# Properties of Quasi-Oscillator in Position-Dependent Mass Formalism

Authors: S. Zare; H. Hassanabadi
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Schrödinger equation is considered within position-dependent mass formalism with a quasi-oscillator interaction term. Wave functions and energy spectra have been obtained analytically. Thermodynamic properties, information entropy, and uncertainty in coordinate and momentum spaces are calculated. To provide a better physical insight into the solutions, some figures are included.


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## Body

## 1. Introduction

The position-dependent mass (PDM) approach has a wide range of applications in various areas of science [1–8] and has motivated many recent researches [9–17]. A very notable application of the field is in the micro fabrication techniques such as molecular-beam epitaxy and nanolithography [18–20]. In quantum mechanics, the PDM Schrödinger equation has been already studied via path integral approach [21], supersymmetric quantum mechanics [22], Darboux transformation [23], de Broglie-Bohm technique [24], and Hamiltonian factorization [25]. On the other hand, in a search for possible alternatives ordinary entropy, some authors have investigated Shannon and Fisher information entropies. There are also attempts which connect the PDM with alternative entropies. Sun et al. [26] presented the Shannon entropy for the position and momentum eigenstates of the PDM Schrödinger equation for a particle with a nonuniform solitonic mass density in the case of a hyperbolic-type potential. Falaye et al. [27] considered a particle in PDM formalism with a nonuniform solitonic mass density and a special squared hyperbolic cosecant interaction. Amir and Iqbal [28] quantized the classical oscillator by considering the symmetric ordering of operator equivalents of momentum and PDM, respectively, and obtained a quantum Hamiltonian which was manifestly Hermitian in the configuration space. With inspiration of above points, we consider the Schrödinger equation within PDM formalism with a quasi-harmonic interaction term. We investigate this system by obtaining the wave function, energy spectra, Shannon information entropy, and corresponding thermodynamics properties.We have organized this article as follows. We first introduce the PDM Schrödinger equation and consider it with a quasi-oscillator in detail in Section2. We next obtain the corresponding exact analytical solutions. Having calculated the energy spectra and the wave functions, in Section 3, we obtain the thermodynamic properties of the system. In Section 4, some discussions regarding the associated information entropy are included. In Section 5, some expectation values are reported and the related uncertainty principle is investigated. The results are discussed via various figures throughout the text.


## 2. One-Dimensional Schrödinger Equation with Position-Dependent Mass

To obtain Schrödinger equation, we first define the Lagrangian density as in [29] (1)L=iℏ2Φx,t∂tΨx,t+ℏ28ddx1mxΦx,t∂xΨx,t+ℏ24mxΦx,t∂x2Ψx,t-12VxΦx,tΨx,t-iℏ2Φ∗x,t∂tΨ∗x,t+ℏ28ddx1mxΦ∗x,t∂xΨ∗x,t+ℏ24mxΦ∗x,t∂x2Ψ∗x,t-12VxΦ∗x,tΨ∗x,t,where Ψx,t and Φx,t are fields with Ψ∗x,t and Φ∗x,t being their conjugates, ℏ is the Planck constant, and mx denotes a position-dependent mass. On the other hand, the Euler-Lagrange equation can be written as(2)∂L∂Φ-∂∂x∂L∂∂xΦ-∂∂t∂L∂∂tΦ=0or in generalized form(3)∂L∂Φ-∂∂x∂L∂∂xΦ-∂∂t∂L∂∂tΦ+∂2∂x2∂L∂∂x2Φ=0.Therefore, the position-dependent mass Schrödinger equation appears as(4)-ℏ22mx∂2Ψx,t∂x2-3ℏ24ddx1mx∂Ψx,t∂x+Vx-ℏ24d2dx21mxΨx,t=-iℏddtΨx,t,where(5)H^=-ℏ22mx∂2∂x2-3ℏ24ddx1mx∂∂x-ℏ24d2dx21mx+Vxwith the atomic units ℏ=1 and c=1 being used. Performing the following transformations in (4)(6a)Ψx=mx1/2ϕyx,(6b)dydx=mx1/2,(6c)mx=m01+γx22,we have [30–32](7)yx=m0γarctan⁡xγ.Thus, we have to deal with(8)-12d2ϕydy2+Vyϕy=Eϕy.We now consider(9)Vx=V0arctan⁡xγ2.The Taylor expansion of the potential is (10)Vx=V0x2γ-23V0x4γ2+2545V0x6γ3+⋯;when γ tends to zero, the higher order terms can be neglected and we may write (11)Vx=V0γx2,or, in terms of y, (12)Vy=V0γm0y2.By considering ω=2V0γ/m0, the potential is more familiarly written as(13)Vy=12m0ω2y2,where γ is the mass deformation parameter and V0 denotes the potential parameter. In Figure 1, we have plotted the potential versus x for some different γ values.Figure 1
V x versus x.


