Research Article Dirac Particle for the Position Dependent Mass in the Generalized Asymmetric Woods-Saxon Potential

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1 Advances in High nergy Physics Volume 24, Article ID , pages Research Article Dirac Particle for the Position Dependent Mass in the Generalized Asymmetric Woods-Saxon Potential Soner AlpdoLan andali Havare Department of Physics, Mersin University, Mersin, urkey Correspondence should be addressed to Soner Alpdoğan; Received 4 April 24; Accepted 9 June 24; Published July 24 Academic ditor: Hong-Jian He Copyright 24 S. Alpdoğan and A. Havare. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. he publication of this article was funded by SCOAP 3. he one-dimensional Dirac equation with position dependent mass in the generalized asymmetric Woods-Saxon potential is solved in terms of the hypergeometric functions. he transmission and reflection coefficients are obtained by considering the onedimensional electric current density for the Dirac particle and the equation describing the bound states is found by utilizing the continuity conditions of the obtained wave function. Also, by using the generalized asymmetric Woods-Saxon potential solutions, the scattering states are found out without making calculation for the Woods-Saxon, Hulthen, cusp potentials, and so forth, which are derived from the generalized asymmetric Woods-Saxon potential and the conditions describing transmission resonances and supercriticality are achieved. At the same time, the data obtained in this work are compared with the results achieved in earlier studies and are observed to be consistent.. Introduction he scattering and bound states in nonrelativistic and relativistic quantum mechanics within an external potential are studied in order to describe the behavior of particles, atoms, and molecules in physics. hus, the nonrelativistic and relativistic particle equations in an external potential have been widely handled for the scattering and bound states [ 6]. In the recent years, these external potentials have been generalized to asymmetric potentials in theoretical and experimental physics [3 22]. Also, last year, an international team at the ISOLD radioactive-beam facility at CRN reported that atomic nuclei could suppose asymmetric pear shapes [9]. Another important issue is the concept of position dependent mass. his concept is studied in different applications of physics, for example, electronic properties of semiconductor materials [23], quantum liquids [24], quantum dots [25], and the nuclear forces of relativistic two nucleons [26, 27]. In this paper, the generalized asymmetric Woods-Saxon potential (GAWSp), that includes the Woods-Saxon potential, is considered to obtain the solutions of the Dirac equation with position dependent mass. he Woods-Saxon potential [28] has been widely used in the nonlinear theory of scalar mesons [29, 3], the nuclear shell model, and the distribution of nuclear densities [3, 32]. he GAWSp is introduced in [6] θ ( x) V (x) =V [ q+pe + θ (x) ], a(x+l) () q+ peb(x L) θ(x) is the Heaviside step function and V is potential intensity and real. Parameters L, p, q, L, p, q, a, andb are real. he GAWSp is shown in Figure and in special cases, thegawspturnsintothepotentialswhichareusedinthe nonrelativistic and relativistic quantum physics. Some special cases ofthegawspareshown inable. he paper is organized as follows. In Section 2, the transmission and reflection coefficients are found for the effective mass Dirac particle. In Section 3, anequationis obtained for energy eigenvalues. In Section 4, theresults and discussions and the plots constructed by using Mathematica Software are commented and the energy values are given numerically; moreover, the transmission resonance condition for the scattering states and supercriticality are derived and the data obtained in this work are compared with

