Thermodynamics Quantities for the Klein-Gordon Equation with a Linear plus Inverse-linear Potential: Biconfluent Heun functions
aa r X i v : . [ qu a n t - ph ] A ug Thermodynamics Quantities for the Klein-Gordon Equation witha Linear plus Inverse-linear Potential: Biconfluent Heun functions
Altu˘g Arda, ∗ Cevdet Tezcan, and Ramazan Sever † Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey Faculty of Engineering, Baskent University, Baglica Campus, Ankara,Turkey Department of Physics, Middle East Technical University, 06531, Ankara,Turkey
Abstract
We study some thermodynamics quantities for the Klein-Gordon equation with a linear plusinverse-linear, scalar potential. We obtain the energy eigenvalues with the help of the quantizationrule coming from the biconfluent Heun’s equation. We use a method based on the Euler-MacLaurinformula to compute the thermal functions analytically by considering only the contribution of pos-itive part of spectrum to the partition function.Keywords: thermodynamic quantity, Klein-Gordon equation, linear potential, inverse-linear po-tential, biconfluent Heun’s equation, exact solution
PACS numbers: 03.65.-w, 03.65.Pm, 11.10.Wx ∗ Present adress: Department of Mathematical Science, City University London, Northampton Square,London EC1V 0HB, UK † E-mails: [email protected], [email protected], [email protected] . INTRODUCTION If the one-dimensional linear potential having a form proportional to | x | is considered asthe time-like component of a Lorentz vector then this potential becomes related with theCoulomb potential [1, references therein]. The linear potential is also a basic ground forconfinement of the particles having an odd half-integer spin in the view of quantum fieldtheory. If we consider the linear potential as a Lorentz scalar, then it becomes important forthe structure of quarkonium. So, the one-dimensional linear potential has received a greatinterest in literature. The solutions of the Dirac equation and the non-relativistic limit for alinear scalar potential have been studied in Ref. [2]. The bound state solutions of the Diracequation have been analyzed for the one-dimensional linear potential with Lorentz scalar andvector couplings [1]. Some other relativistic equations such as the Duffin-Kemmer-Petiau[3], and the Klein-Gordon (KG) equation [4] have been also studied for the linear potential.For the non-relativistic case, namely the Schrodinger equation, it is well known that theanalytical solutions are obtained in terms of Airy functions [4].The potential writing as inversely linear ( ∼ | x | − ) denotes another interesting interaction.This is because this potential represents the hydrogen atom in one-dimensional space [5].The non-relativistic results for this potential show that the ground-state solution has aninfinite energy with an eigen function written in terms of delta function near the origin [5].Analyzing this potential for the Klein-Gordon equation presents unacceptable solutions withthe help of continuous dimensionality technique [6]. As a result, it could be interesting tosolve the Klein-Gordon equation for the combination of the above potentials writing as V ( x ) = a + a | x | + a | x | , to find the statistical quantities for the whole system.The study of the thermodynamics quantities for quantum systems in different potentialshas been received a special interest for last few decades. In Ref. [7], the one-dimensionalDirac-oscillator has been analyzed in a thermal bath, and then the three dimensional casehas been computed in Ref. [8]. The Dirac/Klein-Gordon oscillators have been analyzed inthermodynamics point of view by using a different method in Ref. [9]. The Dirac equation ongraphene has been solved to study the thermal functions in Ref. [10]. The non-commutativeeffects on thermodynamics quantities have been also discussed on graphene in literature [11,12]. The spin-one DKP oscillator has been analyzed for the statistical functions by taking2nto account the non-commutative