## 3. The Wave Function and Energy for Oscillator Term

Substituting (13) into (8), we have Weber’s differential equation as (14)d2ϕydy2+2E-ω2y2ϕy=0.The change of variable s=ωy2 brings (14) into the form (15)sd2ϕsds2+12dϕsds+E2ω-s4ϕ=0.For further convenience, we apply the gauge transformation ϕs=e-s/2χ(s) which leads to(16)sd2χsds2+12-sdχsds+E2ω-14χ=0.We identify the above equation as the Kummer differential equation (16) and its eigenfunctions may be expressed in terms of regular confluent hypergeometric functions Ma,c,s as [33](17)χs=A~Ma,12,s+B~s1/2Ma+12,32,s,where A~ and B~ are arbitrary constants and(18)a=-E2ω-14.In terms of variable y, the wave functions can be written as(19)ϕy=A~e-1/2ωy2Ma,12,ωy2+B~ωy21/2Ma+12,32,ωy2.Recalling (6a), (6c), and (7), Ψx becomes [33](20)Ψx=m01+γx22A~e-ωm0/2γarctan2⁡xγMa,12,ωm0γarctan2⁡xγ+B~ωm0γarctan⁡xγMa+12,32,ωm0γarctan2⁡xγ.The confluent Hypergeometric function are related to the Hermit polynomials via(21)Hnevenξ=-1n2n!n!M-n,12,ωy2,Hnoddξ=-1n22n+1!n!M-n,32,ωy2.In view of the above equations, the even and odd eigenfunctions may be, respectively, expressed as (22a)Ψeven=Nnm01+γx22e-ωy2/2Hnyω,(22b)Ψodd=Nnm01+γx22e-ωy2/2Hnyω,where Nn is the normalization constant. However, the even odd eigenfunctions may be combined and the stationary states of the relativistic oscillator are (23)Ψnx=Nnm01+γx22e-ωm0/2γarctan2⁡xγHnωm0γarctan⁡xγ.The energy eigenvalues of spin-zero particles bound in this oscillator potential may be found using (14). Therefore, the energy for Even and odd states can be written as(24)En=n+12ω=2V0γn+12.