2 2 Advances in High nergy Physics able : he special potentials: the symmetric potentials and asymmetric (asy.) potentials (pot.). he GAWSp he asy. Hulthen pot. he asy. cusp pot. q q, q q L = L =, θ( x) V(x) = V [ q+pe + θ(x) a(x+l) q+ pe ] L= L =, p= p =, q= q =, p= p =, b(x L) V(x) = V [ θ( x) e ax q + θ(x) e bx q ] V(x) = V [ θ( x) e ax he Woods-Saxon pot. he Hulthen pot. he cusp pot. a=b, L= L, a=b, q q, q q a=b, L= L =, q= q =, p= p =, L= L =, p= p =, q= q =, p= p =, θ( x) V(x) = V [ +e + a(x+l) θ(x) +e a(x L) ] V(x) = V [ θ( x) e ax q + θ(x) e ax q ] V(x) = V [ θ( x) e ax + θ(x) e bx ] + θ(x) e ax ] q =.6, q =.8 p=, p =8 a = 2, b = 4 L=4, L =3 Figure : he different values of the GAWSp. shows V =, a=5, b=5, p=, p =, L=2,and L =2(solid lines) and V =, q=, q =, a=2, b=2, L=2,and L =2(dash-dotted lines). shows V =, p=, p =, q=, q =, L=2,and L =2(solid lines) and V =, q=, q =, a=5, b=5, p=,and p =(dash-dotted lines). the results achieved in earlier studies. Finally, the present paper is summarized in Section Scattering Solutions of the GAWSp for the Variable Mass Dirac Particle he Dirac equation for the relativistic free-particle is given as the following form [2] (in natural units ħ=c=): [iγ μ μ m(x)]ψ(x) =, (2) m(x), the Dirac particle mass, depends on one spatially coordinate x. In order to obtain the one-dimensional Dirac equation under an external potential V(x), the gamma matrices γ x and γ are reduced to the Pauli matrices iσ x and iσ z,respectively, {( ) d [ V(x)] ( )+m(x) ( )} ( U (x) U 2 (x) )=, U (x) and U 2 (x) aredecomposedintoupperand lowercomponentsofthetwo-componentwavefunctionψ(x) (3) and (3) turnsintothetwocoupleddifferentialequationsas follows: du (x) du 2 (x) = [m (x) + V(x)] U 2 (x), = [m (x) +V(x)] U (x). hen the following two expressions are written as they were described by Flügge as [3] Θ (x) =U (x) +iu 2 (x), Φ (x) =U (x) iu 2 (x). Putting these expressions into (4), the following equationsare achieved: (4) (5) dθ (x) =i[ V(x)] Θ (x) im(x) Φ (x), (6) dφ (x) = i[ V(x)] Φ (x) +im(x) Θ (x). (7)

3 Advances in High nergy Physics 3 By using the two equations given above, for Θ(x) and Φ(x), the following two independently second-order differential equations are obtained: d 2 Θ (x) 2 m (x) dm (x) dθ (x) +{[ V(x)] 2 m 2 dv (x) (x) +i d 2 Φ (x) 2 +i[ V(x)] m (x) m (x) dm (x) dφ (x) dm (x) }Θ(x) =, +{[ V(x)] 2 m 2 dv (x) (x) i i[ V(x)] m (x) dm (x) }Φ(x) =. he Dirac particle mass m(x) is supposed in [2] (8) m (x) =m +m f (x), (9) the function f(x) is given by f(x) = θ( x)/(q + pe a(x+l) )+θ(x)/( q+ pe b(x L) ). he parameter m is the rest mass of Dirac particle and m is a real positive and small parameter; therefore, the derivative terms including the mass m(x) function are ignored in (8). So, (8)becomes d 2 Θ (x) 2 d 2 Φ (x) 2 +{[ V(x)] 2 m 2 (x) +i + {[ V(x)] 2 m 2 (x) i dv (x) }Θ(x) =, () dv (x) } Φ (x) =. () It is clearly seen from (6) and(7) thatitwillbesufficientto