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Dirac particles interacting with the improved Frost–Musulin potential within the effective mass formalism
Accepted Manuscript Dirac particles interacting with the improved Frost-Musulin potential within the effective mass formalism Ahmet Tas, Oktay Aydogdu, Mustafa Salti PII: DOI: Reference:
S0003-4916(17)30058-1 http://dx.doi.org/10.1016/j.aop.2017.02.010 YAPHY 67337
To appear in:
Annals of Physics
Received date: 20 January 2017 Accepted date: 17 February 2017 Please cite this article as: A. Tas, O. Aydogdu, M. Salti, Dirac particles interacting with the improved Frost-Musulin potential within the effective mass formalism, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.02.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Dirac particles interacting with the improved Frost-Musulin potential within the effective mass formalism Ahmet Tas∗ , Oktay Aydogdu† , Mustafa Salti‡ Department of Physics, Faculty of Arts and Sciences, Mersin University, Mersin-33343, Turkey. We mainly investigate the dynamics of spin- 12 particles with position-dependent mass for the improved Frost-Musulin potential under spin-pseudospin symmetry. First, we find an approximate analytical solution of the Dirac equation both for bound and scattering states under spin-pseudospin symmetry and then we see that the normalized solutions are given in terms of the Gauss hypergeometric functions. In further steps, we analyze our results numerically. PACS numbers: 03.65.Nk, 03.65.Ge, 03.65.Pm, 03.65.Db, 34.20.Gj Keywords: Dirac equation, Position-dependent mass, Bound states, Scattering phase shifts.
I.
INTRODUCTION
Investigating solutions of the non-relativistic or relativistic wave equations for the real physical potentials is one of the important issues in quantum mechanics. On this purpose, the variable mass formalism gives meaningful and useful theoretical estimations of many experimental features for quantum many-body systems [1]. In literature, the effective mass concept has been used for many significant topics such as quantum liquids, 3 He clusters, nuclei, electronic properties of semiconductors, and metallic clusters [2–6]. Also, in the non-relativistic Schr¨odinger equation, coordinate dependent mass and momentum operator do not commute each other which causes ordering ambiguity that’s why it is hard to construct a true kinetic energy operator within the position-dependent mass framework [7–10]. However, this ordering ambiguity problem emerged in the non-relativistic Schr¨odinger equation is automatically eliminated in the relativistic Dirac and Klein-Gordon equations. Besides, it is useful to consider the position-dependent mass Dirac equation when quantum mechanical systems that include heavy atoms are dealt with [11]. Moreover, studying exact solutions of the Dirac equation with a position-dependent mass has become an important research area, because such investigations may yield a testing ground for some approximation schemes and meaningful conclusions to understand some physical phenomena [2, 12]. Therefore, the exact solvability of relativistic (and also non-relativistic) quantum mechanical problems within the framework of positiondependent mass has always been very attractive [13–19]. On the other hand, finding a universal potential energy function is another important puzzle in this area and various potential energy functions have been proposed in order to discuss diatomic molecules and other physical systems. The P¨oschl-Teller [20], Hulth´en [21], Woods-
∗ [email protected]
† [email protected] ‡ [email protected]
(corresponding author)
Saxon [22], Rydberg [23], Rosen-Morse [24], DengFan [25], Schi¨ oberg [26], Tietz [27], Wei [28] and FrostMusulin [29] type potentials are some of the well-known models (see Ref. [30] and references within for a good review). Furthermore, some improved potentials have been recently introduced to study dynamics and other physical properties of diatomic molecules [31–35]; e.g. thermodynamic properties of the real diatomic lithium dimer molecules have been investigated in Ref. [36] within the improved Manning-Rosen potential. The above works motivated us to investigate the approximate solutions of 3-dimensional Dirac equation with spatially dependent mass for the improved FrostMusulin potential offered by Jia and his coworkers [37] to discuss diatomic molecules. In literature, Brown and Musulin [38] solved the Born-Oppenheimer nuclear equation in the presence of Frost-Musulin potential energy function asymptotically. Next, using the improved FrostMusulin potential, Adepoju and Eweh [39] obtained approximate bound state solution of the Schr¨ odinger equation. In another recent paper, Onate et al. [40] have worked on pseudospin and spin symmetric solutions of the Dirac equation in the presence of the Hellmann-FrostMusulin type potential energy function. We see that, in literature, the bound and scattering states solutions of relativistic spin- 21 particles interacting with external improved Frost-Musulin potential have not been studied so far within the spatially dependent mass framework. That’s why, we focus on this problem in order to get analytic solutions for the Dirac particles in view of position dependent mass formalism with the improved FrostMusulin potential. Our work is structured in five sections. The first one includes the scope and purpose of our investigation. The second one introduces the spin symmetric solutions of bound and scattering states for the relativistic spin- 21 particles. The pseudospin symmetric solutions of bound and scattering states are investigated in the third section. We devote the fourth section to the numerical analysis of our theoretical results. The last section gives final remarks.
2 II. SPIN SYMMETRIC SOLUTIONS OF BOUND AND SCATTERING STATES
d2 κ(κ + 1) − − [m + Enκ − ∆(r)]Σ(r) dr2 r2 −(m − Enκ )[m + Enκ − ∆(r)]}Fnκ (r) = 0
(1)
with ¯h = c = 1. For the spin symmetry (see Ref. [41] for detailed information), ∆(r) = Cs where Cs is a constant and Σ(r) = V (r) where V (r) denotes the improved FrostMusulin potential given as [37] λr2 V (r) = Ed {1 − (1 + λrb )e−λ(r−rb ) + b e−λ(r−rb ) }, (2) r where Ed , λ and rb are three positive parameters and they represent the dissociation energy, range of potential well and the equilibrium bound length, respectively. The variation of improved Frost-Musulin potential versus r is plotted for different values of the dissociation energy, range of potential well and the equilibrium bound length in FIG. 1. Additionally, the position-dependent definition of mass is written generally as given below m(r) = m0 + m1 V (r).
(10)
κ(κ + 1) = l(l + 1).
(11)
with
Dynamics of relativistic spin- 12 particles is discussed by using the Dirac equation. Due to the potential and mass have radial dependence, we should focus on the following spherical form of Dirac equation {
ξ6 = −m1 (1 + m1 )Ed2 (λrb + 1)2 e2λrb ,
(3)
In order to solve the Dirac equation analytically and reduce our results to the usual constant mass case, we need to assume such definition of mass. Hence, making use of relations (2) and (3) transforms equation (1) into the following form { d2 Fnκ (r) ξ3 κ(κ + 1) + ξ1 + (ξ2 + )e−λr − dr2 r r2 } 1 ξ5 + (ξ4 + + rξ6 )e−2λr Fnκ (r) = 0, (4) r r
In the new form of radial Dirac equation written in (4), we have terms including 1r and r12 which make this equation imposible to solve analytically. From this point of view, we can consider the following approximations in order to get a second order solvable differential equation [42]:
ξ2 = [m0 (1 + 2m1 ) + Enκ − (1 + m1 )Cs +2m1 (1 + m1 )Ed ]Ed (1 + λrb )eλrb ,
d2 Fnκ (r) ξ3 λ −λr + ξ1 Fnκ (r) + (ξ2 + 1−e Fnκ (r) −λr )e dr 2 ξ λ + 1−eλ−λr (ξ4 + 1−e5−λr )e−2λr Fnκ (r) + ξ6 e−2λr Fnκ (r) 2
κ(κ+1)λ − (1−e −λr )2 Fnκ (r) = 0.
2
Fnκ x(1 − x) d dx + λ1 (ξ3 + ϵ4 x + 2
+(1 − x) dFdxnκ +
x(1 − x)
−
(8)
ξ5 = −m1 (1 + m1 )Ed2 λ2 rb4 e2λrb ,
(9)
−
κ(κ+1)λ x(1−x) )Fnκ
+ ξ2 + ξ6 x)Fnκ = 0. (14)
d2 Fnκ dFnκ + (1 − x) dx2 dx
1 (Υ1 + Υ2 x − Υ3 x2 )Fnκ = 0, x(1 − x)
(15)
where 1 [κ(κ + 1)λ2 + λξ4 − ξ1 − ξ2 − ξ6 ], λ2
(16)
1 [2(ξ1 + ξ2 + ξ6 ) − λ(ξ3 + 3ξ4 )], λ2
(17)
1 [ξ1 + ξ2 + ξ6 + λ2 ξ5 − λ(ξ3 + 2ξ4 )]. λ2
(18)
Υ2 =
ξ4 = 2m1 (1 + m1 )Ed2 λrb2 (1 + λrb )e2λrb ,
(1−x) ξ1 λ2 ( x
xλξ5 1−x
Due to we have λr ≪ 1, some of the terms including orders of (x−1)3 and (x−1)4 can be eliminated. Thence, after e few mathematical steps, we obtain that
(6)
(7)
(13)
Bound State Solution: To find the possible bound state solution, here we introduce a new variable such as x = e−λr . Thence, equation (13) can be transformed into the following form
Υ1 =
ξ3 = −[m0 (1 + 2m1 ) + Enκ − (1 + m1 )Cs +2m1 (1 + m1 )Ed ]Ed λrb2 eλrb ,
(12)
It is important to mention here that the above approximations are valid for λr ≪ 1. The characteristic of ap2 proximations 1r , 1−eλ−λr , r12 and (1−eλ−λr )2 are analyzed numerically in FIG. 2. Then, one can find the following second order differential equation
where 2 ξ1 = Enκ − m20 − (Enκ − m0 )Cs − [m0 (1 + 2m1 ) +Enκ − (1 + m1 )Cs ]Ed − m1 (1 + m1 )Ed2 , (5)
1 λ2 ≈ . 2 r (1 − e−λr )2
λ 1 ≈ , r 1 − e−λr
Υ3 =
We see that equation (15) has two singular points at x = 0 and x = 1. It is known that singular points of
3 a differential equation determine the form of solutions. Thus, one can rearrange the function Fnκ as
should find again Fnκ → 0, thence, a quantum condition is obtained as written below
Fnκ (x) = xµ (1 − x)ν fnκ (x).
a+ = µ + ν + σ = −n.
(19)
Under the boundary conditions, we get the following energy eigenvalues equation and eigenfunctions from equations (31) and (30), respectively;
Therefore, we get d2 fnκ dfnκ + [2µ + 1 − (2µ + 2ν + 1)x] dx2 dx 1 + [µ2 − Υ1 − (2µ2 + 2µν + ν + Υ2 )x x(1 − x)
x(1 − x)
+(µ2 + 2µν + ν 2 + Υ3 )x2 ]fnκ = 0. (20)
Now, at this step, with the following definitions √ µ = ± Υ1 , √ 1 ν = [1 ± 1 − 4(Υ3 − Υ2 − Υ1 )], 2 σ=± equation (20) becomes
√
−Υ3 ,
[ ]2 √ √ 1 Υ1 − n + (1 + 1 − 4(Υ3 − Υ2 − Υ1 ) − −Υ3 = 0, 2 (32) Fnκ (r) = A
(1 − e−λr )ν −λr ). 2 F1 (a+ , a− ; 1 + 2µ; e eµλr
(21) (22)
(23)
d2 fnκ dfnκ x(1 − x) + [2µ + 1 − (2µ + 2ν + 1)x] dx2 dx −(µ + ν + σ)(µ + ν − σ)fnκ = 0. (24) This is one of the well-known differential equation types, and the corresponding solution is given in terms of the Gauss hypergeometric functions [43] as given below fnκ (x) = A 2 F1 (a+ , a− ; 1 + 2µ; x) +Bx−2µ 2 F1 (b+ , b− ; 1 − 2µ; x),
(25)
a+ = µ + ν + σ,
(26)
a− = µ + ν − σ,
(27)
b+ = −µ + ν + σ,
(28)
b− = −µ + ν − σ.
