ARTICLE
Relativistic spin-1 particles with position-dependent mass under the
Coulomb interaction: Exact analytical solutions of the DKP equation
M.K. Bahar and F. YasukAbstract: The Duffin–Kemmer–Petiau equation with position-dependent mass for relativistic spin-1 particles under equal vector
and scalar Coulomb interaction is studied analytically. The energy eigenvalues and corresponding eigenfunctions are obtained using the asymptotic iteration method.
PACS Nos.: 03.65.–w, 03.65.Ge, 03.65.Pm.
Résumé : Nous étudions analytiquement l'équation de Duffin–Kemmer–Petiau avec une masse dépendant de la position pour des
particules relativistes de spin 1 sous égale interaction de Coulomb, vecteur et scalaire. Nous obtenons les valeurs propres et les fonctions propres correspondantes a` l'aide de la méthode d'itération asymptotique. [Traduit par la Rédaction]
1. Introduction
The problem of a particle with arbitrary spin has a long history. The first-order relativistic wave equations for arbitrary spin were first investigated by Lubánski, Madhavarao, and Bhabha and have come to be known as Bhabha wave equations [1,2]. The simplest special cases of these equations reduce to the Dirac and Duffin– Kemmer–Petiau (DKP) equations for spin-1/2 and spins-0 and -1, respectively. Following the success of the Dirac equation in rela-tivistically describing particles of spin-1/2, the search began for similar first-order wave equations for spin-0 and spin-1 particles. Apart from spin-1/2, none of the other spins obeys a single relativ-istic wave equation. For instance, it was generally believed that for spins-0 and -1, the second-order Klein–Gordon [3] and Proca [4] equations were unique. But the Dirac-like first-order relativistic DKP equation was found about ten years later [5–8]. The first-order DKP equation is be defined for both spin-0 and spin-1 particles and has been used to analyze relativistic interactions of spin-0 and spin-1 hadrons. The general way to understand the interaction between relativistic quantum mechanical particles is to obtain the solutions of their wave equations. In the recent years, the solutions of the relativistic DKP equation have been investigated with different potentials such as Coulomb, harmonic oscillator, sextic oscillator, deformed Hulthen, Aharonov–Bohm, Woods– Saxon, and step potential [9–24]. However, the problems with position-dependent mass have been studied more efficiently in the recent literature. Systems with position-dependent mass have been found very useful in studying the physical and electronic properties of semiconductors, quantum wells and quantum dots, quantum liquids,3He clusters, graded alloys, and semiconductor heterostructures [25–27]. The relativistic extension of this prob-lem is also of interest and remains unexplored. The nonrelativis-tic Schrödinger, relativisnonrelativis-tic Klein–Gordon, and Dirac equations with position-dependent mass in any dimension for different po-tentials and suitable mass distrubution functions have been stud-ied using various methods [28–40]. Also, another relativistic wave equation, namely the DKP equation, with position-dependent mass for spin-0 and spin-1 particles is investigated in the recent literature [41,42]. In this study, our aim is to obtain solutions to
the DKP equation with position-dependent mass for spin-1 parti-cles under the equal vector and scalar Coulomb interaction by using the asymptotic iteration method (AIM). This method has been proposed by Ciftci et al. to solve second-order differential equations in recent years [43–45]. This paper is arranged as fol-lows. In Sect. 2, the DKP formalism is given briefly. Then, the method used in our calculations is presented in Sect. 3. After that, the solutions of the DKP equation with position-dependent mass in the Coulomb field are obtained in Sect. 4. Finally, conclusions and discussion are given in Sect. 5.
