Klein-Gordon and Dirac Equations with Thermodynamic Quantities
aa r X i v : . [ qu a n t - ph ] O c t Klein-Gordon and Dirac Equations withThermodynamic Quantities
Altu˘g Arda a ∗ Cevdet Tezcan b and Ramazan Sever c † a Department of Physics Education, Hacettepe University , b Faculty of Engineering, Baskent University, Baglica Campus, Ankara,Turkey c Physics Department, Middle East Technical University06531, Ankara, Turkey
Abstract
We study the thermodynamic quantities such as the Helmholtz free energy, themean energy and the specific heat for both the Klein-Gordon, and Dirac equations.Our analyze includes two main subsections: ( i ) statistical functions for the Klein-Gordon equation with a linear potential having Lorentz vector, and Lorentz scalarparts ( ii ) thermodynamic functions for the Dirac equation with a Lorentz scalar,inverse-linear potential by assuming that the scalar potential field is strong ( A ≫ ∗ Present adress: Department of Mathematical Science, City University London, UK † [email protected], [email protected], [email protected] Introduction
The study of the thermodynamic functions for different potential fields within non-relativisticand/or relativistic regimes has been received a special interest for a few decades. Pachecoand co-workers have analyzed the one-dimensional Dirac-oscillator in a thermal bath [1],and then extended the same subject to three dimensional case [2]. In Refs. [3, 4], theDirac/Klein-Gordon oscillators have been analyzed in thermodynamics point of view, andthe Dirac equation on graphene has been solved to study the thermal functions, respec-tively. In Refs. [5, 6], the non-commutative effects on thermodynamics quantities have beendiscussed in graphene. The spin-one DKP oscillator has been analyzed for the statisticalfunctions by taking into account the non-commutative effects with an external magneticfield [7]. In Ref. [8], the authors have studied the thermodynamics properties of a harmonicoscillator plus an inverse square potential within the non-relativistic region.In the present work, we analyze some thermodynamic quantities such as the free energy,the mean energy, and the specific heat for the relativistic wave equations, namely Klein-Gordon (KG), and Dirac equations. Our basic function will be the partition function, Z ( β ),of whole system. We then compute other thermal quantities, and also give the results forthe case of high temperatures ( β ≪ µ which is written in terms of a quantity analogous to the Debye temperature in solidstate physics [2]. In the second part of the present paper, we analyze the same quantitiesfor the Dirac equation with a Lorentz scalar, inversely linear potential. We compute thepartition function of the system by using a method based on the Euler and the RiemannZeta functions, and their properties [9]. This approach has been used to handle thermal2unctions for different physical systems [3, 4, 11]. We consider the case where the scalarpotential coupling is a large constant for getting an analytical expression for the partitionfunction, and analyzing also the quantity numerically. As in the first part, we write allof thermal functions in terms of a dimensionless parameter, ¯ µ . Throughout the paper, weconsider only the case where the particle-particle interactions are excluded for handling thepartition function which means that the positive-energy levels give a contribution only, andthe partition function does not involve a sum over negative-energy levels [12].The paper is organized as follows. In Section II, we first compute the analytical solutionsof the KG equation for a one-dimensional, linear potential mixing of vector and scalar partwith unequal magnitudes. After determining the partition function for the case where onlyscalar part of the potential gives a contribution, we give some thermodynamics functionssuch as the Helmholtz free energy ( F ( β )), the mean energy ( U ( β )), the entropy ( S ( β )),and the specific heat ( C ( β )) with figures showing the variation of these functions versustemperature. In Section III, we solve the Dirac equation for an inverse linear potential,and derive the analytical results which will be used to write the partition function. Weobtain the partition function with the help of the Euler and the Riemann Zeta functions,and then the other thermal