(29)
where
Thus, the general solution will be Fnκ (x) = Axµ (1 − x)ν 2 F1 (a+ , a− ; 1 + 2µ; x) +Bx−µ (1 − x)ν 2 F1 (b+ , b− ; 1 − 2µ; x).
(31)
(30)
We have two boundary conditions for this solution. First, when r → ∞ (on the other hand x = e−λr → 0) we must have Fnκ → 0 in order to obtain a normalizable solution at center of the potential. Thence, it is seen that we have B = 0. Second, when r → 0 (or x = e−λr → 1) we
(33)
Moreover, the normalization constant can be calculated by using the normalization condition given as ∫ ∞ ∫ 1 1 |Fnκ (x)|2 dx = 1. (34) |Fnκ (r)|2 dr = λ 0 x 0 Here, making use of Fnκ (x) gives |A|2 I = 1, λ
(35)
where I=
∫
1 0
[2 F1 (−n, n + 2(µ + ν); 1 + 2µ; x)]2 dx. x1−2µ (1 − x)−2v
(36)
For different n values, the integral I has the following results[44] Γ(2µ)Γ(1 + 2ν) Γ(1 + 2µ + 2ν) (1 + ν)(1 + 2µ + 2ν) Γ(2 + 2µ)Γ(1 + 2ν) µ(1 + 2µ)2 Γ(3 + 2µ + 2ν) (2 + ν)(1 + 3µ + 2µ2 )−2 Γ(3 + 2µ)Γ(2 + 2ν) 4µ(2 + µ + ν) Γ(2 + 2µ + 2ν) 3(3 + ν) Γ(4 + 2µ)Γ(3 + 2ν) (3 + 11µ + 12µ2 + 4µ3 )2 Γ(3 + 2µ + 2ν)
for n = 0, for n = 1, for n = 2, for n = 3, ...
m!(ν + m) Γ(2µ)Γ(2µ + 1)Γ(2ν + m) for n = m. µ + ν + m Γ(2µ + m + 1)Γ(2µ + 2ν + m) (37) Then, we obtain √ λ(µ + ν + n)Γ(2µ + n + 1)Γ(2µ + 2ν + n) A= . n!(ν + n)Γ(2µ)Γ(1 + 2µ)Γ(2ν + n) (38) Scattering State Solution: On the contrary of bound state cases, the scattered particles are labeled with a wave function which mostly behaves like the one for plane waves at large distances. Starting from equation
4 (13) one can obtain the scattering state solutions. After defining y = 1 − e−λr , in equation (13), we find y(1 − y)
d2 Fnκ dFnκ λ−2 −y + [ξ2 (1 − y)y 2 2 dy dx y(1 − y)
+ξ1 y 2 + ξ3 λy(1 − y) + ξ4 λy(1 − y)2 + ξ5 λ2 (1 − y)2 +ξ6 y 2 (1 − y)2 − κ(κ + 1)λ2 ]Fnκ = 0.
√ where ρ = −α3 . Similar to the bound state case, solution of the above differential equation is written in terms of the Gauss hypergeometric functions[43]: gnκ (y) = C 2 F1 (c+ , c− ; 2τ ; y) +Dy 1−2τ 2 F1 (d+ , d− ; 2 − 2τ ; y), where
(39)
Now, ignoring y and higher order terms because of that λr ≪ 1, we obtain 3
dFnκ d2 Fnκ −y y(1 − y) dy 2 dy
+
1 (−α1 − α2 y + Υ3 y 2 )Fnκ = 0, y(1 − y)
(40)
where α1 = κ(κ + 1) − ξ5 , α2 =
1 [2λξ5 − ξ3 − ξ4 ], λ
1 α3 = 2 [ξ1 + ξ2 + ξ6 − (ξ3 + 2ξ4 )λ + λ2 ξ5 ]. λ
(41)
(42)
(43)
ik
(44)
√ 1 (1 ∓ 1 + 4α1 ) 2
(45)
√ k = λ α3 − α2 − α1 ,
(46)
τ=
Fnκ (y) =
and
ik + ρ, λ
(50)
c− = τ −
ik − ρ, λ
(51)
d+ = −τ −
ik + ρ, λ
(52)
d− = −τ −
ik − ρ. λ
(53)
Cy τ ik
(1 − y) λ
Dy 1−τ
ik
(1 − y) λ
2 F1 (c+ , c− ; 2τ ; y)
2 F1 (d+ , d− ; 2
− 2τ ; y).
(54)
Now, we focus on boundary conditions to specify the general solution given above. First, when r → 0 (or y → 0) it should be Fnκ (r) = 0, then we have D = 0 and the solution transforms into the form of Fnκ (y) =
with
c+ = τ −
Hence, for the general solution, we have
+
Next, assuming Fnκ (y) = y τ (1 − y)− λ gnκ (y)
(49)
Cy τ ik
(1 − y) λ
2 F1 (c+ , c− ; 2τ ; y).
(55)
On the other hand, Landau and Lifshitz [45] showed that a wave function written for scattering states should have the following form when r → ∞ (or y → 1): lim Fnκ (r) ≈ 2 sin(kr −
r→∞
lπ + ϕ), 2
(56)
where ϕ represents the phase shifts, then we have
yields d2 gnκ ik dgnκ + [2τ − (2τ − + 1)y] dy 2 λ dy 1 2ikτ 2 2 [τ − τ − α1 + (τ − 2τ − α2 + )y + y(1 − y) λ k2 2ikτ −( 2 + − τ 2 − α3 )y 2 ]gnκ = 0. (47) λ λ y(1 − y)
After performing a few simple mathematical steps, we can rewrite the above equation in a new form: y(1 − y)
d2 gnκ ik dgnκ + [2τ − (2τ − + 1)y] dy 2 λ dy ik ik + ρ][τ − − ρ]gnκ = 0. −[τ − λ λ
(48)
Fnκ (r) = C
(1 − e−λr )τ −λr ). (57) 2 F1 (c+ , c− ; 2τ ; 1 − e e−ikr
Therefore, it is obtained lim Fnκ (r) ≈ Ceikr 2 F1 (c+ , c− ; 2τ ; 1).
r→∞
(58)
Using recurrence relation Γ(2τ )Γ(2τ − c− − c+ ) Γ(2τ − c+ )Γ(2τ − c− ) ×2 F1 (c+ , c− ; c+ + c− + 1 − 2τ ; 1 − z) 2τ −c− −c+ Γ(2τ )Γ(c+ + c− − 2τ ) + (1 − z) Γ(c+ )Γ(c− ) ×2 F1 (2τ − c+ , 2τ − c− ; 2τ − c+ − c− + 1; 1 − z), (59) 2 F1 (c+ , c− ; 2τ ; z)
=
5 and feature 2 F1 (Λ1 , Λ2 ; Λ3 ; 0)
=1
(60)
of the Gauss hypergeometric functions [43], we find Γ(2τ )Γ(2τ − c− − c+ ) 2 F1 (c+ , c− , 2τ ; 1) = Γ(2τ − c+ )Γ(2τ − c− ) Γ(2τ )Γ(η1 + c− − 2τ ) −2ikr + e . Γ(c+ )Γ(c− )
The spherical Dirac equation under pseudospin symmetry (see Ref. [41] for detailed information), is given as (61)
Next, taking into account c+ + c− − 2τ = (2τ − c− − c+ )∗ , c+ = (2τ − c− )∗ and c− = (2τ − c+ )∗ gives Γ(2τ )Γ(2τ − c− − c+ ) 2 F1 (c+ , c− , 2τ ; 1) = Γ(2τ − c+ )Γ(2τ − c− ) [ ]∗ Γ(2τ − c− − c+ ) + Γ(2τ )e−2ikr . Γ(2τ − c+ )Γ(2τ − c− )
(62)
It is known that[43]
Γ(2τ − c− − c+ ) iφ Γ(2τ − c− − c+ ) e . = Γ(2τ − c+ )Γ(c+ − c− ) Γ(2τ − c+ )Γ(2τ − c− ) (63) Using this relation, we obtain Γ(2τ − c− − c+ ) F (c , c , 2τ ; 1) = 2Γ(2τ ) 2 1 + − Γ(2τ − c+ )Γ(2τ − c− ) π ×e−ikr sin(kr + + φ).(64) 2 Thence, the solution (58) at large distance, i.e. r → ∞, turns into the following form π lim Fnκ (r) → 2CΓ(2τ ) sin(kr + + φ) r→∞ 2 Γ(2τ − c− − c+ ) . × (65) Γ(2τ − c+ )Γ(2τ − c− ) After comparing equations (56) and (65), we find 1 Γ(2τ − c+ )Γ(2τ − c− ) C= , Γ(2τ ) Γ(2τ − c− − c+ )
π(l + 1) +φ 2 π(l + 1) = + arg[Γ(2τ − c− − c+ )] 2 −arg[Γ(2τ − c+ )] − arg[Γ(2τ − c− )].
(66)
ϕ =
(67)
Using the singular pole points of scattering amplitude C (2τ − c+ = −n or 2τ − c− = −n), we can calculate the corresponding energy eigenvalues to check the accuracy of our result. Then, using we get
2τ − c+ = −n
III. PSEUDOSPIN SYMMETRIC SOLUTIONS OF BOUND AND SCATTERING STATES
(68)
[ ]2 √ √ 1 α3 − α2 − α1 + n + (1 + 1 + 4α1 ) − −α3 = 0. 2 (69) It is easy to see that this result can be rewritten in another form which is exactly same as given in equation (32).
d2 κ(κ − 1) − − [Enκ − m − Σ(r)]∆(r) dr2 r2 +(Enκ + m)[Enκ − m − Σ(r)]}Gnκ (r) = 0 {
(70)
with h ¯ = c = 1. Under the pseudospin symmetry, Σ(r) is assumed to be equal to a constant Cps and we also have ∆(r) = V (r) where V (r) is the improved Frost-Musulin potential. After considering the position-dependent mass definition, i.e. m(r) = m0 + m1 V (r), we obtain { d2 Gnκ (r) ξe3 κ(κ + 1) + ξe1 + (ξe2 + )e−λr − 2 dr r r2 } 1 ξe5 + (ξe4 + + rξe6 )e−2λr Gnκ (r) = 0, (71) r r where
2 ξe1 = Enκ − m20 − (Enκ + m0 )Cps + m1 (1 − m1 )Ed2 +[m0 (1 − 2m1 ) − Enκ + (1 − m1 )Cps ]Ed , (72)
ξe2 = [Enκ − m0 (1 − 2m1 ) − (1 − m1 )Cps
(73)
ξe3 = [m0 (1 − 2m1 ) − Enκ + (1 − m1 )Cps
(74)
−2m1 (1 − m1 )Ed ]Ed (1 + λrb )eλrb ,
+2m1 (1 − m1 )Ed ]Ed λrb2 eλrb ,
ξe4 = 2m1 (m1 − 1)Ed2 λrb2 (1 + λrb )e2λrb ,
(75)
ξe6 = m1 (1 − m1 )Ed2 (λrb + 1)2 e2λrb ,
(77)
ξe5 = m1 (1 − m1 )Ed2 λ2 rb4 e2λrb ,
(76)
κ(κ − 1) = e l(e l + 1).
(78)
Considering the approximations given in equation (12), one can rewrite equation (71) in the form of d2 Gnκ (r) dr 2
+ ξe1 Gnκ (r) + (ξe2 +
+ 1−eλ−λr (ξe4 +
e3 λ ξ )e−λr Gnκ (r) 1−e−λr
e5 λ ξ )e−2λr Gnκ (r) 1−e−λr 2
+e ϵ6 e−2λr nκ (r)
κ(κ+1)λ − (1−e −λr )2 Gnκ (r) = 0.