effects with an external magnetic field [13]. In Ref. [14],the thermodynamics properties of a harmonic oscillator plus an inverse square potential havebeen studied within the non-relativistic region.The paper is organized as follows. In Section II, we obtain the bound state solutionsof the Klein-Gordon equation for the above potential with the help of the quantizationcondition giving the biconfluent Heun’s eqution. We will see that the results reveal anenergy-eigenvalue equation which is independent of the potential parameter a . In SectionIII, we compute the partition function, Z ( β ), by using the Euler-MacLaurin formula in termsof a dimensionless parameter ¯ m by restricting ourselves to the case where the particle-particleinteractions appear only. For this case, the partition function does not involve a sum overthe negative-energy states [15]. We search then the other thermal quantities such as thefree energy, the mean energy, and the specific heat numerically. In Section IV, we give ourconclusions. II. THE BOUND STATES
The time-independent one-dimensional Klein-Gordon equation with scalar, V S ( x ), andvector, V V ( x ), potentials reads as [16] (cid:26) − d dx + Q [ mc + V S ( x )] − Q [ V V ( x ) − E ] (cid:27) ψ ( x ) = 0 , (1)with Q = 1 / ~ c , c is the speed of light, m is the rest mass, and E is the energy. Here wetend to the vector potential as V V ( x ) = 0, and the scalar part given as above. So, we have d ψ ( x ) dx − Q (cid:20) ( mc + a ) + 2 a a + 2 a ( mc + a ) 1 | x | + a x + 2 a ( mc + a ) | x | − a x (cid:21) ψ ( x )= − Q E ψ ( x ) , (2)By defining a new variable y = √ Qa | x | , and using the abbreviations ε = Qa [ E − ( mc + a ) − a a ] ; A = − Qa ( mc + a ) r Qa A = − Q a ; A = − r Qa ( mc + a ) , (3)we have d ψ ( y ) dy + (cid:18) ε + A y + A y + A y − y (cid:19) ψ ( y ) = 0 . (4)3n order to get a more suitable form for Eq. (4) we write the wave function as ψ ( y ) = | y | p e − qy − ry φ ( y ) , (5)with p = 12 + 12 p − A , (6)Now substituting Eq. (5) into Eq. (4), the resulting equation reads yφ ′′ ( y ) + (2 p − ry − qy ) φ ′ ( y ) + (cid:2) ( − pq + r + ε ) y − (2 pr − a ) (cid:3) φ ( y ) = 0 , (7)This equation is the biconfluent Heun’s differential equation having a general form [17] ξu ′′ ( ξ ) + (1 + c − c ξ − ξ ) u ′ ( ξ ) + (cid:26) ( c − c − ξ −
12 [ c + c (1 + c )] (cid:27) u ( ξ ) = 0 , (8)with solutions the so-called biconfluent Heun functions, HBφ ( y ) ∼ HB (cid:18)p − A , Q √ a ( mc + a ) , γ + ε , Q a √ Qa ( mc + a ) , y (cid:19) . (9)The biconfluent Heun’s equation has many applications within different subjects to find-ing the quantization condition and the wave functions for the system under consideration[18-21]. The general solution of this equation can be computed by using the Frobeniusmethods, and the biconfluent Heun series results in a polynomial form of degree n when [18] ε + 14 A − p = 2 n , (10)with n = 0 , , , . . . . By using Eq. (3), we obtain the bound states of the system E n = 2 a a + a Q (cid:18) n + 1 + q Q a (cid:19) , (11)with the eigenfunctions ψ ( y ) ∼ | y | + √ − A e ( A y − y ) × HB (cid:18)p − A , Q √ a ( mc + a ) , γ + ε , Q a √ Qa ( mc + a ) , y (cid:19) . (12)We present plots of some eigenfunctions with different quantum number values in Fig.(1). In addition, last two equations makes it possible to handle the single particle level4ensity defined basically as the number of energy levels in the energy interval dE [22], thatis, ρ ( E ) = dEdn , (13)which gives for the system under consideration ρ ( E ) = Qa √ E . (14)where it is clearly seen that the level density depends on the strength of linear and inverse-linear part of potential.In order to have an equation with same dimensions in the left and right hand sides in(11), let us denote the quantity ” a a ” as ε in the rest of computation which makes itpossible to write the Eq. (11) more clearly as E n = ∓ ε q q − (2 n + 1 + p q ) . (15)with a dimensionless parameter q = Qa . In the next Section, we compute the thermalfunctions in terms of a dimensionless parameter ¯ m written with the help of ε . III. THE THERMODYNAMICS QUANTITIES