solve one of () and() to obtain the solutions describing the scattering states. herefore, we start calculation by solving (). Let us now define a new variable y= (q/p)e a(x+l) in ()fortheregionx<,inthisinstance,()becomes a 2 y 2 d 2 Θ L (y) dy 2 + {[+ +a 2 y dθ L (y) dy 2 V y q( y) ] y 2 m 2 m2 q 2 ( y) 2 y +2m m q( y) iay d dy [ V y q( y) ]} Θ L (y) =, (2) and getting a trial wave function Θ L (y) = y ζ ( y) λ w(y), (2) reduces to the Gaussian differential equation [33] y( y) d2 w(y) dy 2 +[+2ζ y(2ζ + 2λ + )] w(y) dy (ζ+λ+γ)(ζ+λ γ)w(y)=, ζ= ik a, λ = 2 + ( 2 + iv aq ) 2 + m2 a 2 q 2, γ = ip a, (3) p 2 =( V q ) 2 (m + m q ) 2, k = 2 m 2. (4) he solution of (3) is found in the form of hypergeometric functions, as follows: w(y)=n 2 F (ζ+λ+γ,ζ+λ γ,+2ζ;y) +N 2 y 2ζ F ( ζ+λ+γ, ζ+λ γ, 2ζ;y), so, the whole left-hand solution becomes Θ L (y) =N y ζ ( y) λ 2 F (ζ+λ+γ,ζ+λ γ,+2ζ;y) +N 2 y ζ ( y) λ F ( ζ+λ+γ, ζ+λ γ, 2ζ;y). (5) (6) Now,wecanmaketheprocesstofindthesolutionforthe region x>.choosinganewvariablez=( q/ p)e b(x L), ()iswrittenas b 2 z 2 d 2 Θ R (z) dz 2 +{[+ +b 2 z dθ R (z) dz 2 V z q ( z) ] m 2 z 2 m2 q 2 ( z) 2 z +2m m q ( z) +ibz d dz [ V z q ( z) ]}Θ R (z) =, (7) andassumingatestwavefunctionθ R (z) = z ζ( z) λv(z), the solution of (7) becomes the following form: V (z) =N 3 2 F ( ζ λ+ γ, ζ λ γ, + 2 ζ; z) +N 4 z 2 ζ 2 F ( ζ λ+ γ, ζ λ γ, 2 ζ; z). (8)

4 4 Advances in High nergy Physics In this case, the whole right-hand solution is Θ R (z) =N 3 z ζ( z) λ 2 F ( ζ λ+ γ, ζ λ γ, + 2 ζ; z) +N 4 z ζ( z) λ F ( ζ λ+ γ, ζ λ γ, 2 ζ; z), (9) ζ = ik b, λ = 2 + ( 2 iv b q ) 2 + m2 b 2 q 2, γ =i p b, p 2 =( V q ) 2 (m + m q ) 2, k = 2 m 2. (2) In order to investigate the transmission () and reflection (R) coefficients, the solutions obtained above have to be used asymptotic behaviors as x and x +.Ifx ischosen,theleft-handsolutioncanbewrittenas Θ L (x) N e π(k/a) ( q i(k/a) p ) e ik(x+l) +N 2 e π(k/a) ( q p ) i(k/a)e ik(x+l), and if x + is chosen, the right-hand solution is (2) Θ R (x) N 4 e π(k/b) ( q p ) i(k/b)e ik(x L). (22) he one-dimensional electric current density for the Dirac particle is given as the following form: J (x) = 2 [ Θ (x) 2 Φ (x) 2 ]; (23) so as to find electric current density, (2)and(22)areinserted into (6). In this case, the asymptotic behaviors for Φ L (x) and Φ R (x) become Φ L (x) N e π(k/a) ( q i(k/a) p ) [ k m (x) ]eik(x+l) +N 2 e π(k/a) ( q p ) i(k/a) [ +k m (x) ]e ik(x+l), Φ R (x) N 4 e π(k/b) ( q p ) i(k/b) [ k m (x) ]eik(x L). (24) Using (2), (22), and (24) for electric current density given in (23), j inc., j trans.,andj ref. arecalculatedandthereflectionand transmission coefficients are found as follows, respectively: R= j ref. j inc. = 2 N 2 N = j trans. j inc. = 2 N 4 ( +k k )e4πk/a, e 2πk(/a+/b), N (25) able 2: he abbreviations taking part in (26), (34), and (35). β= q p, β = q p B = β ζe b ζ L ( βe b L ) λ B 2 =β ζ e aζl ( βe al ) λ B 3 =β ζ e aζl ( βe al ) λ B 4 =b ζ β ζe b ζ L ( βe b L ) λ B 5 = b λ β ζ+ e b L( ζ ) ( βe b L ) λ B 6 = b β ζ+ e b L( ζ ) ( βe b L ) λ ( ζ λ+ γ)( ζ λ γ) 2 ζ B 7 =aζβ ζ e aζl ( βe al ) λ B 8 = aλβ ζ+ e al(ζ+) ( βe al ) λ B 9 =aβ ζ+ e al(ζ+) ( βe al λ (ζ + λ + γ)(ζ + λ γ) ) +2ζ B = aζβ ζ e aζl ( βe al ) λ B = aλβ ζ+ e al(ζ ) ( βe al ) λ B 2 =aβ ζ+ e al(ζ ) ( βe al λ ( ζ + ] + γ)( ζ + λ γ) ) 2ζ F = 2 F ( ζ λ+ γ, ζ λ γ, 2 ζ; βe b L ) F 2 = 2 F (ζ+λ+γ,ζ+λ γ,+2ζ;βe al ) F 3 = 2 F ( ζ+λ+γ, ζ+λ γ, 2ζ;βe