(79)
We see that the differential equations (13) and (79) have the same form. Thus, we perform similar steps while
6 obtaining bound and scattering solutions from equation (79). Hence, at this point, we give those solutions directly. Bound State Solution: For the pseudospin symmetric bound states, we obtain the following energy eigenvalues equation ]2 [ √ √ e 1 − n + 1 (1 + 1 − 4(Υ e3 − Υ e2 − Υ e 1 ) − −Υ e 3 = 0, Υ 2 (80) with e 1 = 1 [κ(κ + 1)λ2 + λξe4 − ξe1 − ξe2 − ξe6 ], Υ λ2
e 2 = 1 [2(ξe1 + ξe2 + ξe6 ) − λ(ξe3 + 3ξe4 )], Υ λ2
e 3 = 1 [ξe1 + ξe2 + ξe6 + λ2 ξe5 − λ(ξe3 + 2ξe4 )], Υ λ2
(81)
e= B
π(e l + 1) ϕe = + arg[Γ(2e τ − eb− − eb+ )] 2 −arg[Γ(2e τ − eb+ )] − arg[Γ(2e τ − eb− )],
e eb+ = τe − ik + ρe, λ
(82)
e eb− = τe − ik − ρe, λ
(83)
τe =
−λr e )ν e (1 − e a+ , e a− ; 1 + 2e µ; e−λr ). (84) Gnκ (r) = A 2 F1 (e µλr ee
νe =
1 [1 ± 2
√ e 1, µ e=± Υ
√
e3 − Υ e2 − Υ e 1 )], 1 − 4(Υ
σ e=±
√
e 3. −Υ
α e2 =
α e3 =
IV.
(90)
Scattering State Solution: Similar to the scattering state solutions in spin symmetric case, here, we get the following results for the wave function, normalization constant, phase shifts and energy eigenvalues, respectively: −λr e )τ e e e (1 − e τ ; 1 − e−λr ), Gnκ (r) = B 2 F1 (b+ , b− ; 2e e−iekr (91)
1 [2λξe5 − ξe3 − ξe4 ], λ
1 e [ξ1 + ξe2 + ξe6 − (ξe3 + 2ξe4 )λ + λ2 ξe5 ], λ2
(88)
(89)
√ α e3 − α e2 − α e1 ,
α e1 = κ(κ + 1) − ξe5 ,
(86) (87)
√ 1 α1 ), (1 + 1 + 4e 2
e k=λ
with the normalization constant √ λ(e µ + νe + n)Γ(2e µ + n + 1)Γ(2e µ + 2e ν + n) e A= , n!(e ν + n)Γ(2e µ)Γ(1 + 2e µ)Γ(2e ν + n) (85) where
e a− = µ e + νe − σ e,
(92)
(93)
[ ]2 √ √ 1 α e3 − α e2 − α e1 + n + (1 + 1 + 4e α1 ) − −e α3 = 0, 2 (94) with
and eigenfunctions
e a+ = −n = µ e + νe + σ e,
1 Γ(2e τ − eb+ )Γ(2e τ − eb− ) , Γ(2e τ ) Γ(2e τ − eb− − eb+ )
ρe =
√ −f α3 .
(95)
(96)
(97)
(98) (99) (100)
(101)
(102)
NUMERICAL ANALYSIS
In the beginning steps, we are already discussed numerically the shape of the improved Frost-Musulin potential and characteristics of approximations in FIGS. 1 and 2. FIG. 1 includes three different graphics. In the upper left one, we analyze the potential according to different values of the dissociation energy. We use Ed = 7f m−1 for the black line and Ed = 9f m−1 for the dot-dashed line with λ = 0.1 and rb = 0.4f m. In the upper right one, we plot the potential according to different values of the potential well range that we use λ = 0.1 for the black line and λ = 0.5 for the dot-dashed line with Ed = 9f m−1 and rb = 0.4f m. The bottom one shows the variation of improved Frost-Musulin potential according to different values of the equilibrium bound length that it is used rb = 0.4f m for the black line and rb = 0.8f m for the
7 dot-dashed line with λ = 0.1 and Ed = 7f m−1 . In the FIG. 2, we discuss two different approximations and give corresponding two graphics. The characteristic of ap2 proximations r12 (black line) and (1−eλ−λr )2 (dashed line) given in the left graphics while the other one presents the characteristic of 1r (black line) and 1−eλ−λr (dashed line). In each case, it is seen from FIG. 2 that approximations used for 1/r and 1/r2 terms are good ones within the limit of λr ≪ 1. Now, we focus on further numerical investigations. Analyzing the energy eigenvalues equations given in equation (32) (or identical one (69)) and equation (80) (or identical one (94)) analytically are challenging due to their transcendental structures. Numerical analysis of both spin and pseudospin symmetric bound states are presented in tables I and II within the atomic units. From Tables I and II, it is seen that two states representing same l(e l) have same energy which are called as spin (pseudospin) doublet [41]. The variation of bound state energies under spin symmetry is plotted versus m0 , m1 , λ, rb , Ed and Cs , respectively, in FIG. 3. In all of the graphics given in FIG. 3, we use the black line for 1p3/2 , 1p1/2 (or n = 1), dotted line for 2p3/2 , 2p1/2 (or n = 2) and the dot-dashed line for 3p3/2 , 3p1/2 (or n = 3) spin doublets. In the upper left figure, we discuss the bound state energies numerically according to different values of m1 with the auxiliary parameters Ed = 7f m−1 , Cs = 2.25f m−1 , m0 = 2f m−1 , λ = 0.1, rb = 0.4f m, k = −2. We see that spin symmetric energy eigenvalues decrease for 0 ≤ m1 ≤ 0.487f m−1 interval while increase for m1 > 0.487f m−1 values. Moreover, it is seen that we find minimum values of energy for each quantum states at m1 = 0.487f m−1 point. In the upper right graphic, we investigate the variation of bound state energies numerically according to m0 with Ed = 7f m−1 , Cs = 2.25f m−1 , m1 = 0.4f m−1 , λ = 0.1, rb = 0.4f m, k = −2. We see here that energy eigenvalues increase for increasing m0 values. In the middle left analysis given in FIG. 3, we plot the variation of bound state energies versus λ with the auxiliary parameters Ed = 7f m−1 , Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. This analysis shows that energy eigenvalues decrease for 0 < λ ≤ 0.163 interval and increase for λ > 0.163 values. On the other hand, the middle right figure shows the variation of bound state energies versus rb with λ = 0.1, Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , Ed = 7f m−1 , k = −2. This figure implies that energy eigenvalues decrease for 0.4f m ≤ rb ≤ 0.6f m values and increase for rb > 0.6f m values. In the bottom left figure, the variation of bound state energies is discussed versus Ed with the auxiliary parameters λ = 0.1, Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. It seen from here that energy eigenvalues always decrease according to Ed values. The final graphics given in the bottom right corner shows the numerical analysis of bound state energies according to Cs with λ = 0.1, Ed = 7f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. This
figure shows that eigenvalues always increase according to the Cs values. Next, we discuss numerical analysis of the spin symmetric phase shifts according to m1 , λ and Cs for three different l states in FIG. 4. We plot the l = 1 case as a black line, the l = 2 state as a dotted line and the l = 3 case as a dot-dashed line. The upper left graphics in FIG. 4 shows the variation of spin symmetric phase shifts according to m1 with Enκ = 3f m−1 , Cs = 2.25f m−1 , λ = 0.2, m0 = 2f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 1. We have increasing phase shift values in this analysis. Next, the variation of spin symmetric phase shifts versus λ is discussed in the upper right graphic of FIG. 4 by taking Enκ = 3f m−1 , Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 1. It is seen that increasing λ values give decreasing phase shifts. Moreover, the variation of spin symmetric phase shifts versus the Cs values is presented in the bottom one with auxiliary parameters Enκ = 3f m−1 , λ = 0.2, m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 2. We see that the spin-symmetric phase shift decreases according to the Cs values. On the pseudospin symmetry side, we get different numerical results. In FIG. 5, we give the variation of bound state energies versus m0 , m1 , λ, rb , Ed and Cps , respectively. In all of the graphics given in FIG. 5, we use the black line for 1p3/2 , 0f5/2 (or n = 1), dotted line for 2p3/2 , 1f5/2 (or n = 2) and the dot-dashed line for 3p3/2 , 2f5/2 (or n = 3) pseudospin doublets. In the upper left graphic, we discuss the bound state energies numerically according to different values of m1 with the auxiliary parameters Ed = 7f m−1 , Cps = −2.25f m−1 , m0 = 2f m−1 , λ = 0.15, rb = 0.4f m, k = −2. It is seen that pseudospin symmetric energy eigenvalues decrease for 0 ≤ m1 ≤ 1.139f m−1 interval and increase for m1 > 1.139f m−1 values. Furthermore, we find minimum energy values for each quantum states at m1 = 1.139f m−1 point. In the upper right graphic, we investigate the variation of bound state energies numerically according to m0 with Ed = 7f m−1 , Cps = −2.25f m−1 , m1 = 0.4f m−1 , λ = 0.15, rb = 0.4f m, k = −2. We get increasing energy eigenvalues for increasing values of m0 . In the analysis given in the middle left side of FIG. 5, we plot the variation of bound state energies versus λ with the auxiliary parameters Ed = 7f m−1 , Cps = −2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. This analysis shows that we have decreasing energy eigenvalues for increasing λ values. Next, the middle right part of FIG. 5 displays the variation of bound state energies versus rb with λ = 0.15, Cps = −2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , Ed = 7f m−1 , k = −2. We find decreasing energy eigenvalues for rb > 0.4f m values. In the bottom left figure, the variation of bound state energies is discussed versus Ed with the auxiliary parameters λ = 0.15, Cps = −2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. It seen from here that pseudospin symmetric energy eigenvalues always decrease according to Ed values. The final graphic given in
8 the bottom right corner of FIG. 5 shows the numerical analysis of bound state energies according to Cps with λ = 0.15, Ed = 7f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. We see that the energy eigenvalues decrease according to the Cps values. Finally, we discussed numerical analysis of the pseudospin symmetric phase shifts according to m1 , λ and Cps for three different e l states in FIG. 6. The e l = 1, e e l = 2 and l = 3 cases are described with the black line, dotted line and the dot-dashed line, respectively. The upper left graphics in FIG. 6 gives the variation of pseudospin symmetric phase shifts according to m1 with Enκ = 3f m−1 , Cps = −2.25f m−1 , λ = 0.2, m0 = 2f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 2. We see from here that pseudospin symmetric phase shift values decrease according to m1 values. Next, the variation of pseudospin symmetric phase shifts versus λ is plotted in the upper right graphic of FIG. 6 by writing Enκ = 3f m−1 , Cps = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 1 in the corresponding relation. Thence, we get decreasing phase shift values for λ dependency. Moreover, the variation of pseudospin symmetric phase shifts versus the Cps values is investigated in the bottom graphic of FIG. 6 with the auxiliary parameters Enκ = 3f m−1 , λ = 0.2, m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 2. This graphic yields that the phase shift values decrease according to the Cps values. V.
pseudospin symmetry limits using functional analysis method. Spin (pseudospin) symmetric bound and scattering states solutions of the Dirac equation for the improved Frost-Musulin potential have been obtained by considering a novel approximations for 1/r and 1/r2 terms. Energy eigenvalues equation under spin (pseudospin) symmetry has been obtained in closed form as an transcendental equation. Spin (pseudospin) symmetric wave functions have been correspondingly calculated in terms of the Gauss hypergeometric functions. And then, the wave functions have∫ been normalized by using 2 the normalization condition |Ψ(r)| dr = 1. We have also investigated the phase shifts of Dirac particles by considering the spin (pseudospin) symmetric solution in the framework of spatially dependent mass. Numerical analysis of the bound states, scattering states and phase shifts of the Dirac particles interacting with the improved Frost-Musulin potential are given in tables and figures within the position-dependent mass in the presence of spin and pseudospin limits. Our results are the first ones to handle with spin (pseudospin) symmetric solution of spin- 21 particles exposed to the improved Frost-Musulin potential in the presence of position-dependent mass. These results can be also reduced into the usual constant mass case by choosing m1 = 0. Moreover, it is expected that results obtained in this study will give some helpful comprehension in studies to atomic and molecular structures.