2. DKP formalism
In this section, the DKP formalism is summarized. Generally, the first-order relativistic DKP equation for a free spin-1 particle with position-dependent mass m(r) is
[
iបc⭸⫺ m(r)c2]
DKP(r)⫽ 0 (1) where( = 0, 1, 2, 3) matrices satisfy the commutation relation ⫹ ⫽ g⫹ g (2) and the metric tensor is g= diag(1, –1, –1, –1), which defines the so-called DKP algebra [47]. The algebra generated by the 4Nhas three irreducible representations: a ten-dimensional one that is related to spin-1 particles.In the spin-1 representation,are 10 × 10 matrices defined as (i = 1, 2, 3) 0⫽
共
0 0 0 0 0T 0 I I 0T I 0 0 0T 0 0 0兲
i⫽共
0 0 ei 0 0T 0 0 ⫺isi ei T 0 0 0 0T ⫺isi 0 0兲
(3)where the matrices siare the usual 3 × 3 spin-1 matrices, given by
Received 18 June 2012. Accepted 8 November 2012.
M.K. Bahar. Deparment of Physics, Erciyes University, 38039, Kayseri, Turkey; Department of Physics, Karamanoglu Mehmetbey University, 70100, Karaman,
Turkey.
F. Yasuk. Deparment of Physics, Erciyes University, 38039, Kayseri, Turkey. Corresponding author: F. Yasuk (e-mail:yasuk@erciyes.edu.tr).
Can. J. Phys. Downloaded from www.nrcresearchpress.com by Depository Services Program on 11/13/14
s1⫽
共
0 0 0 0 0 ⫺i 0 1 0兲
s2⫽共
0 ⫺i 0 i 0 0 0 0 0兲
s3⫽共
0 ⫺i 0 i 0 0 0 0 0兲
(4)and, 0 and I denote, respectively, the zero matrix and the unity matrix. The matrices 0 and eiare given as
0⫽(0 0 0) e1⫽(1 0 0) e2⫽(0 1 0) e3⫽(0 0 1) (5) The dynamical stateDKP(r) is a ten-component spinor for spin-1 particles. We need to use the investigations of earlier works [14,15] for the solutions of the DKP equation for a particle in a central field. It is convenient to recall some general properties of the solution of the DKP equation in a central interaction for spin-1 particles. The central interaction consists of two parts: a Lorentz scalar Usand a time-like vector Uv
0
potential. The DKP equation with position-dependent mass in this case is given by [6]
[
·kc ⫹ m(r)c2⫹ Us⫹ 0
Uv 0
]
(r)⫽ 0E(r) (6) where c is light velocity and k = –iħ momentum operator.In the spin-1 representation, the ten-component DKP spinor
(r)⫽
共
i A(r) B(r) C(r)兲
with A⫽共
A1 A2 A3兲
B⫽共
B1 B2 B3兲
C⫽共
C1 C2 C3兲
(7)so that for stationary states the DKP equations can be written as
k⫽ ⫺iបⵜ (8)
[
m(r)c2⫹ Us]i ⫹ ck·B ⫽ 0 (9)[
m(r)c2⫹ Us]A⫹ c·k∧C ⫺(
E⫺ Uv 0)
B⫽ 0 (10)[
m(r)c2⫹ Us]B⫺ ikc ⫺(
E⫺ Uv 0)
A⫽ 0 (11)[
m(r)c2⫹ Us]C⫹ ik∧A ⫽ 0 (12)The most general solution of(6)is given by [14]
JM(r)⫽ 1 r
共
inJ(r)YJM(⍀)兺
L FnJL(r)YJL1 M(⍀)兺
L GnJL(r)YJL1 M(⍀)兺
L HnJL(r)YJL1 M(⍀)兲
m m ¡ n A ¡共
A1 A2 A3兲
⫽共
F0 F1 F2兲
B ¡共
B1 B2 B3兲
⫽共
G0 G1 G2兲
C ¡共
C1 C2 C3兲
⫽共
H0 H1 H2兲
(13)where YJM(⍀) are the spherical harmonics of order J, YJL1 M
(⍀) are the normalized vector spherical harmonics and nJ(r), FnJL(r), GnJL (r), HnJL(r) are radial wave functions. Here, we use the notation
FnJJ⫽ F0 FnJJ±1⫽ F±1 (14)
and similar definitions for ones G0, G±1, H0, and G±1. Becauseare 10 × 10 matrices defined, the DKP spinorJM(r) in(13)must be ten-component. Here, to be ten-component of the spinor is more related to mathematics of the system. Each of the ten-components of the spinor is a radial wave function that represents the particle. But, considering the tensor notation, to obtain the exact wave function of the particle should be consistedJM(r) spinor.