al ) F 4 = 2 F ( ζ λ+ γ+, ζ λ γ+,2 2 ζ; βe b L ) F 5 = 2 F (ζ+λ+γ+,ζ+λ γ+,2+2ζ;βe al ) F 6 = 2 F ( ζ+λ+γ+, ζ+λ γ+,2 2ζ;βe al ) F = 2 F ( λ+ γ c, λ γ c,; βe b L ) F 2 = 2 F (λ + γ c,λ γ c,;βe al ) F 4 = 2 F ( λ+ γ c +, λ γ c +,2; βe b L ) F 5 = 2 F (λ + γ c +,λ γ c +,2;βe al ) j inc., j trans.,andj ref. represent the incident, transmitted, andreflectedcurrents,respectively.finally,toinvestigate numerically the reflection and transmission coefficients, the continuity condition and the first derivative of the wave function are used. So, N 2 /N and N 4 /N,takingpartinthe above expressions, are found as N 2 =([(B N 4 +B 5 )F +B 6 F 4 ]B 2 F 2 [(B 7 +B 8 )F 2 +B 9 F 5 ]B F ) ([(B +B )F 3 +B 2 F 6 ]B F [(B 4 +B 5 )F +B 6 F 4 ]B 3 F 3 ), N 4 N =([(B 7 +B 8 )F 2 +B 9 F 5 ]B 3 F 3 [(B +B )F 3 +B 2 F 6 ]B 2 F 2 ) ([(B 4 +B 5 )F +B 6 F 4 ]B 3 F 3 [(B +B )F 3 +B 2 F 6 ]B F ), the constants are given in able 2. (26)

5 Advances in High nergy Physics 5 3. he quation of nergy igenvalues he purpose of this section is to derive an equation that is known as the bound state condition for the energy eigenvalues. For this, V is put instead of V in the potential given in () and the new variable is defined as variable y= (q/p)e a(x+l) in the region x<.asaresult,() iswritten in the following form:, R a 2 y 2 d 2 Θ L (y) dy 2 + {[ +a 2 y dθ L (y) dy 2 V y q( y) ] y 2 m 2 m2 q 2 ( y) 2 y +2m m q( y) +iay d dy [ V y q( y) ]} Θ L (y) =. (27) 2 Figure 2: he plot of unitarity condition, R+ =, with the position dependent mass for q =.5, q =.5, p=2, p =2, L=5, L =5, a=3, b=3, V =5, m =,andm. aking a wave function as Θ L (y) = y ζ ( y) λ h(y), the solution of (27)is obtained as Θ L (y) =A y ζ ( y) λ ζ = k b, λ = 2 + ( 2 + iv b q ) 2 + m2 b 2 q 2, γ = i p 2 b, p 2 2 =(+ V 2 q ) (m + m 2 q ), (3) F (ζ +λ +γ,ζ +λ γ,+2ζ ;y) +A 2 y ζ ( y) λ F ( ζ +λ +γ, ζ +λ γ, 2ζ ;y), (28) k = m 2 2. Because of the boundary conditions (which mean the wave functionsgotozeroatinfinity),thewavefunctionsgivenin (28)and(3)arewrittenas Θ L (y) = A y ζ ( y) λ ζ = k a, λ = 2 + ( 2 iv aq ) 2 + m2 a 2 q 2, γ = ip 2 a, p2 2 =(+V q ) 2 (m + m q ) 2, (29) F (ζ +λ +γ,ζ +λ γ,+2ζ ;y), Θ R (z) =A 3 z ζ ( z) λ F ( ζ λ + γ, ζ λ γ,+2 ζ ;z). (32) k = m 2 2. Similarly, choosing the variable z=( q/ p)e b(x L) and wave function Θ R (z) = z ζ ( z) λ f(z) in the region x>,the positive region solution is found as Θ R (z) =A 3 z ζ ( z) λ F ( ζ λ + γ, ζ λ γ,+2 ζ ;z) +A 4 z ζ ( z) λ F ( ζ λ + γ, ζ λ γ, 2 ζ ;z), (3) In order to find an equation for energy eigenvalues, the continuity conditions of the wave function are used, Θ L (x = ) = Θ R (x = ) and dθ L / x= = dθ R / x=.after performing the necessary calculations, an equation for energy eigenvaluesisobtainedasfollows: [(D 3 +D 4 )F 7 +D 5 F 9 ]D 2 F 8 [(D 6 +D 7 )F 8 +D 8 F ]D F 7 =, the constants are given able Results and Discussions (33) 4..InterpretationoftheDrawingsandComparisonofthe Results with Previous Ones. In Figure2,whiletheparameter