FINAL REMARKS Acknowledgement
In the present study, we have investigated the scattering and bound states of Dirac particles interacting with the improved Frost-Musulin potential within the position-dependent mass in the presence of spin and
[1] L. Canderle and A. Plastino, SOP Trans. Theor. Phys. 1 (2014) 99. [2] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Editions de Physique, France, 1988. [3] F.A. de Saavedra, J. Boronat, A. Polls, A. Fabrocini, Phys. Rev. B 50 (1994) 4248. [4] M. Barranco, M. Pi, S.M. Gatica, E.S. Hernandez, J. Navarro, Phys. Rev. B 56 (1997) 8997. [5] P. Ring, P. Schuck, The Nuclear Many Body Problem (Springer-Verlag, New York, 1980), p. 211 [6] L. Dekar, L. Chetouani, T.F. Hammann, J. Math. Phys. 39 (1998) 2551. [7] J. Thomsen, G. T. Einevoll, P. C. Hemmer, Phys. Rev. B 39 (1989) 12783. [8] J. M. Levy-Leblond, Phys.Rev. A 52 (1995) 1845. [9] O. Mustafa, S. H. Mazharimousavi, Int. J. Theor. Phys. 46 (2007) 1786. [10] V. M. Tkachuk, O. Voznyak, Eur. Phys. J Plus 130 (2015) 161.
This study was supported by the Research Fund of Mersin University in Turkey with the project number: 2015-AP4-1244.
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
A.D. Alhaidari, Phys. Rev. A 75 (2007) 042707. L. Serra, E. Lipparini, Europhys. Lett. 40 (1997) 667. I.O. Vakarchuk, J. Phy. A Math. Gen. 38 (2005) 4727. A. de Souza Dutra, C.-S. Jia, Phys. Lett. A 352 (2006) 484. A.D. Alhaidari, Phys. Rev. A 75 (2007) 042707. C.-S. Jia, A. de Souza Dutra, Ann. Phys. 323 (2008) 566. S.M. Ikhadair, R. Sever, Appl. Math. Comput. 216 (2010) 911. O. Panella, S. Biondini, A. Arda, J. Phys. A Math. Theor. 43 (2010) 325302. Y. Chargui, Few-Body Syst. 57 (2016) 289. G. P¨ oschl, E. Teller, Z. Physik 83 (1933) 143. L. Hulthn, Ark. Mat. Astron. Fys. A 28 (1942) 5. R.D. Woods, D.S. Saxon, Phys. Rev. 95 (1954) 577. R. Rydberg, Z. Physik 73 (1931) 376. N. Rosen, P. Morse, Phys. Rev. 42 (1932) 210. Z.H. Deng, Y.P. Fan, Shandong Univ. J. 7 (1957) 162. D. Schiberg, Mol. Phys. 59 (1986) 1123.
9 [27] T. Tietz, J. Chem. Phys. 38 (1963) 3036. [28] H. Wei, Phys. Rev. A 42 (1990) 2524. [29] A.A. Frost, B. Musulin, J. Am. Chem. Soc. 76 (1954) 2045. [30] Y. P. Varshni, Rev. Mod. Phys. 29 (1957) 664. [31] X-T. Hu, J-Y. Liu, C-S. Jia, Comput. Theor. Chem. 1019 (2013) 137. [32] C-S. Jia, G-C. Liang, X-L. Peng, H-M. Tang, L-H. Zhang, Few. Body Syst. 55 (2014) 1159. [33] G-D. Zhang, J-Y. Liu, L-H. Zhang, W. Zhou, C-S. Jia, Phys. Rev. A 86 (2012) 062510. [34] C.S. Jia, T. Chen, S. He, Phys. Lett. A 377 (2013) 682. [35] C.S. Jia, J-W. Dai, L-H. Zhang, J-Y. Liu, X-L. Peng, Phys. Lett. A 379 (2015) 137. [36] C.S. Jia, L-H. Zhang, C-W. Wang, Chem. Phys. Lett. 667 (2017) 211.
VHrL
[37] C-S Jia,Y-F. Diao, X-S. Liu, P-Q. Wang, J-Y. Liu, G-D. Zhang, J. Chem. Phys. 137 (2012) 014101. [38] C.M. Brown, B. Musulin, Trans. Illinois Academy of Science 63 (1970) 251. [39] A.G. Adepoju, E.J. Eweh, Can. J. Phys. 92 (2014) 18. [40] C.A. Onate, M.C. Onyeaju, A.N. Ikot, Ann. Phys. 375 (2016) 239. [41] J. N. Ginocchio, Phys. Rep. 414 (2005) 165. [42] R.L. Greene, C. Aldrich, Phys. Rev. A 14 (1976) 2363. [43] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1965. [44] Mathematica 9, Wolfram Research Inc., 2012. [45] L.D. Landau, E. M. Lifshitz, Quantum Mechanics, NonRelativistic Theory, 3rd ed.; Pergamon: New York, 1977.
VHrL
r
r
VHrL
r FIG. 1: The variation of improved Frost-Musulin potential versus r according to dissociation energy, potential well range and equilibrium bound length, respectively.
1.5
1.2 1.0 0.8 0.6 0.4 0.2 0.0
1.0 0.5 0.0 0
2
4
6
8
0
r FIG. 2: The characteristic of approximations graphical analysis of
1 r
(black line) and
λ 1−e−λr
2
4
6
8
r 1 r2
(black line) and
λ2 (1−e−λr )2
(dashed line) given in the left figure while the
(dashed line) are presented in the right figure.
10 TABLE I: Bound state energy eigenvalues for the improved Frost-Musulin potential in the spin symmetry limit with m0 = 2f m−1 , m1 = 0.3f m−1 , Cs = 2.25f m−1 , Ed = 7f m−1 , λ = 0.1, rb = 0.4, ¯ h = 1 and c = 1. l 1 1 1 1 2 2 2 2 3 3 3 3
n κ<0 0 -2 1 -2 2 -2 3 -2 0 -3 1 -3 2 -3 3 -3 0 -4 1 -4 2 -4 3 -4
(l, j) 0p3/2 1p3/2 2p3/2 3p3/2 0d5/2 1d5/2 2d5/2 3d5/2 0f7/2 1f7/2 2f7/2 3f7/2
(s)
Enκ 1.31209 1.30294 1.28540 1.26050 1.31157 1.29996 1.27860 1.24854 1.31122 1.29796 1.27359 1.23905
n κ>0 0 1 1 1 2 1 3 1 0 2 1 2 2 2 3 2 0 3 1 3 2 3 3 3
(s)
(l, j) Enκ (f m)−1 0p1/2 1.31209 1p1/2 1.30294 2p1/2 1.28540 3p1/2 1.26050 0p3/2 1.31157 1d3/2 1.29996 2d3/2 1.27860 3d3/2 1.24854 0f5/2 1.31122 1f5/2 1.29796 2f5/2 1.27359 3f5/2 1.23905
TABLE II: Bound state energy eigenvalues for the improved Frost-Musulin potential in the pseudospin symmetry limit with m0 = 2f m−1 , m1 = 0.3f m−1 , Cps = −2.25f m−1 , Ed = 7f m−1 , λ = 0.1, rb = 0.4f m, ¯ h = 1 and c = 1. ˜ l 1 1 1 2 2 2 3 3 3
n κ<0 1 -1 2 -1 3 -1 1 -2 2 -2 3 -2 1 -3 2 -3 3 -3
(l, j) 1s1/2 2s1/2 3s1/2 1p3/2 2p3/2 3p3/2 1d5/2 2d5/2 3d5/2
(ps)
(ps)
Enκ n − 1 κ > 0 (l + 2, j + 1) Enκ (f m)−1 -0.451229 0 2 0d3/2 -0.451229 -0.459879 1 2 1d3/2 -0.459879 -0.471497 2 2 2d3/2 -0.471497 -0.452508 0 3 0f5/2 -0.452508 -0.462930 1 3 1f5/2 -0.462930 -0.476732 2 3 2f5/2 -0.476732 -0.453336 0 4 0g7/2 -0.453336 -0.465128 1 4 1g7/2 -0.465128 -0.480776 2 4 2g7/2 -0.480776
11
EnHsL,Κ
1.40
1.7
1.35
1.6 EnHsL,Κ
1.30 1.25
1.5 1.4 1.3
1.20 0.0
0.2
0.4
0.6
1.2 2.0
0.8
2.1
m1
EnHsL,Κ
2.2
1.4 1.3 1.2 1.1 1.0 0.9
2.4
0.7
0.8
1.30 1.25 EnHsL,Κ
1.20 1.15 1.10
0.10
0.15
0.20
0.25
1.05 0.4
0.30
0.5
0.6 rb
Λ
EnHsL,Κ
2.3
m0
1.30 1.25 1.20 1.15 1.10 1.05 1.00
EnHsL,Κ
7
8
9
10
11
12
1.25 1.24 1.23 1.22 1.21 1.20 1.19 1.18 0.0
0.5
1.0
Ed
1.5
2.0
2.5
Cs
FIG. 3: The variation of bound state energies under spin symmetric case.
6.0 5.9 5.8 5.7 Φ 5.6 5.5 5.4 5.3 5.2 0.0
0.2
0.4
0.6
0.8
1.0
m1
6.4 6.2 6.0 Φ 5.8 5.6 5.4 5.2 5.0 0.0
0.1
0.2
0.3
0.4
Λ
6.0 5.9 5.8 5.7 Φ 5.6 5.5 5.4 5.3 5.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Cs FIG. 4: The variation of phase shifts versus m1 , λ and Cs for three different l states under spin symmetry.
12
HpsL
En ,Κ
-0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4
HpsL
En ,Κ
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 2.0
2.1
2.2
m1
HpsL
En ,Κ
-0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 0.10
HpsL
En ,Κ
0.15
0.20
0.25
0.30
-0.6 -0.7 -0.8 -0.9 -1.0 -1.1 -1.2 -1.3 -1.4 0.4
0.5
0.6
2.4
0.7
0.8
rb
Λ
HpsL En ,Κ
2.3
m0
-0.7
-0.8
-0.8
-1.0
-0.9
HpsL En ,Κ -1.2 -1.4
-1.0 -1.1
-1.6 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4
-1.2 7
8
9
10
11
12
Cps
Ed
FIG. 5: The variation of phase shifts versus m1 , λ and Cps for three different e l states under pseudospin symmetry.
6.0 5.9 ~ 5.8 Φ 5.7 5.6 5.5 5.4 5.3 5.2 0.0
6.5 ~
6.0
Φ 5.5 5.0 0.2
0.4
0.6
0.8
1.0
m1
4.5 0.0
0.1
0.2
0.3
0.4
Λ
6.0 5.9 ~ 5.8 Φ 5.7 5.6 5.5 5.4 5.3 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Cps FIG. 6: The variation of phase shifts versus m1 , λ and Cps for three different e l states under pseudospin symmetry.
Highlights 1. Spin-1/2 particles with PDM are investigated under pseudospin and spin symmetry. 2. A proper approximation is used to deal with centrifugal term. 3. Bound and scattering state solutions are obtained for the Frost-Musulin potential. 4. Phase shifts are also found in closed form. 5. Results are presented numerically and graphically.
