^{-\kappa a}.$$

$$\kappa =\mathrm{k}\tan \mathrm{k}a,(\mathrm{even}).$$

$$\kappa =\frac{1}{\hslash }\sqrt{2m(U-E)},E<U,$$

$$\mathrm{k}=\frac{1}{\hslash }\sqrt{2mE)},$$

$${\varphi }_{\mathrm{ii}}^{(\mathrm{odd})}(x)=D\sin \mathrm{k}x.$$

$${\varphi }_{\mathrm{i}}^{(\mathrm{odd})}(x)=B{e}^{-\kappa x},x>a,$$

$${\varphi }_{\mathrm{iii}}^{(\mathrm{odd})}(x)=A{e}^{\kappa x},x<-a.$$

Full text

The presented results open up new possibilities for emulating fractional quantum mechanics using quantum wells.

*PACS *numbers: 03.65.Ge; 78.67.De; 78.55.Et.

Keywords: Quantum well, fractional quantum mechanics, 1D fractional Schrödinger equation, Energy spectrum, Semiconductor layered structures

1 Introduction

A simple and exactly solvable problem of quantum mechanics is the problem of finding the spectrum of a particle confined in a one-dimensional (1D) potential field having a rectangular shape, see §22 in [1]. Such a potential field is called a rectangular potential well. A potential well can have infinite or finite depth.

Interest in 1D quantum mechanical models is driven by the study, design and developments of quantum wells. Quantum well (QW) is a semiconductor layered structure that fabricated by sandwiching a thin layer of semiconductor ”well material” between two layers of a semiconductor ”barrier material”. Comparing to well material the barrier material has a wider band gap. As an example of such layered semiconductors structure let’s mention layer of gallium arsenide GaAs (well material) sandwiched between two layers of aluminum gallium arsenide AlGaAs (barrier material). Another example is a thin layer of indium gallium nitride InGaN (well material) sandwiched between two layers of gallium nitride GaN (barrier material). Due to the difference in band gaps between InGaN and GaN, there is a 1D quantum potential well that confines electrons and holes, resulting in quantum confinement. As a result of the confinement, the energy levels of the confined particles are quantized, that is, they become discrete. The number of discrete levels in QW and their corresponding energies can be tuned by adjusting the depth and thickness of QW. Which in turn allows the emission of light at specific wavelengths to be controlled. GaN-based devices such as light-emitting diodes (LEDs), laser diodes (LDs), and photodetectors are widely used in many fields.

It is well known that impurities and their interactions with electrons and holes affect the electronic and optical properties of semiconductor materials. The impurities are distributed randomly in a semiconductor material. Therefore, the problem of modeling the energy spectrum in QW requires the use of the Schrödinger equation with a potential modified by stochastic component induced by impurities. In [2] it was suggested to mock-up the disorder in QW implementing fractional Schrödinger equation with regular (non stochastic) potential. The fractality parameter $\alpha$ $(1<\alpha\leq 2)$, the Levy index [3], allows modeling the impact of the impurities distribution in a semiconductor on the energy spectrum of a quantum particle in QW. As shown in the work [2], ”weak disorder” is characterized by a Gaussian probability distribution, which is local and occurs when the parameter $\alpha=2$. The case of ”strong disorder” occurs when $\alpha$ deviates from 2, leading to long-range non-local distributions. Thus, the idea is to use framework of fractional quantum mechanics [3] to model energy spectrum in QW.

In the fractional quantum mechanics [3] the problem of finding the spectrum of a particle confined in 1D infinite potential well was solved first in [4]. Despite the initial controversy surrounding this exact solution, it is now a generally accepted result thanks to S. Bayin [5], who proved the correctness of the solution obtained in [4] by directly substituting this solution into the 1D fractional Schrödinger equation and performing the calculations.

It turns out that in fractional quantum mechanics the problem of finding the spectrum of quantum mechanical particle moving in symmetrical 1D finite potential well is also exactly solvable. Moreover, the problem is reduced to finding the roots of a transcendental equations, which are exactly the same as in standard quantum mechanics. The only difference between fractional and standard quantum mechanics is the dispersion law, that is, the fundamental relationship between the energy and momentum of a quantum particle.

