# Analytical Solution of the Schrödinger Equation with Spatially Varying Effective Mass for Generalised Hylleraas Potential

Authors: Sanjib Meyur; Smarajit Maji; S. Debnath
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

We have obtained exact solution of the effective mass Schrödinger equation for the generalised Hylleraas potential. The exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunctions are obtained in terms of the hypergeometric functions. Results are also given for the special case of potential parameter.


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## Body

## 1. Introduction

The study of quantum mechanical systems within the framework of effective position dependent mass has been the subject of much activity in recent years. The Schrödinger equation with position-dependent (nonconstant) mass provides an interesting and useful model for the description of many physical problems. The most extensive use of such an equation is in the physics of semiconductor nanostructures [1, 2], quantum dots [3], 3He clusters [4], quantum liquids [5], semiconductor heterostructures [6, 7], and so forth.The solutions of nonrelativistic wave equations with constant mass have been extended to the position dependent mass in recent studies [8–11]. A general formalism for energy spectra and wave functions was found in nonrelativistic problems by using point canonical transformation [8]. Compared to the constant mass wave equation, the position-dependent mass Schrödinger equation is more complex. It is difficult to obtain its analytical solution as usual. Several authors have studied the effects of the position-dependent mass on the solutions of the Schrödinger equation. A position-dependent effective mass, m(x)=m1·m(x), associated with a quantum mechanical particle constitutes a useful model for the study of various potentials such as Morse potential [12–18], hard-core potential [18], Scarf potential [19–21], Pöschl-Teller potential [22, 23], spherically ring-shaped potential [24], Hulthén potential [25], Kratzer potential [26], and Coulomb-like potential [27, 28]. Different techniques have been developed to obtain its exact solutions, such as factorization methods [29], Nikiforov-Uvarov (NU) methods [30], and supersymmetric quantum mechanics [31]. The position-dependent effective mass might have impact on high-energy physics [31].The objective of this paper is to investigate the position-dependent effective mass Schrödinger equation for the generalised Hylleraas potential [32, 33] by using the Nikiforov-Uvarov (NU) method (Figure 1) [30]. Hylleraas potential is used to describe the interaction between two atoms in a diatomic molecule. We have also investigated the solutions of Hulthén potential and Woods-Saxon potential.Figure 1
Hylleraas potential forV1=50, V2=100, a=10, b=0.5, and d=20.The plan of the present paper is as follows. In Section2, the Nikiforov-Uvarov method is summarized. Section 3 is devoted to the solution of the position-dependent effective mass Schrödinger equation. In Sections 4 and 5, the Hulthén potential and Woods-Saxon potential are discussed, respectively. The paper is ended with a summary.


## 2. Nikiforov-Uvarov Method

The NU method is a useful technique to solve the second-order linear differential equations with special orthogonal functions [34]. In this method, after employing an appropriate coordinate transformation s=s(x), the nonrelativistic Schrödinger equation d2ψ/dx2+(E-V(x))ψ=0, (ħ=2m=1) can be written in the following form:
(1)ψ′′(s)+τ~(s)σ(s)ψ′(s)+σ~(s)σ2(s)ψ(s)=0,
where the prime denotes the differentiation with respect to s, σ(s) and σ~(s) are polynomials, at most of second degree, and τ~(s) is a polynomial, at most of first degree. In order to obtain a particular solution to (1), we set the following wave function as a multiple of two independent parts:
(2)ψ(s)=ϕ(s)yn(s).
Equation (2) transformed (1) to a hypergeometric-type equation:
(3)σ(s)yn′′(s)+τ(s)yn′(s)+λyn(s)=0,
where first part of (2), ϕ(s), has a logarithmic derivative:
(4)ϕ′(s)ϕ(s)=π(s)σ(s),
and second part of (2), yn(s), is the hypergeometric-type function whose polynomial solution satisfies the Rodrigues relation:
(5)yn(s)=Cnρ(s)dndsn[σn(s)ρ(s)],
where Cn is normalization constant and the weight function ρ(s) satisfies the relation as
(6)dds[σ(s)ρ(s)]=τ(s)ρ(s).
The function π(s) and the eigenvalue λ required in this method are defined as
(7)π(s)=(σ′-τ~2)±(σ′-τ~2)2-σ~+kσ,(8)k=λ-π′(s).
Hence, the determination of k is the essential point in the calculation of π(s), for which the discriminant of the square root in (7) is set to zero. Also, the eigenvalue equation defined in (8) takes the following new form:
(9)λ=λn=-nτ′(s)-n(n-1)2σ′′(s),n=0,1,2,…,(10)τ(s)=τ~(s)+2π(s),τ′(s)<0.
Since ρ(s)>0 and σ(s)>0, the derivative of τ(s) should be negative [30], which helps to generate the essential condition for any choice of proper bound state solutions. In addition, the energy eigenvalues are obtained from (8) and (9).