S0003-4916(17)30058-1 http://dx.doi.org/10.1016/j.aop.2017.02.010 YAPHY 67337
To appear in:
Annals of Physics
Received date: 20 January 2017 Accepted date: 17 February 2017 Please cite this article as: A. Tas, O. Aydogdu, M. Salti, Dirac particles interacting with the improved Frost-Musulin potential within the effective mass formalism, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.02.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Dirac particles interacting with the improved Frost-Musulin potential within the effective mass formalism Ahmet Tas∗ , Oktay Aydogdu† , Mustafa Salti‡ Department of Physics, Faculty of Arts and Sciences, Mersin University, Mersin-33343, Turkey. We mainly investigate the dynamics of spin- 12 particles with position-dependent mass for the improved Frost-Musulin potential under spin-pseudospin symmetry. First, we find an approximate analytical solution of the Dirac equation both for bound and scattering states under spin-pseudospin symmetry and then we see that the normalized solutions are given in terms of the Gauss hypergeometric functions. In further steps, we analyze our results numerically. PACS numbers: 03.65.Nk, 03.65.Ge, 03.65.Pm, 03.65.Db, 34.20.Gj Keywords: Dirac equation, Position-dependent mass, Bound states, Scattering phase shifts.
I.
INTRODUCTION
Investigating solutions of the non-relativistic or relativistic wave equations for the real physical potentials is one of the important issues in quantum mechanics. On this purpose, the variable mass formalism gives meaningful and useful theoretical estimations of many experimental features for quantum many-body systems [1]. In literature, the effective mass concept has been used for many significant topics such as quantum liquids, 3 He clusters, nuclei, electronic properties of semiconductors, and metallic clusters [2–6]. Also, in the non-relativistic Schr¨odinger equation, coordinate dependent mass and momentum operator do not commute each other which causes ordering ambiguity that’s why it is hard to construct a true kinetic energy operator within the position-dependent mass framework [7–10]. However, this ordering ambiguity problem emerged in the non-relativistic Schr¨odinger equation is automatically eliminated in the relativistic Dirac and Klein-Gordon equations. Besides, it is useful to consider the position-dependent mass Dirac equation when quantum mechanical systems that include heavy atoms are dealt with [11]. Moreover, studying exact solutions of the Dirac equation with a position-dependent mass has become an important research area, because such investigations may yield a testing ground for some approximation schemes and meaningful conclusions to understand some physical phenomena [2, 12]. Therefore, the exact solvability of relativistic (and also non-relativistic) quantum mechanical problems within the framework of positiondependent mass has always been very attractive [13–19]. On the other hand, finding a universal potential energy function is another important puzzle in this area and various potential energy functions have been proposed in order to discuss diatomic molecules and other physical systems. The P¨oschl-Teller [20], Hulth´en [21], Woods-
∗ [email protected]
† [email protected] ‡ [email protected]
(corresponding author)
Saxon [22], Rydberg [23], Rosen-Morse [24], DengFan [25], Schi¨ oberg [26], Tietz [27], Wei [28] and FrostMusulin [29] type potentials are some of the well-known models (see Ref. [30] and references within for a good review). Furthermore, some improved potentials have been recently introduced to study dynamics and other physical properties of diatomic molecules [31–35]; e.g. thermodynamic properties of the real diatomic lithium dimer molecules have been investigated in Ref. [36] within the improved Manning-Rosen potential. The above works motivated us to investigate the approximate solutions of 3-dimensional Dirac equation with spatially dependent mass for the improved FrostMusulin potential offered by Jia and his coworkers [37] to discuss diatomic molecules. In literature, Brown and Musulin [38] solved the Born-Oppenheimer nuclear equation in the presence of Frost-Musulin potential energy function asymptotically. Next, using the improved FrostMusulin potential, Adepoju and Eweh [39] obtained approximate bound state solution of the Schr¨ odinger equation. In another recent paper, Onate et al. [40] have worked on pseudospin and spin symmetric solutions of the Dirac equation in the presence of the Hellmann-FrostMusulin type potential energy function. We see that, in literature, the bound and scattering states solutions of relativistic spin- 21 particles interacting with external improved Frost-Musulin potential have not been studied so far within the spatially dependent mass framework. That’s why, we focus on this problem in order to get analytic solutions for the Dirac particles in view of position dependent mass formalism with the improved FrostMusulin potential. Our work is structured in five sections. The first one includes the scope and purpose of our investigation. The second one introduces the spin symmetric solutions of bound and scattering states for the relativistic spin- 21 particles. The pseudospin symmetric solutions of bound and scattering states are investigated in the third section. We devote the fourth section to the numerical analysis of our theoretical results. The last section gives final remarks.
2 II. SPIN SYMMETRIC SOLUTIONS OF BOUND AND SCATTERING STATES
d2 κ(κ + 1) − − [m + Enκ − ∆(r)]Σ(r) dr2 r2 −(m − Enκ )[m + Enκ − ∆(r)]}Fnκ (r) = 0
(1)
with ¯h = c = 1. For the spin symmetry (see Ref. [41] for detailed information), ∆(r) = Cs where Cs is a constant and Σ(r) = V (r) where V (r) denotes the improved FrostMusulin potential given as [37] λr2 V (r) = Ed {1 − (1 + λrb )e−λ(r−rb ) + b e−λ(r−rb ) }, (2) r where Ed , λ and rb are three positive parameters and they represent the dissociation energy, range of potential well and the equilibrium bound length, respectively. The variation of improved Frost-Musulin potential versus r is plotted for different values of the dissociation energy, range of potential well and the equilibrium bound length in FIG. 1. Additionally, the position-dependent definition of mass is written generally as given below m(r) = m0 + m1 V (r).
(10)
κ(κ + 1) = l(l + 1).
(11)
with
Dynamics of relativistic spin- 12 particles is discussed by using the Dirac equation. Due to the potential and mass have radial dependence, we should focus on the following spherical form of Dirac equation {
ξ6 = −m1 (1 + m1 )Ed2 (λrb + 1)2 e2λrb ,
(3)
In order to solve the Dirac equation analytically and reduce our results to the usual constant mass case, we need to assume such definition of mass. Hence, making use of relations (2) and (3) transforms equation (1) into the following form { d2 Fnκ (r) ξ3 κ(κ + 1) + ξ1 + (ξ2 + )e−λr − dr2 r r2 } 1 ξ5 + (ξ4 + + rξ6 )e−2λr Fnκ (r) = 0, (4) r r
In the new form of radial Dirac equation written in (4), we have terms including 1r and r12 which make this equation imposible to solve analytically. From this point of view, we can consider the following approximations in order to get a second order solvable differential equation [42]:
ξ2 = [m0 (1 + 2m1 ) + Enκ − (1 + m1 )Cs +2m1 (1 + m1 )Ed ]Ed (1 + λrb )eλrb ,
d2 Fnκ (r) ξ3 λ −λr + ξ1 Fnκ (r) + (ξ2 + 1−e Fnκ (r) −λr )e dr 2 ξ λ + 1−eλ−λr (ξ4 + 1−e5−λr )e−2λr Fnκ (r) + ξ6 e−2λr Fnκ (r) 2
κ(κ+1)λ − (1−e −λr )2 Fnκ (r) = 0.
2
Fnκ x(1 − x) d dx + λ1 (ξ3 + ϵ4 x + 2
+(1 − x) dFdxnκ +
x(1 − x)
−
(8)
ξ5 = −m1 (1 + m1 )Ed2 λ2 rb4 e2λrb ,
(9)
−
κ(κ+1)λ x(1−x) )Fnκ
+ ξ2 + ξ6 x)Fnκ = 0. (14)
d2 Fnκ dFnκ + (1 − x) dx2 dx
1 (Υ1 + Υ2 x − Υ3 x2 )Fnκ = 0, x(1 − x)
(15)
where 1 [κ(κ + 1)λ2 + λξ4 − ξ1 − ξ2 − ξ6 ], λ2
(16)
1 [2(ξ1 + ξ2 + ξ6 ) − λ(ξ3 + 3ξ4 )], λ2
(17)
1 [ξ1 + ξ2 + ξ6 + λ2 ξ5 − λ(ξ3 + 2ξ4 )]. λ2
(18)
Υ2 =
ξ4 = 2m1 (1 + m1 )Ed2 λrb2 (1 + λrb )e2λrb ,
(1−x) ξ1 λ2 ( x
xλξ5 1−x
Due to we have λr ≪ 1, some of the terms including orders of (x−1)3 and (x−1)4 can be eliminated. Thence, after e few mathematical steps, we obtain that
(6)
(7)
(13)
Bound State Solution: To find the possible bound state solution, here we introduce a new variable such as x = e−λr . Thence, equation (13) can be transformed into the following form
Υ1 =
ξ3 = −[m0 (1 + 2m1 ) + Enκ − (1 + m1 )Cs +2m1 (1 + m1 )Ed ]Ed λrb2 eλrb ,
(12)
It is important to mention here that the above approximations are valid for λr ≪ 1. The characteristic of ap2 proximations 1r , 1−eλ−λr , r12 and (1−eλ−λr )2 are analyzed numerically in FIG. 2. Then, one can find the following second order differential equation
where 2 ξ1 = Enκ − m20 − (Enκ − m0 )Cs − [m0 (1 + 2m1 ) +Enκ − (1 + m1 )Cs ]Ed − m1 (1 + m1 )Ed2 , (5)
1 λ2 ≈ . 2 r (1 − e−λr )2
λ 1 ≈ , r 1 − e−λr
Υ3 =
We see that equation (15) has two singular points at x = 0 and x = 1. It is known that singular points of
3 a differential equation determine the form of solutions. Thus, one can rearrange the function Fnκ as
should find again Fnκ → 0, thence, a quantum condition is obtained as written below
Fnκ (x) = xµ (1 − x)ν fnκ (x).
a+ = µ + ν + σ = −n.
(19)
Under the boundary conditions, we get the following energy eigenvalues equation and eigenfunctions from equations (31) and (30), respectively;
Therefore, we get d2 fnκ dfnκ + [2µ + 1 − (2µ + 2ν + 1)x] dx2 dx 1 + [µ2 − Υ1 − (2µ2 + 2µν + ν + Υ2 )x x(1 − x)
x(1 − x)
+(µ2 + 2µν + ν 2 + Υ3 )x2 ]fnκ = 0. (20)
Now, at this step, with the following definitions √ µ = ± Υ1 , √ 1 ν = [1 ± 1 − 4(Υ3 − Υ2 − Υ1 )], 2 σ=± equation (20) becomes
√
−Υ3 ,
[ ]2 √ √ 1 Υ1 − n + (1 + 1 − 4(Υ3 − Υ2 − Υ1 ) − −Υ3 = 0, 2 (32) Fnκ (r) = A
(1 − e−λr )ν −λr ). 2 F1 (a+ , a− ; 1 + 2µ; e eµλr
(21) (22)
(23)
d2 fnκ dfnκ x(1 − x) + [2µ + 1 − (2µ + 2ν + 1)x] dx2 dx −(µ + ν + σ)(µ + ν − σ)fnκ = 0. (24) This is one of the well-known differential equation types, and the corresponding solution is given in terms of the Gauss hypergeometric functions [43] as given below fnκ (x) = A 2 F1 (a+ , a− ; 1 + 2µ; x) +Bx−2µ 2 F1 (b+ , b− ; 1 − 2µ; x),
(25)
a+ = µ + ν + σ,
(26)
a− = µ + ν − σ,
(27)
b+ = −µ + ν + σ,
(28)
b− = −µ + ν − σ.