This paper presents exact solutions for the energy spectrum of a particle confined in symmetric 1D rectangular potential well in the framework of fractional quantum mechanics.

The paper is organized as follows. Section 2 presents the 1D fractional Schrödinger equation and its general solutions for a particle moving in the field of a one-dimensional finite symmetric potential well. Section 3 discusses in detail the boundary conditions at the edges of the potential well in order to obtain transcendental equations for finding the energy spectra of even and odd quantum states of a particle in a quantum well. Section 4 introduces dimensionless variables to simplify the transcendental equations and describes a simple graphical algorithm for solving the resulting system of transcendental equations to find number of energy levels in the well and the energy values associated with them.

In Conclusion, the obtained results are summarized and discussed.

2 Fractional Schrödinger equation for a particle moving in 1D

finite symmetric potential well

The 1D fractional Schrödinger equation first introduced by Laskin has the form [3], [6], [7]

[TABLE]

where $\psi(x,t)$ is the wave function, $\hbar$ is Planck’s constant, $D_{\alpha}$ is the scale factor with units of $[D_{\alpha}]=\mathrm{erg}^{1-\alpha}\mathrm{cm}^{\alpha}\sec^{-\alpha}$ (when $\alpha=2$, $D_{2}=1/2m$, where $m$ is the mass of the quantum particle), and the 1D quantum Riesz fractional derivative $(\hbar\nabla)^{\alpha}$ was defined as follows [3], [6]

[TABLE]

here $\varphi(p,t)$ is the Fourier transform of the wave function $\psi(x,t)$

[TABLE]

and respectively

[TABLE]

Let us consider the problem of finding energy levels for a quantum particle moving in a symmetric potential field $V(x)$ having a rectangular shape

[TABLE]

[TABLE]

[TABLE]

We are interested in solutions of the fractional Schrödinger equation (1) for stationary states when $E<U$. A stationary state with an energy $E$ is described by a wave function $\psi(x,t)$, which can be written as $\psi(x,t)=\exp\{-iEt/\hbar\}\phi(x),$ where $\phi(x)$ is time independent wave function that satisfies the time independent fractional Schrödinger equation

[TABLE]

Outside of the potential well, regions $(\mathrm{i})$ and $(\mathrm{iii}),$ Eq.(6) takes the form

[TABLE]

and inside of the well, in the region $(\mathrm{i}),$ it takes the form

[TABLE]

where $\phi_{\mathrm{i}}(x),$ $\phi_{\mathrm{ii}}(x)$ and $\phi_{\mathrm{iii}}(x)$ are the wave functions in the regions $(\mathrm{i}),(\mathrm{ii})$ and $(\mathrm{iii})$ respectively.

Using the property of the quantum fractional Riesz derivative given by Eq.(3.42) in [3], we express the general solutions of the Eq.(7) for the wave function outside the well $(|x|>a)$ in the form

[TABLE]

where $A$ and $B$ are normalization constants and

[TABLE]

and inside the well $(|x|<a)$ the general solution of Eq.(8) has the form

[TABLE]

where $C$ and $D$ are normalization constants and

[TABLE]

3 Boundary conditions and energy spectrum

Imposing a boundary condition at $x=\pm a$ on the wave function allows one to obtain transcendental equations for finding the energy spectrum of a quantum particle in a finite symmetric potential well. From the requirement of continuity of the wave function at the edges of the well, we obtain the following boundary conditions:

[TABLE]

and

[TABLE]

here the following notations were introduced

[TABLE]

Due to the symmetry of the well potential $V(x)=V(-x)$, the solution of the fractional Schrödinger equation (1) will be inside the well either symmetric (even) or antisymmetric (odd) relative to the well center. Considering the even solution $\phi(x)=\phi(-x)$, we come to the conclusion that $D=0$ in Eq.(11). Thus, the even solution for the wave function inside the well has the form

[TABLE]

Outside the potential well the wave function decays exponentially

[TABLE]

and

[TABLE]

Imposing the boundary condition at $x=a$, we have

[TABLE]

and

[TABLE]