functions. We find all thermodynamic quantities in terms ofthe coupling constant. So, we summarize our results in figures showing the variation of theabove thermal functions versus the temperature for different values of A . In last Section,we give the conclusion. The Klein-Gordon equation for a particle with mass m in the presence of a vector, V ( x ),and a scalar potential, V ( x ), is written as [13] d ψ ( x ) dx + Q (cid:2) ( E − V ( x )) − ( µ − V ( x )) (cid:3) ψ ( x ) = 0 , (2.1)with Q = 1 / ~ c , and µ = mc . By taking the vector and scalar part of the linear potentialas ~ ca | x | , and ~ ca | x | , respectively, and inserting into Eq. (2.1), we can write Eq. (2.1)3s a Schrodinger-like equation (cid:20) − µ d dx + Q (cid:18) k x + k | x | + ǫ (cid:19)(cid:21) ψ ( x ) = 0 , (2.2)where k = a µQ (1 − a ) ; k = a µ √ Q ( Ea − µ ) ; ǫ = µ − E µ , (2.3)with a = a /a . Throughout the first part of the paper, we consider only the case where a = a , otherwise we have the Coulomb problem which will not be the subject of thepresent work. The reader can find a detailed analyze about the Coulomb case in Ref. [10].Defining a new variable y = | x | + k /k in Eq. (2.2) gives us (cid:20) − µ d dy + Q ( 12 k y − ǫ ) (cid:21) ψ ( y ) = 0 , (2.4)where ǫ = − ǫ + k / k . Eq. (4) corresponds to the Schrodinger equation for the harmonicoscillator with the energy ~ ω ( n + 1 /
2) which together with Eq. (2.3) gives the bound statesolutions of the linear potential as E = aµ ± p (1 − a ) µ ~ ω (2 n + 1) , (2.5)Here, if one takes the frequency of the system as ω = p k c /µ , then our result in Eq. (2.5)is consistent with the ones obtained in literature [10].In next section, we compute some thermal quantities of the system starting from thepartition function while we deal with only particles corresponding to the positive energylevels to write the thermodynamics quantities. The partition function of the system is given as [1] Z ( β ) = ∞ X n =0 e − β ( E n − E ) = e µβ ∞ X n =0 e − βE n , (2.6)where β = 1 /k B T , k B Boltzmann constant, and T is temperature in Kelvin. With thehelp of Eq. (2.5), we study the following thermal quantities such as the Helmholtz free4nergy, the mean energy, the entropy, and the specific heat defined in terms of the partitionfunction as following F ( β ) = − β ln Z ( β ); U ( β ) = − ∂∂β ln Z ( β ); S ( β ) = k B β ∂∂β F ( β ); C ( β ) = − k B β ∂∂β U ( β ) , (2.7)Although the summation over n in Eq. (2.6) is infinite, one can check the divergence ofthe summation by using the integral formula [1] Z ∞ e − β √ β ′ n + β ′′ dn = 2 β ′ β (1 + β p β ′′ ) e − β √ β ′′ , (2.8)which means that the partition function is convergent. Because Eq. (2.6) is convergent,the partition function can be computed by using the Euler-MacLaurin formula [1, 2, 5] ∞ X m =0 f ( m ) = 12 f (0) + Z ∞ f ( x ) dx − ∞ X i =1 i )! B i f (2 i − (0) , (2.9)where B i are the Benoulli numbers, B = 1 / B = − / . . . . Up to i = 2, Eqs. (2.9)and (2.10) gives the partition function of the system as Z (¯ µ ) = 12 + ¯ µ (¯ µ + 1) + 1240¯ µ (19¯ µ − ¯ µ − , (2.10)With the help of the above result, we write explicitly the thermal functions studyinghere in terms of a dimensionless parameter ¯ µ = 1 /µβ = k B T /M c ¯ F = Fµ = − ¯ µ ln Z (¯ µ ) , ¯ U = Uµ = 3¯ µ (1 + 2¯ µ − µ + 240¯ µ + 480¯ µ ) − − µ + 57¯ µ + 360¯ µ + 720¯ µ (1 + ¯ µ ) , ¯ S = Sk B = 3(1 + 2¯ µ − µ + 240¯ µ + 480¯ µ ) − − µ + 57¯ µ + 360¯ µ + 720¯ µ (1 + ¯ µ ) + ln Z (¯ µ ) , ¯ C = Ck B = 1[ − − µ + 57¯ µ + 360¯ µ + 720¯ µ (1 + ¯ µ )] × (cid:8) − − µ − µ − µ − µ − µ + 43200¯ µ + 309600¯ µ +691200¯ µ (1 + ¯ µ ) + 345600¯ µ ) (cid:9) , (2.11)Let us first analyze the case of high temperatures corresponding to β ≪
1. Eq. (2.11)gives the following results for high temperatures Z ∼ ¯ µ ; U ∼ µ ; C ∼ µ for M c / ~ ω = 1. This relation between energyand frequency of the system corresponds to the relativistic case [3] where the thermalfunctions of the Dirac and KG oscillators have been analysed by taking into account thedifferent values of this ratio. In Figs. 1, 3, and 4, we can see that ln Z (¯ µ ), the meanenergy, and the entropy increase with temperature. In Fig. 2, we observe that the freeenergy decreases with temperature as expected. At this point, we should point out thatthe entropy of the system has a ’turning point’ about ¯ µ ∼ .