## 4. Thermodynamic Properties

In order to consider the thermodynamic properties of a neutral particle, we concentrate, at first, on the calculation of the partition function [34](25)Q1=∑n=0∞e-βEn=e-βω/2+e-3βω/2+⋯=1eβω/2-e-βω/2=2sinh⁡2V0γβ2-1,where β=1/KT. The partition function for noninteractingN-body system is (26)QN=Q1N=2sinh⁡2V0γβ2-N,where(27)ln⁡QN=-Nln⁡2sinh⁡2V0γβ2.In Figure 2, Q1β is plotted in terms of β by setting different constant values for m0,V0, and γ. As it can be seen, by increasing γ and β, Q1β decreases as ina, somehow, the same manner.Figure 2
Q 1 β versus β.Once the Helmholtz free energy is obtained, the other statistical quantities are obtained in a straightforward way. We therefore start from(28)A=-1βln⁡QN=Nβln⁡2sinh⁡2V0γβ2.The chemical potential is obtained from (28) by a simple differentiation as (29)μ=∂A∂N=KTln⁡2sinh⁡2V0γβ2.The mean energy is obtained from (30)U=-∂∂βln⁡QN=2V0γN2coth⁡2V0γβ2.We have plotted Uβ/N in terms of β. In Figure 3, for small values of β, we face with convergences, but as β grows up, a small smooth divergence is observed. Uβ/N, as γ and β have higher values, gains higher values as well.Figure 3
U β / N versus β.The main statistical quantity, that is, the entropy, is related to other quantities via(31)SK=β2∂A∂β=-Nln⁡2sinh⁡2V0γβ2+2V0γβN2coth⁡2V0γβ2.In Figure 4, Sβ/KN is shown as β varies. Although a divergent behavior is seen at initial values of β, the curves tend to zero at large values. The specific heat capacity at constant volume is obtained from(32)CK=-β2∂U∂β=2V0γβ2N4csch⁡2V0γβ22.In Figure 5, we have plotted Cβ/KN versus β. In this figure, somehow, like the last figure, we have convergence but depending on the value of γ, values of Cβ/KN plunge into zero values with different slops.Figure 4
S β / K N versus β.Figure 5
C β / K N versus β.


## 5. Information Entropy

The position space Shannon and Fisher information entropies for a one-dimensional system can be calculated asSx=-∫Ψnx2ln⁡Ψnx2dx and Fx=-∫Ψnx2d/dxln⁡Ψnx2dx. In general, explicit derivation of the Shannon information entropy is quite difficult. In particular the derivation of analytical expression for the Sx is almost impossible. We represent the position Shannon and Fisher information entropy densities, respectively, by ρSx=Ψn(x)2ln⁡Ψn(x)2 and ρFx=Ψnx2d/dxln⁡Ψnx2 [35–40]. In our case, these quantities are obtained as(33)ρSx=m0e-ωy2Hnyω2ln⁡m0e-ωy2Hnyω2/1+γx221+γx22,ρFx=-4m0e-ωy2xγHnyω21+γx23.To understand the essential features of the entropies, some related figures are included. At first case we plotted the position space Shannon information entropies Sx considering γ varying for m0=1,ω=1. In Figures 6 and 7, we depicted the behavior of the position space Shannon information entropies Sx as function of n and γ by considering m0=1,ω=1. The ascending nature can be seen easily. In Figures 8 and 9 we have plotted the position Shannon information entropy densities versusx.Figure 6
Position Shannon information entropiesSx versus γ.Figure 7
Position Shannon information entropySx versus n.Figure 8
Position Shannon information entropy densitiesρx versus x.Figure 9
Position Shannon information entropy densityρx versus x.


## 6. Some Expectation Values and the Uncertainty Principle

The asymmetry caused by the parameterγ can be adequately quantified in terms of the average position calculated as x=∫-∞∞Ψn∗xxΨnxdx and average for x2 is x2=∫-∞∞Ψn∗xx2Ψnxdx. From (23), the average of the modified momentum is p=0 and average for p2 is p2=-ℏ2∫-∞∞Ψn∗xd2/dx2Ψnxdx [41, 42]. We have plotted the uncertainty for x,p and the uncertainty principle. Figure 10 shows how uncertainty for x can change. Also treatments of uncertainty in momentum are shown in Figure 11 for m0=1andω=1 for different n in terms of γ. Figure 12, assuming m0=1andω=1, illustrates the uncertainty principle versus γ.Figure 10
Uncertainty in position versusγ.Figure 11
Uncertainty in momentum versusγ.Figure 12
Combined uncertainty versusγ.


## 7. Conclusion

We studied the physical characteristics of a nonrelativistic quasi-oscillator interaction within position-dependent mass formalism. We first obtained the wave functions and the energy spectra of the system in an exact analytical manner. Next, the thermodynamic properties, information entropy, some expectation values, and some uncertainty principles were evaluated. In addition, we included some figures to illustrate the physical characteristics and asymptotic behavior of the results.



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