6 6 Advances in High nergy Physics able 3: he constants given in (33)and(37). β= q p, β = q p D = β ζ e b ζ L ( βe b L ) λ D 2 =β ζ e aζ L ( βe al ) λ D 3 = b ζ β ζ e b ζ L ( βe b L ) λ D 4 = b λ β ζ + e b L( ζ +) ( βe b L ) λ D 5 = b β ζ + e b L( ζ +) ( βe b L ) λ ( ζ λ + γ )( ζ λ γ ) +2 ζ D 6 =aζ β ζ e aζ L ( βe al ) λ D 7 = aλ β ζ + e al(ζ +) ( βe al ) λ D 8 =aβ ζ + e al(ζ +) ( βe al ) λ (ζ +λ +γ )(ζ +λ γ ) +2ζ F 7 = 2 F ( ζ λ + γ, ζ λ γ,+2 ζ ; βe b L ) F 8 = 2 F (ζ +λ +γ,ζ +λ γ,+2ζ ;βe al ) F 9 = 2 F ( ζ λ + γ +, ζ λ γ +,2+2 ζ ; βe b L ) F = 2 F (ζ +λ +γ +,ζ +λ γ +,2+2ζ ;βe al ) F 7 = 2 F ( λ + γ c, λ γ c,; βe b L ) F 8 = 2 F (λ +γ c,λ γ c,;βe al ) F 9 = 2 F ( λ + γ c +, λ γ c +,2; βe b L ) F = 2 F (λ +γ c +,λ γ c +,2;βe al ) m goes to zero and the GAWSp is a symmetric potential, one can clearly see that the unitary condition, R+ =, is provided in the GAWSp for the position dependent mass Dirac particle. On the other hand, if the parameter m does notgotozeroandthegawspisanasymmetricone,the unitary condition is not provided. Figure 3 shows the effect of position dependent mass for the Dirac particle within the GAWSp on the transmission coefficient. In this figure, it is seen that the presence of variable mass causes the formation of a wider line according to the previous graph that is one of the constant masses. One of the advantages of this work is that, by using the GAWSp solutions for the Dirac particle with variable mass, the transmission and reflection coefficients are directly obtained without doing calculation for the potentials which are given in able. For example, in the numerical results of the GAWSp, if one chooses a=b, L= L, p= p =, and q = q =, one can acquire the scattering states of the Woods-Saxon potential, by putting a=b, L= L =, p= p =, q q,andq q, the scattering states of the Hulthen potential are obtained, substituting a=b, L= L, p= p =, and q= q =, the scattering states of the cusp potential are found. Figures 4 6 are drawn using the GAWSp solutions for the Dirac particle. Figure 4 displays a plot of the transmission coefficient describing the behavior of the Dirac particle in the Woods-Saxon potential versus the energy and the potential strength for both the position dependent mass and the constant mass. From Figure 4, itisseen,in the presence of position dependent mass, that the zero region of the transmission coefficient expands. At the same time, the results of Figure 4 are compatible with [7, 2]. Figure 5 shows the effect of the position dependent Dirac particle for the Hulthen potential on the transmission coefficient. he data drawn from Figure 5 prove that the values of the transmission coefficient in the case of position dependent mass decrease. Furthermore, these results are similar to [2, 22]. Figure 6 displays the effect of the Dirac particle for the cusp potential on the transmission coefficient. Figure7 shows a plot of the transmission coefficients obtained for the Dirac particle with the position dependent mass against the energy and the potential strength for the GAWS, the Woods-Saxon, the Hulthen, and the cusp potentials. In the left panel which is located in this figure, while the transmission resonance appears for the GAWS and Woods-Saxon potentials at lower values of energy, it does not exist for the Hulthen and the cusp potentials at same values of energy. In the right panel taking partinthesamefigure,whilethetransmissionresonance peaks are more for the Woods-Saxon potential at higher values of potential strength (V ), the transmission resonance does not occur for the cusp potential at same values of V. Moreover, if one wants to achieve the scattering solutions of the asymmetric Hulthen and the asymmetric cusp potentials, one can use the solutions describing the scattering states, of the GAWSp, found in this study. In addition, by using (33), energy eigenvalues