5, the values of ¯ µ smaller than ∼ . The Dirac equation is written in terms of the time-independent Lorentz scalar, V ( x ), andLorentz vector, V ( x ), potentials as [14] (cid:2) cα ˆ p + β ( mc + V ( x )) + V ( x ) (cid:3) ψ ( x ) = Eψ ( x ) , (3.1)where E is the energy of the fermion with mass m , c is the speed of light, and ˆ p is themomentum operator. α and β are traceless matrices satisfying the relations α = β = 1,and { α, β } = 0, separately. By choosing them as Pauli matrices as α = σ , and β = σ [15]6nd taking the spinor with upper and lower components as ψ ( x ) = f ( x ) g ( x ) , (3.2)Eq. (3.1) gives two coupled equations in the absence of the vector potential( mc + V ( x )) g ( x ) − ~ c dg ( x ) dx = Ef ( x ) , ( mc + V ( x )) f ( x ) + ~ c df ( x ) dx = Eg ( x ) , (3.3)We obtain two second order differential equations as (cid:26) − d dx + Q ( µ − E ) + 2 Q µV ( x ) + Q V ( x ) − Q dV ( x ) dx (cid:27) f ( x ) = 0 , (cid:26) − d dx + Q ( µ − E ) + 2 Q µV ( x ) + Q V ( x ) + Q dV ( x ) dx (cid:27) g ( x ) = 0 , (3.4)with Q = 1 / ~ c , and µ = mc . Inserting the Lorentz scalar potential as V ( x ) = − ~ cA/ | x | with a dimensionless coupling A into Eq. (3.4) gives us (cid:26) − d dx + Q ( M − E ) − QM A | x | + ξ ( f,g ) x (cid:27) f ( x ) g ( x ) = 0 . (3.5)The upper indices ( f, g ) of the constant ξ indicates the upper component with ξ f = A ( A − ξ g = A ( A + 1).We obtain a second order equation for the upper component by changing the variableas y = 2 p Q ( µ − E ) | x | ( d dy −
14 + µA/ p µ − E y − ξ ( f,g ) y ) f ( y ) = 0 , (3.6)The upper component can be taken of the form e − y/ y B h ( y ) because of the asymptoticbehaviours should be satisfied by a wave function. So, inserting this form into Eq. (3.6)gives for h ( y ) ( y d dy + (2 B − y ) ddy − B − µA p µ − E !) h ( y ) = 0 . (3.7)Eq. (3.7) is a confluent hypergeometric-type equation having the solutions [11] h ( y ) ∼ F ( B − µA p µ − E ; 2 B ; y ) (3.8)7hich can also be written in terms of the Laguerre polynomials L kn ( y ) [14]. As stated in Ref.[15], by a detailed analyse on the constant ξ ( f,g ) , we can give the bound states solutions,and the upper and lower components as f ( y ) = N y A e − y/ L A − n ( y ) ,g ( y ) = N y A +1 e − y/ L A +1 n − ( y ) , (3.9)with E = ∓ µ s − A ( n + A ) ; n = 0 , , , . . . (3.10)We are now ready to write the eigenvalues for the case where the potential field is strongwhich means A ≫ E ∼ ∓ µ q nA , and if the potential field becomesextremely strong the eigenvalues go to zero [15]. In the next section, we will use the aboveresult to write the partition function for particle eigenvalues. The partition function of the system can be written as a summation over all the quantumstates [1] Z ( β ) = ∞ X n =0 e − µ E n , (3.11)with ¯ µ = 1 /βµ , and β = 1 /k B T where k B Boltzmann constant, and T is temperature inKelvin.In order to evaluate the partition function analytically, we can use the following integralequality over the contour C [6, 7, 9] e − y = 12 πi Z C y − t Γ( t ) dt , (3.12)where Γ( z ) is the Euler Gamma function [11]. With the help of Eq. (3.12), and by usingthe Riemann Zeta function defined as ζ ( t ) = P ∞ n =0 /n t in Eq. (3.11) [11], we write ∞ X n =0 e − µ √ an = 12 πi Z C (¯ µ ) t (2 a ) − t/ ζ ( t/ t ) dt , (3.13)8ith a = 1 /A . We see that Eq. (3.13) has two simple poles at t = 2, and t = 0. Byapplying the residue theorem, the wanted partition function is obtained as Z (¯ µ ) = ¯ µ a + 12 . (3.14)Before going further, from this result, we tend to write here the reduced thermal func-tions explicitly ¯ F (¯ µ ) = F ¯ µ = − ¯ µ ln (cid:18) ¯ µ + a a (cid:19) , ¯ U (¯ µ ) = Uµ = 2¯ µ ¯ µ + a , ¯ C (¯ µ ) = Ck B = 2¯ µ (¯ µ + 3 a )(¯ µ + a ) , (3.15)Let us first analyze the case of high temperatures corresponding to β ≪
1. Eq. (3.14)gives the following results for high temperatures Z ∼ ¯ µ / a ; U ∼ µ ; C ∼ . (3.16)We observe that the partition function still depends also to the potential parameter a whilethe dependency of the mean energy, and the specific heat disappear for high temperatures,which can be seen in Figs. (6)-(8) below.We observe that the thermodynamic quantities in Eq. (3.15) depend also the couplingparameter a . So, we give our all numerical results as the variation of them versus thetemperature for three different values of potential parameter, namely, a = 1, a = 0 . a = 0 .