InsertingJM(r) as given in(13)into(9)–(12)by using the proper-ties of vector spherical harmonics [45] and using(8)one gets the following set of first-order coupled relativistic differential radial equations
[
E⫺ Uv 0]
F0(r)⫽[
m(r)c 2⫹ U s]G0(r) (15) បc关
dF0(r) dr ⫺ J⫹ 1 r F0(r)兴
⫽ ⫺ 1 j[
m(r)c2⫹ Us]H1(r) (16) បc关
dF0(r) dr ⫹ J rF0(r)兴
⫽ ⫺ 1 ␣j[
m(r)c2⫹ Us]H⫺1(r) (17) 1 បc兵[
m(r)c 2⫹ U s]F0(r)⫺(
E⫺ Uv 0)
G0其
⫽ ⫺␣j关
dH⫺1(r) dr ⫺ J rH⫺1(r)兴
⫺j关
dH1(r) dr ⫹ J⫹ 1 r H1 (r)兴
(18) បc关
dH0(r) dr ⫺ J⫹ 1 r H0 (r)兴
⫽ ⫺1 j兵[
m(r)c2⫹ Us]F 1(r)⫺(
E⫺ Uv 0)
G1(r)其
(19) បc关
dH0(r) dr ⫹ J rH0 (r)兴
⫽ ⫺1 ␣j兵[
m(r)c2⫹ Us]F ⫺1(r)⫺(
E⫺ Uv 0)
G ⫺1(r)其
(20) 1 បc[
m(r)c 2⫹ U s]H0(r)⫽ ⫺ j关
dF1(r) dr ⫹ J⫹ 1 r F1(r)兴
⫺␣j关
dF⫺1(r) dr ⫺ J rF⫺1(r)兴
(21) បc关
d(r) dr ⫺ J⫹ 1 r (r)兴
⫽ ⫺1 ␣j兵[
m(r)c2⫹ Us]G 1(r)⫺(
E⫺ Uv 0)
F1(r)其
(22) បc关
d(r) dr ⫹ J r(r)兴
⫽ 1 j兵[
m(r)c2⫹ Us]G⫺1(r)⫺(
E⫺ Uv 0)
F⫺1(r)其
(23) ⫺␣j 1 បc[
m(r)c 2⫹ U s](r)⫽关
dG1(r) dr ⫹ J⫹ 1 r G1(r)兴
⫹j关
dG⫺1(r) dr ⫺ J rG⫺1 (r)兴
(24) with the definition of␣j= [(j + 1)/(2j + 1)]1/2andj= [J/(2J + 1)]1/2. Using(15)–(23)and(24), we obtain the following:
兵
d2 dr2⫺ 1 m(r)c2⫹ U s d[
m(r)c2⫹ Us] dr d dr⫺ J(J⫹ 1) r2 ⫹(
E⫺ Uv 0)
2⫺[
m(r)c2⫹ Us]2 (បc)2其
F0 (r)⫽ 0 (25)Can. J. Phys. Downloaded from www.nrcresearchpress.com by Depository Services Program on 11/13/14
F1(r)⫽ បc
(
E⫺ Uv 0)
2⫺[
m(r)c2⫹ Us]2兵
j[
m(r)c2⫹ Us] ×共
d dr ⫺ J⫹ 1 r兲
H0(r)⫹␣j(
E⫺ Uv 0)
共
d dr ⫺ J⫹ 1 r兲
(r)其
(26) G1(r)⫽ បc(
E⫺ Uv 0)
2⫺[
m(r)c2⫹ Us]2兵
␣j[
m(r)c2⫹ Us] ×共
d dr ⫺ J⫹ 1 r兲
(r)⫹j(
E⫺ Uv 0)
共
d dr⫺ J⫹ 1 r兲
H0(r)其
(27) F⫺1(r)⫽ បc(
E⫺ Uv 0)
2⫺[
m(r)c2⫹ Us]2兵
␣j[
m(r)c2⫹ Us]共
d dr⫹ J r兲
H0(r) ⫺j(
E⫺ Uv 0)
共
d dr⫹ J r兲
(r)其
(28) G⫺1(r)⫽ បc(
E⫺ Uv 0)
2⫺(
m(r)c2⫹ Us)2兵
␣j(
E⫺ Uv 0)
共
d dr⫹ J r兲
H0(r) ⫺j[
m(r)c 2⫹ U s]共
d dr ⫹ J r兲
(r)其
(29)3. The AIM
We briefly outline the AIM here; the details can be found in refs. 43–45. The AIM was proposed to solve second-order differential equations of the form
yn(x)⫽0(x)yn(x)⫹ s0(x)yn(x) (30) where0(x) ≠ 0 and s0(x),0(x) are in C∞[a, b]. The functions s0(x) and 0(x) are sufficiently differentiable. Differential equation(30)has a general solution yn(x)⫽ exp
关
⫺冕
x ␣(u)du兴
共
C2⫹ C1冕
x exp兵
冕
u [0(v)⫹ 2␣(v)]dv其
du兲
(31)if, for sufficiently large values of k, sk(x) k(x) ⫽ sk⫺1(x) k⫺1(x) ⫽␣(x) (32) where k(x)⫽k⫺1(x)⫹ sk⫺1(x)⫹0(x)k⫺1(x) sk(x)⫽ sk⫺1(x)⫹ s0(x)k⫺1(x) k⫽ 1, 2, 3, …, n (33)