## 3. Position-Dependent Effective Mass Schrödinger Equation

In general, working on position-dependent effective mass Hamiltonians is inspired by the von Roos Hamiltonian [35] proposal with ħ=2m0=1:
(11)[-12(mα(x)∂xmβ(x)∂xmγ(x)+mγ(x)∂xmβ(x)∂xmα(x))+V(x)12]φ(x)=Eφ(x),
where ħ=2m0=1 and m(x) is the dimensionless form of the function m(x)=m1·m(x). The ambiguity parameters are constrained by the relation α+β+γ=-1 and we have the following time-independent Schrödinger equation from (11):
(12)Hφ(x)≡[-∂x(1m(x))∂x+Veff(x)-E]φ(x)=0,
where the effective potential is
(13)Veff(x)=V(x)+12(β+1)m′′(x)m2(x)-[α(α+β+1)+(β+1)]m′2(x)m3(x),
where primes denote derivatives. Thus Schrödinger equation takes the form
(14)(-1m(x)d2dx2+m′(x)m2(x)ddx+Veff(x)-E)φ(x)=0.
Using the transformation [36], φ(x)=mν(x)ψ(x) in (14), we have
(15){-d2dx2-(2ν-1)m′(x)m(x)ddx-(ν(ν-2)+α(α+β+1)+β+1)m′2m2+(12(β+1)-ν)m′′(x)m(x)+m(x)(V(x)-E)}ψ(x)=0,
where V(x) is the Hylleraas potential [29, 30] given by
(16)V(x)=V1a+eλxb+eλx-V2d+eλxb+eλx,
where a(≠b), b, and d(≠b) are the Hylleraas parameters, V1, V2 are the potential depths, and -∞<x<∞.Here, we consider the following mass distribution:(17)m(x)=m11+be-λx.
The most extensive use of such kind of mass is in the physics of semiconductor quantum well structures [11]. The motion of electrons in them can often be described by the envelope function effective-mass Schrödinger equation, where m1 is a constant mass (Figure 2).Figure 2
Plot for mass functionm(x) for b=0.5 and m1=1.Obviously it has the exponential form. The above mass function is convergentm(x)→m1 (finite), when x→∞. In order to reduce the above equation (15) into Nikiforov-Uvarov equation, we make the transformation s=1/(1+be-λx), (0≤s≤1):
(18)A=ν(ν-2)+α(α+β+1)+β+1,B=(12(β+1)-ν),ζ=12-ν,P-ζ2=-A+2B-m1(a-b)V1bλ2+m1(d-b)V2bλ2,Q-2ζ2=-2A+3B-m1(aV1-dV2-bE)bλ2,R-ζ2=-A+B,P-Q+R=m1(V1-V2-E)λ2=ε2;
also
(19)m′(x)m(x)=λ(1-s),m′′(x)m(x)=λ2(1-s)(1-2s).
Using (15)–(19), we have
(20)d2ψds2+2ν-(2ν+1)ss(1-s)dψds+1s2(1-s)2[(ζ2-P)s2+(Q-2ζ2)s+(ζ2-R)]ψ=0.
Comparing (20) with (1), we have
(21)σ(s)=s(1-s),σ~(s)=(ζ2-P)s2+(Q-2ζ2)s+(ζ2-R),τ~(s)=2ν-(2ν+1)s.
Substituting these polynomials into (7), we have
(22)π(s)=(12-ν)(1-s)±{(R-ɛ)s-R;k=Q-2R+2ɛR(R+ɛ)s-R;k=Q-2R-2ɛR.
For physical solutions, it is necessary to choose
(23)π(s)=ζ(1-s)-(R+ɛ)s+R111111ifk=Q-2R-2ɛR.
The origin of the Nikiforov-Uvarov method is negative sign of derivative of τ′(s) because the condition τ′(s)<0 helps to generate energy eigenvalues and corresponding eigenfunctions. Therefore τ(s) becomes
(24)τ(s)=1+2R-2(2+2R+2ɛ)s.
Therefore, from (8) and (9), we have
(25)λ=Q-2R-2ɛR-ζ-(R+ɛ),λ=λn=2n(R+ɛ)+n(n+1).
Comparing (25), we have
(26)R+ɛ=-(n+12)+P-ν+14,R-ɛ=Q-P-(n+1/2)+P-ν+1/4.
Using (26) and (18), we have
(27)ε2=14[2n+1+2R-P-ν+14]2.
Hence the energy becomes
(28)En=-λ24m1×[-ν(ν-1)-α(α+β+1)-(β+1)2+(12-ν)22n+1-2m1(V1-V2)λ2-m1(V1a-V2d)bλ2-α(α+β+1)+2-ν(ν-1)-α(α+β+1)-(β+1)2+(12-ν)2]2+(V1-V2),0≤n<∞.
The last term of the square bracket must be positive for BenDaniel and Duke’s model [37] (α=ν=0, β=-1) (Figure 3).Figure 3
Hylleraas energyEn for α=0, β=-1, a=10, b=0.5, d=20, m1=1, V1=50, and V2=100.From (6), (21), and (24) we obtain the weight function
(29)ρ(s)=s2R(1-s)2ɛ,
and from (4), (21), and (23) we have
(30)ϕ(s)=sR+ζ(1-s)ɛ.
Now we use the properties of Jacobi polynomial [35]:
(31)Pn(ζ,ξ)(x)=(-1)n(1-x)-ζ(1+x)-ξ2nn!×dndxn[(1-x)n+ζ(1+x)n+ξ],Pn(2ζ,2ξ)(1-2s)=(-2)n(s)-2ζ(1-s)-2ξ2nn!×dndxn[sn+2ζ(1-s)n+2ξ],
where Pn(a,b)(x) (a>-1, b>-1) is the Jacobi polynomial. The wave functions (Figure 4) are obtained from (2), (5), and ((29)–(31)):
(32)ψn(s)=NnsR+ζ(1-s)ɛPn(2R,2ɛ)(1-2s),
where Nn is normalization constant to be determined from the normalization condition:
(33)∫-∞∞‍|ψn(x)|2dx=1=∫01‍|ψn(s)|2ds.
For acceptable solution it is required that |R+ζ|≥ɛ when R+ζ<0, ɛ>0 and R+ζ≤|ɛ| when R+ζ>0, ɛ<0.Figure 4
Wave functionsψn(s) for α=0, β=-1, ν=1/2, and ɛ=2.