The partition function given as a summation over all the quantum states can be writtenas [7] Z ( β ) = ∞ X n =0 e − ( E n − E ) β = e βE ∞ X n =0 e − βε √ σ n + σ , (16)where β = 1 /k B T , k B Boltzmann constant, T temperature in Kelvin with the constants σ = 2 /q , and σ = 2 + (1 /q )(1 + p q ). We tend to compute the following thermalquantities such as the free energy, the mean energy, and the specific heat written in termsof the partition function F ( β ) = − β ln Z ( β ) ,U ( β ) = − ∂∂β ln Z ( β ) ,C ( β ) = − k B β ∂∂β U ( β ) , (17)5he following integral equation [7, 8] Z ∞ e − β √ β n + β dn = 2 β β e − β √ β (1 + β p β ) , (18)shows that the partition function in Eq. (16) is convergent. The result in Eq. (18) makes itpossible to compute the partition function with the help of the Euler-MacLaurin formula ∞ X n =0 f ( n ) = 12 f (0) + Z ∞ f ( x ) dx − ∞ X i =1 i )! B i f (2 i − (0) , (19)where B i are the Benoulli numbers, B = 1 / B = − / . . . [7, 8]. Up to i = 2, Eq. (16)with the help of (13) gives the partition function of the system written in a dimensionlessparameter βε = 1 / ¯ m as Z ( ¯ m ) = 12 + 2 ¯ m σ (1 + √ σ ¯ m ) + σ
24 ¯ m √ σ − σ mσ / (3 + 3 √ σ ¯ m + σ ¯ m ) . (20)We observe that the thermodynamic quantities in Eq. (17) depend the parameter q including the potential parameter. So, we give our all numerical results as the variationof them versus the temperature for three different values of parameter, namely, q = 0 . q = 1 . q = 1 .
5, in Figs. (2)-(4). Fig. (2) shows that the Helmholtz free energy increasewith increasingly value of a . In Fig. (3), we see that the effect of the parameter q on themean energy is more apparent for nearly low temperatures. On the other hand, the plotsfor different q -values for the mean energy are closing to each other. We give the variation ofthe specific heat according to the temperature in Fig. (4) where it has an upper value whilethe temperature increases.Now we give the results briefly for the thermal functions for high temperatures whichcorresponds to β ≪
1. For this case, Eq. (20) gives the results Z ( ¯ m ) ∼ m σ ∼ ¯ m Qa ,U ( ¯ m ) ∼ m ,C ( ¯ m ) ∼ . (21)where the upper limit for the specific heat can be seen clearly in Fig. (4).Studying the partition function in Eq. (20) according to the potential parameters showsthat the descent contribution, which is inverse-linear, comes from the part of the potentialproportional to | x | . The other part of the potential proportional to | x | gives a weaker6ontribution, which is linear in some terms and inverse-linear in others. On the other hand,Eq. (21) gives that only the potential parameter a gives an inverse-linear contribution topartition function while both parameters a and a give an inverse-squared contribution tothe mean energy for high temperature. IV. CONCLUSIONS
We have obtained the thermodynamics quantities for the Klein-Gordon equation with alinear plus inverse-linear potential by using the quantization condition appeared in bicon-fluent Heun’s equation. The variation of a few eigenfunctions versus spatially coordinatehas been given in a figure, and the single particle level density analyzed briefly. The ther-modynamics quantities such as the free energy, the mean energy, and the specific heat havebeen computed by a method based on the Euler-MacLaurin formula. We have obtained thevariation of thermal functions according to temperature, and also discussed the results forhigh temperatures.
V. ACKNOWLEDGMENTS
One of authors (A.A.) thanks professor A. Fring from City University London and theDepartment of Mathematics for hospitality. This research was partially supported by theScientific and Technical Research Council of Turkey and through a fund provided by Uni-versity of Hacettepe.The authors also thank the referee for comments which have improved the manuscript.7
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