1, in Figs. (6)-(8). Fig. (6) shows that the Helmholtz free energy increase withincreasingly value of a . In Fig. (7), we see that the effect of the coupling parameter onthe mean energy is more apparent for nearly low temperatures. On the other hand, theplots for different a -values for the mean energy are closing to each other which means thatthe dependence of the mean energy on a disappears for high temperatures. We give thevariation of the specific heat according to the temperature in Fig. (8) where it has an uppervalue while the temperature increases. 9 Conclusions
We have studied the thermal functions for the Klein-Gordon equation with a Lorentz scalarpotential having a linear form, and for the Dirac equation with a Lorentz scalar, inverse-linear potential for the case where the potential field is strong. For this aim, we haveconsidered the case where the positive part of the energy spectrum gives a contribution tothe partition function. All important thermodynamics quantities have been computed by amethod based on the Euler-MacLaurin formula for the KG solutions, and a method basedon Euler Gamma and the Riemann Zeta functions for the Dirac solutions, respectively. Theresults for high temperatures are also given for the specific heat, and the mean energy.
One of authors (A.A.) thanks Prof Dr Andreas Fring from City University London andthe Department of Mathematics for hospitality. This research was partially supported bythe Scientific and Technical Research Council of Turkey and through a fund provided byUniversity of Hacettepe. 10 eferences [1] M. H. Pacheco, R. R. Landim, and C. A. S. Almeida, Phys. Lett. A , 93 (2003).[2] M. H. Pacheco, R. V. Maluf, C. A. S. Almeida, and R. R. Landim, EPL , 10005(2014).[3] A. Boumali, EJTP , 1 (2015).[4] A. Boumali, [arXiv: 1411.1353v1 [cond-math.mes-hall]].[5] V. Santos, R. V. Maluf, and C. A. S. Almeida, Ann. Phys. , 402 (2014).[6] A. Boumali, and H. Hassanabadi, Eur. Phys. J. Plus , 124 (2013).[7] S. Hassanabadi, and M. Ghominejad, Advances in High Energy Physics Vol. 2014,Article ID 185169.[8] S. H. Dong, M. Lozada-Cassou, J. Yu, F. Gimenez-Angeles, A. L. Rivera, Int. J. Quant.Chem. , 366 (2007).[9] M. A. Dariescu, and C. Dariescu, Chaos, Solitons and Fractals , 776 (2007).[10] A. S. de Castro, Phys. Lett. A , 71 (2005).[11] M. Abramowitz, and I. A. Stegun (Eds.), Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables (New York: Dover Publications, 2007).[12] W. T. Grandy Jr.,
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Relativistic Quantum Mechanics-Wave Equations (Berlin: Springer Ver-lag, 1987).[15] A. S. de Castro, Phys. Lett. A , 289 (2004).11 l n Z – µ Figure 1: The variation of ln Z versus ¯ µ . -50-45-40-35-30-25-20-15-10-5 0 5 0 1 2 3 4 5 6 7 8 9 10 – F – µ Figure 2: The free energy for linear potential versus ¯ µ .12 – U – µ Figure 3: The mean energy for linear potential versus ¯ µ . – S – µ Figure 4: The entropy for linear potential versus ¯ µ .13 – C – µ Figure 5: The specific heat for linear potential versus ¯ µ . -70-60-50-40-30-20-10 0 10 0 1 2 3 4 5 6 7 8 9 10 – F – M a=1.0a=0.5a=0.1
Figure 6: The Helmholtz free energy for the inverse-linear potential versus ¯ µ .14 – U – Ma=1.0a=0.5a=0.1
Figure 7: The mean energy for the inverse-linear potential versus ¯ µ . – C – M a=1.0a=0.5a=0.1
Figure 8: The specific heat for linear potential versus ¯ µµ