The termination condition of the method, together with(32), can be also be written as follows:
␦k(x)⫽k⫺1(x)sk(x)⫺k(x)sk⫺1(x)⫽ 0 (34) For a given potential, the idea is to convert the relativistic wave equation to the form of(30). Then, s0and0are determined and sk andkparameters are calculated. The energy eigenvalues are ob-tained by the termination condition given by(34). The general solution of(30)is given by(31). The first part of(31)gives us the
polynomial solutions that are convergent and physical, whereas the second part of(31) gives us nonphysical solutions that are divergent. Although(31)is the general solution of(30), we take the coefficient of the second part (C1) as zero, to square integrable solutions. Therefore, the exact eigenfunctions can be derived from the following wave function generator:
yn(x)⫽ C2exp
共
⫺冕
x sn(u) n(u)
du
兲
(35)where n = 0, 1, 2, ... radial quantum number and k is the iteration step number and usually greater than n.
4. Solutions of the DKP equation with
position-dependent mass in Coulomb field
Firstly, let us now solve(25) to find the eigenvalues and the eigenfunctions of the radial DKP equation with position-dependent mass under the Coulomb potential. The mass function of the DKP particle, which depends on the spatial coordinate, is assumed as the following:
m(r)⫽ m0⫹Ze
r (36)
where m0is a constant (real and positive) and corresponds to the rest mass of the DKP particle. In(36), m(r) is the mass function rather than real mass. The second term in(36)corresponds to the location dependence of the effective mass. By substituting (36) into(6), an effective potential, composed of a mass function, and scalar and vector potentials, occurs in the system. Namely, the mass function creates a shift in the potential profile of the system. Therefore, localizations of the wave functions change. The men-tioned effect of the position-dependent mass is seen more clearly in semiconductor physics applications [27].
We consider the Coulomb potential defined as –(Ze/r), where e is the electron charge; Z is the Coulomb charge; and ħ = c = 1 atomic units are used for all subsequent calculations. In this study, the vector potential has been taken equal to the scalar potential (Uv
0
(r)⫽ Us(r)), and Uv 0
(r), Us(r) and m(r) are inserted into(25), we obtain the following equation:
共
d2 dr2⫺ ␣0 r2 ⫹ ␣1 r ⫺␣2兲
F0(r)⫽ 0 (37)where␣0= J(J + 1) – Z2e2,␣1= 2EnJZe, and␣2⫽ m0 2⫺ E
nJ
2 .