can be calculated numerically. As the bound states are considered, k must be real which means <m in (29) and(3) and also values of energy must be V(x) <. Besides, the energy takes the values providing the two conditions that V /(q + pe al ) < < m in the region x < and V /( q + pe b L ) <<m in the region x>due to the asymmetric form of the GAWS well. As the scattering states are handled, by using the GAWS well solutions for the Dirac particle, the numerical energy values are directly obtained without doing calculation for the asymmetric Hulthen and asymmetric cusp wells. Additionally, the energy takes the values providing these conditions that V /( q) < < m in the region x<and V /( q) < < m in the region x>for the asymmetric Hulthen well and V <<m in the regions x<and x>for the asymmetric cusp well. he numerical energy eigenvalues containing all cases which are given above for asymmetric potentials are found in able Conditions for ransmission Resonance and Supercriticality. he aim of this section is to obtain the conditions describing a transmission resonance (providing the requirement that the transmission coefficient is unity) and supercriticality (while the bound state of the Dirac particle is at = m ). o find a condition describing the transmission resonance for the scattering states, N 2 /N taking part in (26), must be equal to zero. From this situation, the following expression is attained: [(B 4 +B 5 )F +B 6 F 4 ]B 2 F 2 [(B 7 +B 8 )F 2 +B 9 F 5 ]B F =, (34)

7 Advances in High nergy Physics V m =. m =.4 m =. m =.4 Figure 3: he effect of position dependent mass for the GAWSp on transmission coefficient. shows V =4, L =.2, L =., a=, b=, p = 2.2, p = 2., q=, q =,andm =.shows=2, L =.2, L =., a=, b=, p = 2.2, p = 2., q=, q =,andm = V m =. m =. m =. m =. Figure 4: he effect of position dependent mass for the Woods-Saxon potential on transmission coefficient. shows V =.2, L=, L =, a=3, b=3, p=, p =, q=, q =,andm =.4.shows=.8, L=, L =, a=3, b=3, p=, p =, q=, q =,and m =.4. the constants are given in able 2.heaboveequation takesthefollowingforminthelimitoflowmomentum( m which leads to ζ= ζ =): + aβe al ( βe al ) λ (λ + γ c )(λ γ c ) F 5 ] ( βe b L ) λ F =, [( λb) βe b L ( βe b L ) λ F (35) b βe b L ( βe b L ) λ ( λ+ γc )( λ γ c ) F 4 ] ( βe al ) λ F 2 [( aλ) βe al ( βe al ) λ F 2 γ c = ip c a, p2 c = V2 m2 V q 2 2m +m, q γ c = i p c b, p 2 c = V2 m2 V q 2 2m +m. q he abbreviations taking part in (35)are given in able 2. (36)

8 8 Advances in High nergy Physics V m =. m =.4 m =. m =.4 Figure 5: he effect of position dependent mass for the Hulthen potential on transmission coefficient. shows V =4, L=, L =, a=, b=, p=, p =, q =.5, q =.5,andm =.shows=2, L=, L =, a=, b=, p=, p =, q =.5, q =.5,andm = V m =. m =.4 m =. m =.4 Figure 6: he effect of position dependent mass for the cusp potential on transmission coefficient. shows V =4, L=, L =, a=, b=, p=, p =, q= q,andm =.shows=2, L=, L =, a=, b=, p=, p =, q= q,andm =. o achieve the supercritical condition for the bound states, the low-momentum limit ( m which leads to ζ = ζ = ) is used. In this case, (33) giving the energy eigenvalues reduces to the following equation: + aβe al ( βe al ) λ (λ +γ c )(λ γ c ) F ] [( βe b L ) λ F 7 ]=, [( λ b) βe b L ( βe b L ) λ F 7 b βe b L ( βe b L ) λ ( λ + γ c )( λ γ c ) F 9 ] [( βe al ) λ F 8 ] [( aλ )σe al ( βe al ) λ F 8 γ c = ip c a, p2 c = V2 m2 V q 2 2m +m, q γ c = i p c b, p 2 c = V2 m2 V q 2 2m +m. q F 7, F 8, F 9,and F in (37)aregiveninable 3. (37) (38)