(29)
where
Thus, the general solution will be Fnκ (x) = Axµ (1 − x)ν 2 F1 (a+ , a− ; 1 + 2µ; x) +Bx−µ (1 − x)ν 2 F1 (b+ , b− ; 1 − 2µ; x).
(31)
(30)
We have two boundary conditions for this solution. First, when r → ∞ (on the other hand x = e−λr → 0) we must have Fnκ → 0 in order to obtain a normalizable solution at center of the potential. Thence, it is seen that we have B = 0. Second, when r → 0 (or x = e−λr → 1) we
(33)
Moreover, the normalization constant can be calculated by using the normalization condition given as ∫ ∞ ∫ 1 1 |Fnκ (x)|2 dx = 1. (34) |Fnκ (r)|2 dr = λ 0 x 0 Here, making use of Fnκ (x) gives |A|2 I = 1, λ
(35)
where I=
∫
1 0
[2 F1 (−n, n + 2(µ + ν); 1 + 2µ; x)]2 dx. x1−2µ (1 − x)−2v
(36)
For different n values, the integral I has the following results[44] Γ(2µ)Γ(1 + 2ν) Γ(1 + 2µ + 2ν) (1 + ν)(1 + 2µ + 2ν) Γ(2 + 2µ)Γ(1 + 2ν) µ(1 + 2µ)2 Γ(3 + 2µ + 2ν) (2 + ν)(1 + 3µ + 2µ2 )−2 Γ(3 + 2µ)Γ(2 + 2ν) 4µ(2 + µ + ν) Γ(2 + 2µ + 2ν) 3(3 + ν) Γ(4 + 2µ)Γ(3 + 2ν) (3 + 11µ + 12µ2 + 4µ3 )2 Γ(3 + 2µ + 2ν)
for n = 0, for n = 1, for n = 2, for n = 3, ...
m!(ν + m) Γ(2µ)Γ(2µ + 1)Γ(2ν + m) for n = m. µ + ν + m Γ(2µ + m + 1)Γ(2µ + 2ν + m) (37) Then, we obtain √ λ(µ + ν + n)Γ(2µ + n + 1)Γ(2µ + 2ν + n) A= . n!(ν + n)Γ(2µ)Γ(1 + 2µ)Γ(2ν + n) (38) Scattering State Solution: On the contrary of bound state cases, the scattered particles are labeled with a wave function which mostly behaves like the one for plane waves at large distances. Starting from equation
4 (13) one can obtain the scattering state solutions. After defining y = 1 − e−λr , in equation (13), we find y(1 − y)
d2 Fnκ dFnκ λ−2 −y + [ξ2 (1 − y)y 2 2 dy dx y(1 − y)
+ξ1 y 2 + ξ3 λy(1 − y) + ξ4 λy(1 − y)2 + ξ5 λ2 (1 − y)2 +ξ6 y 2 (1 − y)2 − κ(κ + 1)λ2 ]Fnκ = 0.
√ where ρ = −α3 . Similar to the bound state case, solution of the above differential equation is written in terms of the Gauss hypergeometric functions[43]: gnκ (y) = C 2 F1 (c+ , c− ; 2τ ; y) +Dy 1−2τ 2 F1 (d+ , d− ; 2 − 2τ ; y), where
(39)
Now, ignoring y and higher order terms because of that λr ≪ 1, we obtain 3
dFnκ d2 Fnκ −y y(1 − y) dy 2 dy
+
1 (−α1 − α2 y + Υ3 y 2 )Fnκ = 0, y(1 − y)
(40)
where α1 = κ(κ + 1) − ξ5 , α2 =
1 [2λξ5 − ξ3 − ξ4 ], λ
1 α3 = 2 [ξ1 + ξ2 + ξ6 − (ξ3 + 2ξ4 )λ + λ2 ξ5 ]. λ
(41)
(42)
(43)
ik
(44)
√ 1 (1 ∓ 1 + 4α1 ) 2
(45)
√ k = λ α3 − α2 − α1 ,
(46)
τ=
Fnκ (y) =
and
ik + ρ, λ
(50)
c− = τ −
ik − ρ, λ
(51)
d+ = −τ −
ik + ρ, λ
(52)
d− = −τ −
ik − ρ. λ
(53)
Cy τ ik
(1 − y) λ
Dy 1−τ
ik
(1 − y) λ
2 F1 (c+ , c− ; 2τ ; y)
2 F1 (d+ , d− ; 2
− 2τ ; y).
(54)
Now, we focus on boundary conditions to specify the general solution given above. First, when r → 0 (or y → 0) it should be Fnκ (r) = 0, then we have D = 0 and the solution transforms into the form of Fnκ (y) =
with
c+ = τ −
Hence, for the general solution, we have
+
Next, assuming Fnκ (y) = y τ (1 − y)− λ gnκ (y)
(49)
Cy τ ik
(1 − y) λ
2 F1 (c+ , c− ; 2τ ; y).
(55)
On the other hand, Landau and Lifshitz [45] showed that a wave function written for scattering states should have the following form when r → ∞ (or y → 1): lim Fnκ (r) ≈ 2 sin(kr −
r→∞
lπ + ϕ), 2
(56)
where ϕ represents the phase shifts, then we have
yields d2 gnκ ik dgnκ + [2τ − (2τ − + 1)y] dy 2 λ dy 1 2ikτ 2 2 [τ − τ − α1 + (τ − 2τ − α2 + )y + y(1 − y) λ k2 2ikτ −( 2 + − τ 2 − α3 )y 2 ]gnκ = 0. (47) λ λ y(1 − y)
After performing a few simple mathematical steps, we can rewrite the above equation in a new form: y(1 − y)
d2 gnκ ik dgnκ + [2τ − (2τ − + 1)y] dy 2 λ dy ik ik + ρ][τ − − ρ]gnκ = 0. −[τ − λ λ
(48)
Fnκ (r) = C
(1 − e−λr )τ −λr ). (57) 2 F1 (c+ , c− ; 2τ ; 1 − e e−ikr
Therefore, it is obtained lim Fnκ (r) ≈ Ceikr 2 F1 (c+ , c− ; 2τ ; 1).
r→∞
(58)
Using recurrence relation Γ(2τ )Γ(2τ − c− − c+ ) Γ(2τ − c+ )Γ(2τ − c− ) ×2 F1 (c+ , c− ; c+ + c− + 1 − 2τ ; 1 − z) 2τ −c− −c+ Γ(2τ )Γ(c+ + c− − 2τ ) + (1 − z) Γ(c+ )Γ(c− ) ×2 F1 (2τ − c+ , 2τ − c− ; 2τ − c+ − c− + 1; 1 − z), (59) 2 F1 (c+ , c− ; 2τ ; z)
=
5 and feature 2 F1 (Λ1 , Λ2 ; Λ3 ; 0)
=1
(60)
of the Gauss hypergeometric functions [43], we find Γ(2τ )Γ(2τ − c− − c+ ) 2 F1 (c+ , c− , 2τ ; 1) = Γ(2τ − c+ )Γ(2τ − c− ) Γ(2τ )Γ(η1 + c− − 2τ ) −2ikr + e . Γ(c+ )Γ(c− )
The spherical Dirac equation under pseudospin symmetry (see Ref. [41] for detailed information), is given as (61)
Next, taking into account c+ + c− − 2τ = (2τ − c− − c+ )∗ , c+ = (2τ − c− )∗ and c− = (2τ − c+ )∗ gives Γ(2τ )Γ(2τ − c− − c+ ) 2 F1 (c+ , c− , 2τ ; 1) = Γ(2τ − c+ )Γ(2τ − c− ) [ ]∗ Γ(2τ − c− − c+ ) + Γ(2τ )e−2ikr . Γ(2τ − c+ )Γ(2τ − c− )
(62)
It is known that[43]
Γ(2τ − c− − c+ ) iφ Γ(2τ − c− − c+ ) e . = Γ(2τ − c+ )Γ(c+ − c− ) Γ(2τ − c+ )Γ(2τ − c− ) (63) Using this relation, we obtain Γ(2τ − c− − c+ ) F (c , c , 2τ ; 1) = 2Γ(2τ ) 2 1 + − Γ(2τ − c+ )Γ(2τ − c− ) π ×e−ikr sin(kr + + φ).(64) 2 Thence, the solution (58) at large distance, i.e. r → ∞, turns into the following form π lim Fnκ (r) → 2CΓ(2τ ) sin(kr + + φ) r→∞ 2 Γ(2τ − c− − c+ ) . × (65) Γ(2τ − c+ )Γ(2τ − c− ) After comparing equations (56) and (65), we find 1 Γ(2τ − c+ )Γ(2τ − c− ) C= , Γ(2τ ) Γ(2τ − c− − c+ )
π(l + 1) +φ 2 π(l + 1) = + arg[Γ(2τ − c− − c+ )] 2 −arg[Γ(2τ − c+ )] − arg[Γ(2τ − c− )].
(66)
ϕ =
(67)
Using the singular pole points of scattering amplitude C (2τ − c+ = −n or 2τ − c− = −n), we can calculate the corresponding energy eigenvalues to check the accuracy of our result. Then, using we get
2τ − c+ = −n
III. PSEUDOSPIN SYMMETRIC SOLUTIONS OF BOUND AND SCATTERING STATES
(68)
[ ]2 √ √ 1 α3 − α2 − α1 + n + (1 + 1 + 4α1 ) − −α3 = 0. 2 (69) It is easy to see that this result can be rewritten in another form which is exactly same as given in equation (32).
d2 κ(κ − 1) − − [Enκ − m − Σ(r)]∆(r) dr2 r2 +(Enκ + m)[Enκ − m − Σ(r)]}Gnκ (r) = 0 {
(70)
with h ¯ = c = 1. Under the pseudospin symmetry, Σ(r) is assumed to be equal to a constant Cps and we also have ∆(r) = V (r) where V (r) is the improved Frost-Musulin potential. After considering the position-dependent mass definition, i.e. m(r) = m0 + m1 V (r), we obtain { d2 Gnκ (r) ξe3 κ(κ + 1) + ξe1 + (ξe2 + )e−λr − 2 dr r r2 } 1 ξe5 + (ξe4 + + rξe6 )e−2λr Gnκ (r) = 0, (71) r r where
2 ξe1 = Enκ − m20 − (Enκ + m0 )Cps + m1 (1 − m1 )Ed2 +[m0 (1 − 2m1 ) − Enκ + (1 − m1 )Cps ]Ed , (72)
ξe2 = [Enκ − m0 (1 − 2m1 ) − (1 − m1 )Cps
(73)
ξe3 = [m0 (1 − 2m1 ) − Enκ + (1 − m1 )Cps
(74)
−2m1 (1 − m1 )Ed ]Ed (1 + λrb )eλrb ,
+2m1 (1 − m1 )Ed ]Ed λrb2 eλrb ,
ξe4 = 2m1 (m1 − 1)Ed2 λrb2 (1 + λrb )e2λrb ,
(75)
ξe6 = m1 (1 − m1 )Ed2 (λrb + 1)2 e2λrb ,
(77)
ξe5 = m1 (1 − m1 )Ed2 λ2 rb4 e2λrb ,
(76)
κ(κ − 1) = e l(e l + 1).
(78)
Considering the approximations given in equation (12), one can rewrite equation (71) in the form of d2 Gnκ (r) dr 2
+ ξe1 Gnκ (r) + (ξe2 +
+ 1−eλ−λr (ξe4 +
e3 λ ξ )e−λr Gnκ (r) 1−e−λr
e5 λ ξ )e−2λr Gnκ (r) 1−e−λr 2
+e ϵ6 e−2λr nκ (r)
κ(κ+1)λ − (1−e −λr )2 Gnκ (r) = 0.