To make an ansatz for the solution it is useful to first investigate the limits r ¡ ∞ and r ¡ 0. To solve(37)with the AIM, the reason-able physical wave function we now propose is as follows:
Table 1. Bound state energies for the
spin-1 particle with position-dependent mass (m0= e = Z = 1). Position-dependent mass |n, l, j典 Energy |0, 0, 1典 0.850 651 |1, 0, 1典 0.934 172 |2, 1, 2典 0.978 906 |3, 2, 3典 0.989 524 |4, 3, 4典 0.993 729 |5, 3, 4典 0.995 824 |6, 5, 6典 0.997 019
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F0,(n)(r)⫽ e
⫺r兹␣2r(1/2)
(
1⫹兹1⫹4␣0)
fn(r) (38)
If we insert this wave function into(37), we have the second-order homogeneous linear differential equations in the following form:
fn(r)⫹ 1⫹
兹
1⫹ 4␣0⫺ 2r兹
␣2 r fn (r) ⫹␣1⫺(
1⫹兹
1⫹ 4␣0)
兹
␣2 r fn (r)⫽ 0 (39)which is now amenable to an AIM solution. By comparing (39) with(30), we can determine the0and s0values and by means of
(33), we may calculatenand sn. This gives:
0(r)⫽ ⫺ 1⫹
兹
1⫹ 4␣0 r ⫹ 2r兹
␣2 (40a) s0(r)⫽⫺␣1⫹(
1⫹兹
1⫹ 4␣0)
兹
␣2 r (40b) 1(r)⫽ 4␣0⫹ 3(
1⫹兹
1⫹ 4␣0)
⫺ r[
␣1⫹ 3(
1⫹兹
1⫹ 4␣0)
兹
␣2]
⫹ 4r 2␣ 2 r2 (40c) s1(r)⫽(
2⫹兹
1⫹ 4␣0⫺ 2r兹
␣2)[
␣1⫺(
1⫹兹
1⫹ 4␣0)
兹
␣2]
r2 É (40d)Combining these results with the condition given by(34)yields
s01⫺ s10⫽ 0 ⇒ ␣10⫽
(
1⫹兹
1⫹ 4␣0)
兹
␣2 (41a)s12⫺ s21⫽ 0 ⇒ ␣11⫽
(
3⫹兹
1⫹ 4␣0)
兹
␣2 (41b)s23⫺ s32⫽ 0 ⇒ ␣12⫽
(
5⫹兹
1⫹ 4␣0)
兹
␣2É (41c)
When the preceding expressions are generalized, the indirect eigenvalues turn out as
␣1n⫽
(
2n⫹ 1 ⫹兹
1⫹ 4␣0)
兹
␣2 (42)Fig. 1. Plot of the wavefunctions for n = 0, 1, 2 quantum states of the spin-1 particle with position-dependent mass (Z = 1, e = 1, m0= 1).
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By inserting ␣0 and ␣2 into (42) and comparing with ␣1, we obtain the exact bound-state eigenvalues for relativistic spin-1 particles: EnJ⫽
兵
2n⫹ 1 ⫹兹
1⫹ 4[J(J⫹ 1)⫺ Z2e2]其
兹
m0 2⫺ E nJ 2 2Ze (43)We can now construct the corresponding eigenfunctions by using the wave function generator given in(35). Using(32)and(35), the eigenfunctions are obtained as follows:
f0(r)⫽ 1 f1(r)⫽ 2
兹
␣2r⫺(
1⫹兹
1⫹ 4␣0)
f2(r)⫽(
2⫹兹
1⫹ 4␣0)(
1⫹兹
1⫹ 4␣0)
⫺ 4兹
␣2(
2⫹兹
1⫹ 4␣0)
r⫹ 4␣2r 2 É (44)Finally, the following general formula for the exact solutions fn(r) is acquired as:
fn(r)⫽(⫺1)n
(
1⫹兹
1⫹ 4␣0)
n1F1(
⫺n, 1 ⫹兹
1⫹ 4␣0, 2兹
␣2r)
(45) If the preceding equation is inserted into(38), we obtain F0,(n)(r)⫽ Ne⫺r兹␣2r(1/2)
(
1⫹兹1⫹4␣0)
(
1⫹兹
1⫹ 4␣0
)
n×1F1
(
⫺n, 1 ⫹兹
1⫹ 4␣0, 2兹
␣2r)
(46) where N is the normalization constant.By using(25)–(28)and(29)the DKP spinor is determined. The eigenvalues of the spin-1 particle with position-dependent mass inTable 1and wavefunctions that are plotted inFig. 1seem as though they could represent the same particle. In(25), if m(r) is the mass function and is selected as(36), and if we use atomic units, m(r) = m0− Us, then we eliminate the second term in(25), and its third term is [(E⫺ Uv
0 )2⫺ m0
2
]/(បc)2. In the absence of a scalar potential, if m(r) is the mass of the particle in(25)and is a constant, then the same result is obtained for(25)as in the preceding case. In other words, because the choice of position-dependent mass function shifted the scalar potential, we can conclude that the physical states of a particle with the selected position-dependent mass in the presence of equal scalar and vector potentials are the same as the physical states of a particle with constant mass in the presence of a vector potential and the absence of a scalar potential.