9 Advances in High nergy Physics 9 able 4: he numerical energy values (in atomic units: fm )forasymmetricwells. he potentials he GAWS well he asymmetric Hulthen well he asymmetric cusp well he parameters he energy values Constant mass ffective mass m = m =. L=2, L = 2., p = p =, =.9545 = q=, q =, a =.7, b =.8, 2 = =.985 m =,V = 3 3 = = = = L= L =,p= p =, m = m =. q q =.5, q q =.5, =.5762 =.5887 a =.7, b =.8, 2 = =.9975 m =,V = 3 3 = = = =.3732 m = m =. L= L =,p= p =, = = q= q, a =.7, b =.8, 2 = = m =,V = 3 3 = = = = V Figure 7: he effect of the potentials for position dependent mass on the transmission coefficient: the GAWSp for q= q =.25, a=b=4, L= L =3, p= p =, m =,andm =. (dot-dashed lines), the Woods-Saxon potential for q= q =, L= L =3, p= p =, m =, and m =. (solid lines), the cusp potential for q= q, a=b=4, L= L =, p= p =, m =,andm =. (dash-double-dotted lines), and the Hulthen potential for q= q =.75, a=b=4, L= L =, p= p =, m =,andm =. (dashed lines). In and, parameters are V =3and =2,respectively. When (35) and(37) are compared, it is clearly seen that these two equations are equal to each other to be λ = λ and λ=λ. his case means that the GAWSp promotes a state which has the zero momentum. At the same time, this state is defined as the half-bound one. 5. Conclusions he effective mass Dirac equation written for the GAWSp is solved in the approach that the parameter m has very small values (as m ). he transmission and reflection coefficients are acquired by utilizing the asymptotic behaviors andtheboundaryconditionsoftheobtainedwavefunctions. Besides, the scattering states are discussed in both the effective mass and the constant mass and the scattering states for the Woods-Saxon, the Hulthen, and the cusp potentials are obtained by using the scattering solutions of thegawsp.heenergyeigenvaluesfoundforthegaws, the asymmetric Hulthen, and the asymmetric cusp wells are calculated numerically by using the regular wave functions. Finally, the transmission resonance condition and the equation for the supercritical state are acquired. For all that, the results obtained in this study are compared with the ones foundinearlierstudiesandareseentobecompatible.once and for all, when the parameter m does not go to zero and the GAWSp is an asymmetric potential, the unitary condition is notprovided.hisresultisoneofthemostimportantresults of this study. his is a problem which is questionable.

10 Advances in High nergy Physics Conflict of Interests he authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments he authors wish to thank Assistant Professor Oktay Aydoğdu for many useful discussions that provided improvements to the present paper. Also, this research was partially supported by the Scientific and echnological Research Council of urkey. his work was supported by Mersin University Scientific Research Unit (BAP-FB F (SA) 2- YL). References []L.D.Landauand.M.Lifshitz,Quantum Mechanics, Non- Relativistic heory, Pergamon, New York, NY, USA, 3rd edition, 977. [2] R. G. Newton, Scattering heory of Waves and Particles, Springer, New York, NY, USA, 2nd edition, 982. [3] S. Flügge, Practical Quantum Mechanics, Springer, 2nd edition, 994. [4] D. Bohm, Quantum Mechanics, Prentice-Hall, New York, NY, USA, 95. [5] N. Domhey, P. Kennedy, and A. 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