(79)
We see that the differential equations (13) and (79) have the same form. Thus, we perform similar steps while
6 obtaining bound and scattering solutions from equation (79). Hence, at this point, we give those solutions directly. Bound State Solution: For the pseudospin symmetric bound states, we obtain the following energy eigenvalues equation ]2 [ √ √ e 1 − n + 1 (1 + 1 − 4(Υ e3 − Υ e2 − Υ e 1 ) − −Υ e 3 = 0, Υ 2 (80) with e 1 = 1 [κ(κ + 1)λ2 + λξe4 − ξe1 − ξe2 − ξe6 ], Υ λ2
e 2 = 1 [2(ξe1 + ξe2 + ξe6 ) − λ(ξe3 + 3ξe4 )], Υ λ2
e 3 = 1 [ξe1 + ξe2 + ξe6 + λ2 ξe5 − λ(ξe3 + 2ξe4 )], Υ λ2
(81)
e= B
π(e l + 1) ϕe = + arg[Γ(2e τ − eb− − eb+ )] 2 −arg[Γ(2e τ − eb+ )] − arg[Γ(2e τ − eb− )],
e eb+ = τe − ik + ρe, λ
(82)
e eb− = τe − ik − ρe, λ
(83)
τe =
−λr e )ν e (1 − e a+ , e a− ; 1 + 2e µ; e−λr ). (84) Gnκ (r) = A 2 F1 (e µλr ee
νe =
1 [1 ± 2
√ e 1, µ e=± Υ
√
e3 − Υ e2 − Υ e 1 )], 1 − 4(Υ
σ e=±
√
e 3. −Υ
α e2 =
α e3 =
IV.
(90)
Scattering State Solution: Similar to the scattering state solutions in spin symmetric case, here, we get the following results for the wave function, normalization constant, phase shifts and energy eigenvalues, respectively: −λr e )τ e e e (1 − e τ ; 1 − e−λr ), Gnκ (r) = B 2 F1 (b+ , b− ; 2e e−iekr (91)
1 [2λξe5 − ξe3 − ξe4 ], λ
1 e [ξ1 + ξe2 + ξe6 − (ξe3 + 2ξe4 )λ + λ2 ξe5 ], λ2
(88)
(89)
√ α e3 − α e2 − α e1 ,
α e1 = κ(κ + 1) − ξe5 ,
(86) (87)
√ 1 α1 ), (1 + 1 + 4e 2
e k=λ
with the normalization constant √ λ(e µ + νe + n)Γ(2e µ + n + 1)Γ(2e µ + 2e ν + n) e A= , n!(e ν + n)Γ(2e µ)Γ(1 + 2e µ)Γ(2e ν + n) (85) where
e a− = µ e + νe − σ e,
(92)
(93)
[ ]2 √ √ 1 α e3 − α e2 − α e1 + n + (1 + 1 + 4e α1 ) − −e α3 = 0, 2 (94) with
and eigenfunctions
e a+ = −n = µ e + νe + σ e,
1 Γ(2e τ − eb+ )Γ(2e τ − eb− ) , Γ(2e τ ) Γ(2e τ − eb− − eb+ )
ρe =
√ −f α3 .
(95)
(96)
(97)
(98) (99) (100)
(101)
(102)
NUMERICAL ANALYSIS
In the beginning steps, we are already discussed numerically the shape of the improved Frost-Musulin potential and characteristics of approximations in FIGS. 1 and 2. FIG. 1 includes three different graphics. In the upper left one, we analyze the potential according to different values of the dissociation energy. We use Ed = 7f m−1 for the black line and Ed = 9f m−1 for the dot-dashed line with λ = 0.1 and rb = 0.4f m. In the upper right one, we plot the potential according to different values of the potential well range that we use λ = 0.1 for the black line and λ = 0.5 for the dot-dashed line with Ed = 9f m−1 and rb = 0.4f m. The bottom one shows the variation of improved Frost-Musulin potential according to different values of the equilibrium bound length that it is used rb = 0.4f m for the black line and rb = 0.8f m for the
7 dot-dashed line with λ = 0.1 and Ed = 7f m−1 . In the FIG. 2, we discuss two different approximations and give corresponding two graphics. The characteristic of ap2 proximations r12 (black line) and (1−eλ−λr )2 (dashed line) given in the left graphics while the other one presents the characteristic of 1r (black line) and 1−eλ−λr (dashed line). In each case, it is seen from FIG. 2 that approximations used for 1/r and 1/r2 terms are good ones within the limit of λr ≪ 1. Now, we focus on further numerical investigations. Analyzing the energy eigenvalues equations given in equation (32) (or identical one (69)) and equation (80) (or identical one (94)) analytically are challenging due to their transcendental structures. Numerical analysis of both spin and pseudospin symmetric bound states are presented in tables I and II within the atomic units. From Tables I and II, it is seen that two states representing same l(e l) have same energy which are called as spin (pseudospin) doublet [41]. The variation of bound state energies under spin symmetry is plotted versus m0 , m1 , λ, rb , Ed and Cs , respectively, in FIG. 3. In all of the graphics given in FIG. 3, we use the black line for 1p3/2 , 1p1/2 (or n = 1), dotted line for 2p3/2 , 2p1/2 (or n = 2) and the dot-dashed line for 3p3/2 , 3p1/2 (or n = 3) spin doublets. In the upper left figure, we discuss the bound state energies numerically according to different values of m1 with the auxiliary parameters Ed = 7f m−1 , Cs = 2.25f m−1 , m0 = 2f m−1 , λ = 0.1, rb = 0.4f m, k = −2. We see that spin symmetric energy eigenvalues decrease for 0 ≤ m1 ≤ 0.487f m−1 interval while increase for m1 > 0.487f m−1 values. Moreover, it is seen that we find minimum values of energy for each quantum states at m1 = 0.487f m−1 point. In the upper right graphic, we investigate the variation of bound state energies numerically according to m0 with Ed = 7f m−1 , Cs = 2.25f m−1 , m1 = 0.4f m−1 , λ = 0.1, rb = 0.4f m, k = −2. We see here that energy eigenvalues increase for increasing m0 values. In the middle left analysis given in FIG. 3, we plot the variation of bound state energies versus λ with the auxiliary parameters Ed = 7f m−1 , Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. This analysis shows that energy eigenvalues decrease for 0 < λ ≤ 0.163 interval and increase for λ > 0.163 values. On the other hand, the middle right figure shows the variation of bound state energies versus rb with λ = 0.1, Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , Ed = 7f m−1 , k = −2. This figure implies that energy eigenvalues decrease for 0.4f m ≤ rb ≤ 0.6f m values and increase for rb > 0.6f m values. In the bottom left figure, the variation of bound state energies is discussed versus Ed with the auxiliary parameters λ = 0.1, Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. It seen from here that energy eigenvalues always decrease according to Ed values. The final graphics given in the bottom right corner shows the numerical analysis of bound state energies according to Cs with λ = 0.1, Ed = 7f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. This
figure shows that eigenvalues always increase according to the Cs values. Next, we discuss numerical analysis of the spin symmetric phase shifts according to m1 , λ and Cs for three different l states in FIG. 4. We plot the l = 1 case as a black line, the l = 2 state as a dotted line and the l = 3 case as a dot-dashed line. The upper left graphics in FIG. 4 shows the variation of spin symmetric phase shifts according to m1 with Enκ = 3f m−1 , Cs = 2.25f m−1 , λ = 0.2, m0 = 2f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 1. We have increasing phase shift values in this analysis. Next, the variation of spin symmetric phase shifts versus λ is discussed in the upper right graphic of FIG. 4 by taking Enκ = 3f m−1 , Cs = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 1. It is seen that increasing λ values give decreasing phase shifts. Moreover, the variation of spin symmetric phase shifts versus the Cs values is presented in the bottom one with auxiliary parameters Enκ = 3f m−1 , λ = 0.2, m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 2. We see that the spin-symmetric phase shift decreases according to the Cs values. On the pseudospin symmetry side, we get different numerical results. In FIG. 5, we give the variation of bound state energies versus m0 , m1 , λ, rb , Ed and Cps , respectively. In all of the graphics given in FIG. 5, we use the black line for 1p3/2 , 0f5/2 (or n = 1), dotted line for 2p3/2 , 1f5/2 (or n = 2) and the dot-dashed line for 3p3/2 , 2f5/2 (or n = 3) pseudospin doublets. In the upper left graphic, we discuss the bound state energies numerically according to different values of m1 with the auxiliary parameters Ed = 7f m−1 , Cps = −2.25f m−1 , m0 = 2f m−1 , λ = 0.15, rb = 0.4f m, k = −2. It is seen that pseudospin symmetric energy eigenvalues decrease for 0 ≤ m1 ≤ 1.139f m−1 interval and increase for m1 > 1.139f m−1 values. Furthermore, we find minimum energy values for each quantum states at m1 = 1.139f m−1 point. In the upper right graphic, we investigate the variation of bound state energies numerically according to m0 with Ed = 7f m−1 , Cps = −2.25f m−1 , m1 = 0.4f m−1 , λ = 0.15, rb = 0.4f m, k = −2. We get increasing energy eigenvalues for increasing values of m0 . In the analysis given in the middle left side of FIG. 5, we plot the variation of bound state energies versus λ with the auxiliary parameters Ed = 7f m−1 , Cps = −2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. This analysis shows that we have decreasing energy eigenvalues for increasing λ values. Next, the middle right part of FIG. 5 displays the variation of bound state energies versus rb with λ = 0.15, Cps = −2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , Ed = 7f m−1 , k = −2. We find decreasing energy eigenvalues for rb > 0.4f m values. In the bottom left figure, the variation of bound state energies is discussed versus Ed with the auxiliary parameters λ = 0.15, Cps = −2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. It seen from here that pseudospin symmetric energy eigenvalues always decrease according to Ed values. The final graphic given in
8 the bottom right corner of FIG. 5 shows the numerical analysis of bound state energies according to Cps with λ = 0.15, Ed = 7f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, k = −2. We see that the energy eigenvalues decrease according to the Cps values. Finally, we discussed numerical analysis of the pseudospin symmetric phase shifts according to m1 , λ and Cps for three different e l states in FIG. 6. The e l = 1, e e l = 2 and l = 3 cases are described with the black line, dotted line and the dot-dashed line, respectively. The upper left graphics in FIG. 6 gives the variation of pseudospin symmetric phase shifts according to m1 with Enκ = 3f m−1 , Cps = −2.25f m−1 , λ = 0.2, m0 = 2f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 2. We see from here that pseudospin symmetric phase shift values decrease according to m1 values. Next, the variation of pseudospin symmetric phase shifts versus λ is plotted in the upper right graphic of FIG. 6 by writing Enκ = 3f m−1 , Cps = 2.25f m−1 , m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 1 in the corresponding relation. Thence, we get decreasing phase shift values for λ dependency. Moreover, the variation of pseudospin symmetric phase shifts versus the Cps values is investigated in the bottom graphic of FIG. 6 with the auxiliary parameters Enκ = 3f m−1 , λ = 0.2, m0 = 2f m−1 , m1 = 0.4f m−1 , rb = 0.4f m, Ed = 7f m−1 and k = 2. This graphic yields that the phase shift values decrease according to the Cps values. V.
pseudospin symmetry limits using functional analysis method. Spin (pseudospin) symmetric bound and scattering states solutions of the Dirac equation for the improved Frost-Musulin potential have been obtained by considering a novel approximations for 1/r and 1/r2 terms. Energy eigenvalues equation under spin (pseudospin) symmetry has been obtained in closed form as an transcendental equation. Spin (pseudospin) symmetric wave functions have been correspondingly calculated in terms of the Gauss hypergeometric functions. And then, the wave functions have∫ been normalized by using 2 the normalization condition |Ψ(r)| dr = 1. We have also investigated the phase shifts of Dirac particles by considering the spin (pseudospin) symmetric solution in the framework of spatially dependent mass. Numerical analysis of the bound states, scattering states and phase shifts of the Dirac particles interacting with the improved Frost-Musulin potential are given in tables and figures within the position-dependent mass in the presence of spin and pseudospin limits. Our results are the first ones to handle with spin (pseudospin) symmetric solution of spin- 21 particles exposed to the improved Frost-Musulin potential in the presence of position-dependent mass. These results can be also reduced into the usual constant mass case by choosing m1 = 0. Moreover, it is expected that results obtained in this study will give some helpful comprehension in studies to atomic and molecular structures.