6. Conclusion
In this study, for the relativistic particles with position-dependent mass and spin-1, we have obtained the energy eigenvalues and eigen-functions for the DKP equation in the presence of the equal vector and scalar Coulomb potential by the AIM in detail. The advantage of the AIM is that it gives the eigenvalues directly by transforming the second-order differential equation into an equation of the form yn(x)⫽0(x)yn(x)⫹ s0(x)yn(x). The wavefunctions are easily constructed by iterating the values of s0and0. The method presented in this study is general and can be extended to find the solution of the other interactions. As a result, we have found analytically the eigenvalues and eigenfunctions by solving the relativistic DKP equation with position-dependent mass in the case of equal vector and scalar Cou-lomb potentials inTable 1andFig. 1, respectively. The corresponding eigenfunctions have been obtained in terms of confluent hypergeo-metric functions. Because the Coulomb potential is a very important
attractive potential, the results obtained can be utilized in various areas of physics from semiconductor physics to nuclear physics.
Acknowledgments
The authors are grateful to the anonymous referees for their illumi-nating criticism and suggestions. We also thank H. Çiftçi for many help-ful discussions and suggested improvements to the paper.
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Appendix A
(i) The properties of the following vector spherical harmonics are used: ⵜ⌳
(
f(r)YJJM⫹11)
⫽ i共
J 2J⫹ 1兲
1/2共
df(r) dr ⫹ J⫹ 2 r f(r)兲
YJJ1 M (A1a) ⵜ⌳(
f(r)YJJM⫺11)
⫽ i共
J 2J⫹ 1兲
1/2共
df(r) dr ⫺ J⫺ 1 r f(r)兲
YJJ1 M (A1b) ⵜ⌳(
f(r)YJJ1M)
⫽ i共
J 2J⫹ 1兲
1/2共
df(r) dr ⫺ J rf(r)兲
YJJ⫹11 M ⫹ i共
J⫹ 1 2J⫹ 1兲
1/2共
df(r) dr ⫹ J⫹ 1 r f(r)兲
YJJ⫺11 M (A1c)(ii) To obtain(25), the following transactions are made. The following equations are obtained from(15),(16), and(17), respectively:
G0(r)⫽ E⫺ Uv 0 m(r)c2⫹ U s F0(r) (A2) H1(r)⫽ ⫺ បcj m(r)c2⫹ Us
关
dF0(r) dr ⫺ J⫹ 1 r F0(r)兴
(A3) H⫺1(r)⫽ ⫺ បc␣j m(r)c2⫹ Us关