## 4. Hulthén Potential

We set the conditionsV2=V1, a=1+d, and b=-q; the potential in (16) reduces to Hulthén potential (Figure 5) [38–41]:
(34)V(x)=-V1e-λx1-qe-λx.Then the energy becomes
(35)En=-λ24m1×[2-ν(ν-1)-α(α+β+1)-(β+1)2+(12-ν)22n+1-2m1V1qλ2-α(α+β+1)+2-ν(ν-1)-α(α+β+1)-(β+1)2+(12-ν)2]2
with 0≤n<∞. It is exactly the same result in the literature [38] for m1=1.Figure 5
Hulthén potential forV1=100.0 and q=0.5.


## 5. Woods-Saxon Potential

For the conditionsV1=-V0, a=0, and V2=0, the potential given in (16) becomes Woods-Saxon potential (Figure 6) [40–45],
(36)V(x)=-V011+be-λx.Then the energy becomes
(37)En=-λ24m1×[-ν(ν-1)-α(α+β+1)-(β+1)2+(12-ν)22n+1-2m1(1-b)V0bλ2-α(α+β+1)+2-ν(ν-1)-α(α+β+1)-(β+1)2+(12-ν)2]2-V0,
where 0≤n<∞.Figure 6
Woods-Saxon potential forV0=100.0 and b=0.5.


## 6. Conclusion

We have applied the NU method derived for the exponential-type potentials to obtain the bound state solutions of the effective Schrödinger equation with position-dependent mass for the Hylleraas potential. Furthermore, a suitable choice of a position mass function of the exponential-like form has also been devised. Also we have shown that our results are consistent with ones obtained before.



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