FINAL REMARKS Acknowledgement
In the present study, we have investigated the scattering and bound states of Dirac particles interacting with the improved Frost-Musulin potential within the position-dependent mass in the presence of spin and
[1] L. Canderle and A. Plastino, SOP Trans. Theor. Phys. 1 (2014) 99. [2] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Editions de Physique, France, 1988. [3] F.A. de Saavedra, J. Boronat, A. Polls, A. Fabrocini, Phys. Rev. B 50 (1994) 4248. [4] M. Barranco, M. Pi, S.M. Gatica, E.S. Hernandez, J. Navarro, Phys. Rev. B 56 (1997) 8997. [5] P. Ring, P. Schuck, The Nuclear Many Body Problem (Springer-Verlag, New York, 1980), p. 211 [6] L. Dekar, L. Chetouani, T.F. Hammann, J. Math. Phys. 39 (1998) 2551. [7] J. Thomsen, G. T. Einevoll, P. C. Hemmer, Phys. Rev. B 39 (1989) 12783. [8] J. M. Levy-Leblond, Phys.Rev. A 52 (1995) 1845. [9] O. Mustafa, S. H. Mazharimousavi, Int. J. Theor. Phys. 46 (2007) 1786. [10] V. M. Tkachuk, O. Voznyak, Eur. Phys. J Plus 130 (2015) 161.
This study was supported by the Research Fund of Mersin University in Turkey with the project number: 2015-AP4-1244.
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
A.D. Alhaidari, Phys. Rev. A 75 (2007) 042707. L. Serra, E. Lipparini, Europhys. Lett. 40 (1997) 667. I.O. Vakarchuk, J. Phy. A Math. Gen. 38 (2005) 4727. A. de Souza Dutra, C.-S. Jia, Phys. Lett. A 352 (2006) 484. A.D. Alhaidari, Phys. Rev. A 75 (2007) 042707. C.-S. Jia, A. de Souza Dutra, Ann. Phys. 323 (2008) 566. S.M. Ikhadair, R. Sever, Appl. Math. Comput. 216 (2010) 911. O. Panella, S. Biondini, A. Arda, J. Phys. A Math. Theor. 43 (2010) 325302. Y. Chargui, Few-Body Syst. 57 (2016) 289. G. P¨ oschl, E. Teller, Z. Physik 83 (1933) 143. L. Hulthn, Ark. Mat. Astron. Fys. A 28 (1942) 5. R.D. Woods, D.S. Saxon, Phys. Rev. 95 (1954) 577. R. Rydberg, Z. Physik 73 (1931) 376. N. Rosen, P. Morse, Phys. Rev. 42 (1932) 210. Z.H. Deng, Y.P. Fan, Shandong Univ. J. 7 (1957) 162. D. Schiberg, Mol. Phys. 59 (1986) 1123.
9 [27] T. Tietz, J. Chem. Phys. 38 (1963) 3036. [28] H. Wei, Phys. Rev. A 42 (1990) 2524. [29] A.A. Frost, B. Musulin, J. Am. Chem. Soc. 76 (1954) 2045. [30] Y. P. Varshni, Rev. Mod. Phys. 29 (1957) 664. [31] X-T. Hu, J-Y. Liu, C-S. Jia, Comput. Theor. Chem. 1019 (2013) 137. [32] C-S. Jia, G-C. Liang, X-L. Peng, H-M. Tang, L-H. Zhang, Few. Body Syst. 55 (2014) 1159. [33] G-D. Zhang, J-Y. Liu, L-H. Zhang, W. Zhou, C-S. Jia, Phys. Rev. A 86 (2012) 062510. [34] C.S. Jia, T. Chen, S. He, Phys. Lett. A 377 (2013) 682. [35] C.S. Jia, J-W. Dai, L-H. Zhang, J-Y. Liu, X-L. Peng, Phys. Lett. A 379 (2015) 137. [36] C.S. Jia, L-H. Zhang, C-W. Wang, Chem. Phys. Lett. 667 (2017) 211.
VHrL
[37] C-S Jia,Y-F. Diao, X-S. Liu, P-Q. Wang, J-Y. Liu, G-D. Zhang, J. Chem. Phys. 137 (2012) 014101. [38] C.M. Brown, B. Musulin, Trans. Illinois Academy of Science 63 (1970) 251. [39] A.G. Adepoju, E.J. Eweh, Can. J. Phys. 92 (2014) 18. [40] C.A. Onate, M.C. Onyeaju, A.N. Ikot, Ann. Phys. 375 (2016) 239. [41] J. N. Ginocchio, Phys. Rep. 414 (2005) 165. [42] R.L. Greene, C. Aldrich, Phys. Rev. A 14 (1976) 2363. [43] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1965. [44] Mathematica 9, Wolfram Research Inc., 2012. [45] L.D. Landau, E. M. Lifshitz, Quantum Mechanics, NonRelativistic Theory, 3rd ed.; Pergamon: New York, 1977.
VHrL
r
r
VHrL
r FIG. 1: The variation of improved Frost-Musulin potential versus r according to dissociation energy, potential well range and equilibrium bound length, respectively.
1.5
1.2 1.0 0.8 0.6 0.4 0.2 0.0
1.0 0.5 0.0 0
2
4
6
8
0
r FIG. 2: The characteristic of approximations graphical analysis of
1 r
(black line) and
λ 1−e−λr
2
4
6
8
r 1 r2
(black line) and
λ2 (1−e−λr )2
(dashed line) given in the left figure while the
(dashed line) are presented in the right figure.
10 TABLE I: Bound state energy eigenvalues for the improved Frost-Musulin potential in the spin symmetry limit with m0 = 2f m−1 , m1 = 0.3f m−1 , Cs = 2.25f m−1 , Ed = 7f m−1 , λ = 0.1, rb = 0.4, ¯ h = 1 and c = 1. l 1 1 1 1 2 2 2 2 3 3 3 3
n κ<0 0 -2 1 -2 2 -2 3 -2 0 -3 1 -3 2 -3 3 -3 0 -4 1 -4 2 -4 3 -4
(l, j) 0p3/2 1p3/2 2p3/2 3p3/2 0d5/2 1d5/2 2d5/2 3d5/2 0f7/2 1f7/2 2f7/2 3f7/2
(s)
Enκ 1.31209 1.30294 1.28540 1.26050 1.31157 1.29996 1.27860 1.24854 1.31122 1.29796 1.27359 1.23905
n κ>0 0 1 1 1 2 1 3 1 0 2 1 2 2 2 3 2 0 3 1 3 2 3 3 3
(s)
(l, j) Enκ (f m)−1 0p1/2 1.31209 1p1/2 1.30294 2p1/2 1.28540 3p1/2 1.26050 0p3/2 1.31157 1d3/2 1.29996 2d3/2 1.27860 3d3/2 1.24854 0f5/2 1.31122 1f5/2 1.29796 2f5/2 1.27359 3f5/2 1.23905
TABLE II: Bound state energy eigenvalues for the improved Frost-Musulin potential in the pseudospin symmetry limit with m0 = 2f m−1 , m1 = 0.3f m−1 , Cps = −2.25f m−1 , Ed = 7f m−1 , λ = 0.1, rb = 0.4f m, ¯ h = 1 and c = 1. ˜ l 1 1 1 2 2 2 3 3 3
n κ<0 1 -1 2 -1 3 -1 1 -2 2 -2 3 -2 1 -3 2 -3 3 -3
(l, j) 1s1/2 2s1/2 3s1/2 1p3/2 2p3/2 3p3/2 1d5/2 2d5/2 3d5/2
(ps)
(ps)
Enκ n − 1 κ > 0 (l + 2, j + 1) Enκ (f m)−1 -0.451229 0 2 0d3/2 -0.451229 -0.459879 1 2 1d3/2 -0.459879 -0.471497 2 2 2d3/2 -0.471497 -0.452508 0 3 0f5/2 -0.452508 -0.462930 1 3 1f5/2 -0.462930 -0.476732 2 3 2f5/2 -0.476732 -0.453336 0 4 0g7/2 -0.453336 -0.465128 1 4 1g7/2 -0.465128 -0.480776 2 4 2g7/2 -0.480776
11
EnHsL,Κ
1.40
1.7
1.35
1.6 EnHsL,Κ
1.30 1.25
1.5 1.4 1.3
1.20 0.0
0.2
0.4
0.6
1.2 2.0
0.8
2.1
m1
EnHsL,Κ
2.2
1.4 1.3 1.2 1.1 1.0 0.9
2.4
0.7
0.8
1.30 1.25 EnHsL,Κ
1.20 1.15 1.10
0.10
0.15
0.20
0.25
1.05 0.4
0.30
0.5
0.6 rb
Λ
EnHsL,Κ
2.3
m0
1.30 1.25 1.20 1.15 1.10 1.05 1.00
EnHsL,Κ
7
8
9
10
11
12
1.25 1.24 1.23 1.22 1.21 1.20 1.19 1.18 0.0
0.5
1.0
Ed
1.5
2.0
2.5
Cs
FIG. 3: The variation of bound state energies under spin symmetric case.
6.0 5.9 5.8 5.7 Φ 5.6 5.5 5.4 5.3 5.2 0.0
0.2
0.4
0.6
0.8
1.0
m1
6.4 6.2 6.0 Φ 5.8 5.6 5.4 5.2 5.0 0.0
0.1
0.2
0.3
0.4
Λ
6.0 5.9 5.8 5.7 Φ 5.6 5.5 5.4 5.3 5.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Cs FIG. 4: The variation of phase shifts versus m1 , λ and Cs for three different l states under spin symmetry.
12
HpsL
En ,Κ
-0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4
HpsL
En ,Κ
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 2.0
2.1
2.2
m1
HpsL
En ,Κ
-0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 0.10
HpsL
En ,Κ
0.15
0.20
0.25
0.30
-0.6 -0.7 -0.8 -0.9 -1.0 -1.1 -1.2 -1.3 -1.4 0.4
0.5
0.6
2.4
0.7
0.8
rb
Λ
HpsL En ,Κ
2.3
m0
-0.7
-0.8
-0.8
-1.0
-0.9
HpsL En ,Κ -1.2 -1.4
-1.0 -1.1
-1.6 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4
-1.2 7
8
9
10
11
12
Cps
Ed
FIG. 5: The variation of phase shifts versus m1 , λ and Cps for three different e l states under pseudospin symmetry.
6.0 5.9 ~ 5.8 Φ 5.7 5.6 5.5 5.4 5.3 5.2 0.0
6.5 ~
6.0
Φ 5.5 5.0 0.2
0.4
0.6
0.8
1.0
m1
4.5 0.0
0.1
0.2
0.3
0.4
Λ
6.0 5.9 ~ 5.8 Φ 5.7 5.6 5.5 5.4 5.3 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Cps FIG. 6: The variation of phase shifts versus m1 , λ and Cps for three different e l states under pseudospin symmetry.
Highlights 1. Spin-1/2 particles with PDM are investigated under pseudospin and spin symmetry. 2. A proper approximation is used to deal with centrifugal term. 3. Bound and scattering state solutions are obtained for the Frost-Musulin potential. 4. Phase shifts are also found in closed form. 5. Results are presented numerically and graphically.