dF0(r) dr ⫹ J rF0(r)兴
(A4)We consider(18)in this revised version: 1 បc[uF0(r)⫺ vG0]⫽ ⫺␣j
共
d dr ⫺ J r兲
H⫺1(r)⫺j共
d dr⫹ J⫹ 1 r兲
H1(r) (A5) where u = m(r)c2+ U sand v⫽ E ⫺ Uv 0. After inserting(A2),(A3), and (A4)into(A5)we have
1 បc
共
u2⫺ v2 u兲
F0(r)⫽ បc(␣j) 2共
d dr ⫺ J r兲
1 u共
d dr ⫹ J r兲
F0(r) ⫺ បc(j)2共
d dr ⫹ J⫹ 1 r兲
1 u共
d dr ⫺ J⫹ 1 r兲
F0(r) (A6) 1 បc共
u2⫺ v2 u兲
F0(r)⫽ ⫺ បc[(␣j)2⫹(j)2] u2 du dr dF0(r) dr ⫹ បc(␣j) 2 u共
d dr⫺ J r兲共
d dr⫹ J r兲
F0(r) ⫹បc(j) 2 u共
d dr⫹ J⫹ 1 r兲共
d dr⫺ J⫹ 1 r兲
F0(r) (A7) 1 បc共
u2⫺ v2 u兲
F0(r)⫽ ⫺ បc u2 du dr dF0(r) dr ⫹ បc(␣j)2 u共
d2 dr2⫺ J r2⫹ J r d dr ⫺ J r d dr⫺ J2 r2兲
F0(r) ⫹បc(j) 2 u关
d2 dr2⫹ J⫹ 1 r2 ⫺ J⫹ 1 r d dr⫹ J⫹ 1 r d dr ⫺ ( J⫹ 1)2 r2兴
F0(r) (A8) 1 បc共
u2⫺ v2 u兲
F0(r)⫽ ⫺ បc u2 du dr dF0(r) dr ⫹ បc(␣j)2 u关
d2 dr2⫺ J( J⫹ 1) r2兴
× F0(r)⫹ បc(j)2 u关
d2 dr2⫺ J( J⫹ 1) r2兴
F0(r) (A9) 1 បc(u 2⫺ v2) F0(r)⫽ ⫺បc u du dr dF0(r) dr ⫹ បc[(␣j)2⫹(j)2]关
d2 dr2⫺ J( J⫹ 1) r2兴
F0(r) (A10)关
d2 dr2⫺ J( J⫹ 1) r2 ⫺ 1 u du dr d dr ⫹ v2⫺ u2 (បc)2兴
F0(r)⫽ 0 (A11) Later, from(22), it is obtained thatF1(r)⫽
关
បc共
d dr ⫺ J⫹ 1 r兲
(r)⫹ uG1(r)兴
v ⫺1 (A12) and from(19) G1(r)⫽关
បc共
d dr ⫺ J⫹ 1 r兲
H0(r)⫹ uF1(r)兴
v ⫺1 (A13)If(A13)is inserted into(A12), the following equation is obtained: F1(r)⫽ បc v2⫺ u2
共
d dr⫺ J⫹ 1 r兲
(␣jv(r)⫹juH0(r)) (A14) From(22) G1(r)⫽关
⫺បc共
d dr ⫺ J⫹ 1 r兲
(r)⫹ vF1(r)兴
u ⫺1 (A15) From(19) F1(r)⫽关
⫺បc共
d dr ⫺ J⫹ 1 r兲
H0(r)⫹ vG1(r)兴
u ⫺1 (A16)Can. J. Phys. Downloaded from www.nrcresearchpress.com by Depository Services Program on 11/13/14
If(A16)is inserted into(A15), it is obtained that G1(r)⫽ បc v2⫺ u2
共
d dr⫺ J⫹ 1 r兲
(␣ju(r)⫹jvH0(r)) (A17) Combining(A14)and(A17)yields共
F1(r) G1(r)兲
⫽ បc v2⫺ u2共
d dr ⫺ J⫹ 1 r兲共
␣jv ␣ju ju jv兲共
(r) H0(r)兲
(A18)Similarly, to obtain F−1(r) and G−1(r), if we repeat the operations to find(A18), we obtain
共
F⫺1(r) G⫺1(r)兲
⫽ បc v2⫺ u2共
d dr ⫹ J r兲共
⫺jv ⫺ju ␣ju ␣jv兲共
(r) H0(r)兲
(A19)Finally, if F1(r), F−1(r), G1(r), and G−1(r) in(A18)and(A19)are sub-stituted into(21)and(24), respectively, the following equations are obtained: ⫺j 2បc