Imposing the boundary condition at $x=-a$, we have

[TABLE]

and

[TABLE]

Dividing either Eq.(18) by Eq.(17) or Eq.(20) by Eq.(19) we obtain

[TABLE]

By solving this transcendental equation, we find energy spectrum of a quantum particle in the finite symmetric potential well in the case when the wave function of the particle is symmetric (even), that is $\phi(x)=\phi(-x)$. Note that Eq.(21) is valid for fractional quantum mechanics when $1<\alpha<2$, and also for standard quantum mechanics when $\alpha=2$. In the case of standard quantum mechanics Eqs.(10) and (12) read [1]

[TABLE]

and

[TABLE]

where $m$ is the mass of the quantum particle.

Considering the odd solution $\phi(x)=-\phi(-x)$, we come to the conclusion that $C=0$ in Eq.(11). Thus, the odd solution for the wave function inside the well has the form

[TABLE]

Outside the potential well the wave function decays exponentially

[TABLE]

[TABLE]

Imposing the boundary condition at $x=a$, we have

[TABLE]

and

[TABLE]

Imposing the boundary condition at $x=-a$, we have

[TABLE]

and

[TABLE]

Dividing either Eq.(27) by Eq.(26) or Eq.(29) by Eq.(28) we obtain

[TABLE]

By solving this transcendental equation, we find energy spectrum of a quantum particle in a finite symmetric potential well in the case when the wave function of the particle is antisymmetric (odd), that is $\phi(x)=-\phi(-x)$. Note that Eq.(30) is valid for fractional quantum mechanics when $1<\alpha<2$, and also for standard quantum mechanics when $\alpha=2$. In the case of standard quantum mechanics, $\kappa$ and $\mathrm{k}$ are given by Eqs.(22) and (23), respectively see §22 in [1].

4 Dimensionless variables

Instead of $\kappa$ and $\mathrm{k}$ defined by Eqs.(10) and (12) we introduce dimensionless variables $\eta$ and $\varsigma$

[TABLE]

and

[TABLE]

respectively. In terms of $\eta$ and $\varsigma$ Eqs.(21) and (30) become

[TABLE]

and

[TABLE]

It is easy to see that

[TABLE]

where $\mathrm{const}$ depends on the well depth $U$, its width $a$, Plank’s constant $\hbar$, the scale factor $D_{\alpha}$ and the fractality parameter $\alpha$.

Thus, a simple graphical algorithm for solving the system of equations (33) - (35) consists of choosing the numerical value of the dimensionless parameter $(a^{\alpha}U)/(\hbar^{\alpha}D_{\alpha})$, plotting three curves and collecting data on the intersection points. The intersection points of the curves (33) - (35) give the values for $\varsigma$ and $\eta$. The number of the intersection points depends on value of the dimensionless parameter $(a^{\alpha}U)/(\hbar^{\alpha}D_{\alpha})$. The value for $\varsigma$ from the intersection points of curves (33) and (35) corresponds via Eq.(32) to a certain energy levels in the spectrum for an even quantum stationary state for a particle confined in a 1D finite symmetric potential well. The value of $\eta$ from the intersection points of curves (34) and (35) corresponds via Eq.(31) to a certain energy level in the spectrum for an odd quantum stationary state for a particle confined in a 1D finite symmetric potential well.

Thus, the solutions of the system of equations (33) - (35) determine the number of energy levels and the corresponding energy values for even and odd quantum stationary states for a particle in 1D finite symmetric potential well.

The wave function of a quantum particle in a well oscillates inside the well and decays exponentially outside it.

5 Conclusion

Within the framework of fractional quantum mechanics, an exact analytical solution has been found for the energy spectrum of a quantum particle confined in a quantum well - a symmetric one-dimensional finite potential well. A simple numerical algorithm for obtaining the number of discrete levels in a well and their associated energy values is proposed and discussed. The obtained solutions are directly applicable for the study and modeling of discrete quantum-mechanical states in sandwich structures of semiconductors, for example, InGaN - GaN - InGaN.

The presented results open up new possibilities for emulating fractional quantum mechanics using quantum wells.