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Eigenspectra of a complex coupled harmonic potential in three dimensions
Computers and Mathematics with Applications 68 (2014) 2068–2079
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Eigenspectra of a complex coupled harmonic potential in three dimensions S.B. Bhardwaj a,b , Ram Mehar Singh a,∗ , S.C. Mishra c a
Department of Physics, Ch. Devi Lal University, Sirsa-125055, India
b
Department of Physics, Jai Parkash Mukand Lal Innovative Engineering and Technology Institute, Radaur (Haryana), India
c
Department of Physics, Kurukshetra University, Kurukshetra-136119, India
article
info
Article history: Received 30 July 2013 Received in revised form 17 July 2014 Accepted 8 September 2014 Available online 23 October 2014 Keywords: Ansatz Complex potential Eigenvalues and eigenfunctions
abstract Within the framework of extended complex phase space approach characterized by position and momentum coordinates, we investigate the quasi-exact solutions of the Schrödinger equation for a coupled harmonic potential and its variants in three dimensions. For this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone. The energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex. However, in case of complex coupling parameters, the imaginary component of energy eigenvalue reduces to zero if the P T -symmetric condition is satisfied. Thus a non-hermitian Hamiltonian possesses real eigenvalue if it is P T -symmetric. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the study of complex potentials has become more desirable [1–3] for the better theoretical understanding of several newly discovered phenomena in different contexts like optical model of a nucleus, delocalization transitions in condensed matter systems such as a vortex flux line depinning in type-II superconductors, population biology, Bose system of hard spheres, energy spectra of complex Toda lattice, quantum cosmology, quantum field theory and super symmetric quantum mechanics etc. [4–9]. As far as viability of the complex Hamiltonian [10,11] is concerned, it has been investigated in the context of quantum mechanics and semiclassical field theories. Complex Hamiltonians have also been studied in several other theoretical contexts—for example, the study of complex trajectories with regard to the calculation of a semiclassical coherent state propagator in the path integral method has attracted particular interest in the laser physics [11]. Besides some general studies of a complex Hamiltonian in a non-linear domain [1,12], some efforts have been made to study both classical [13,14] as well as quantum aspects [15,16] of a system. At the classical level, construction of exact invariants has been carried out using a more general transformation and the analyticity property of the Hamiltonian leads to a class of integrable systems. However, in the present study, we make use of such transformation to study the quantum aspect of a system and look for the analytic solutions of the Schrödinger equation (SE). It is to be noted that a complex Hamiltonian is no longer hermitian and ordinarily does not guarantee for real eigenvalues. However, in P T -symmetric form [3,4,17], the system is found to exhibit real eigenvalue spectrum [18]. The reality of the spectrum is a direct consequence of combined action of parity and time reversal invariance of Hamiltonian [13,14]. The parity (P ) and time reversal (T ) operators defined
∗
Corresponding author. E-mail address: [email protected] (R.M. Singh).
http://dx.doi.org/10.1016/j.camwa.2014.09.006 0898-1221/© 2014 Elsevier Ltd. All rights reserved.
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by the action of position and momentum operators are
Pˆ : (x, y, z , px , py , pz , i) → (−x, −y, −z , −px , −py , −pz , i), Tˆ : (x, y, z , px , py , pz , i) → (x, y, z , −px , −py , −pz , −i), whereas, the combined action of Pˆ Tˆ -operation is given by
Pˆ Tˆ : (x, y, z , px , py , pz , i) → (−x, −y, −z , px , py , pz , −i)
(1)
where Pˆ 2 Tˆ 2 = 1. Such studies reveal that a complex Hamiltonian can give real and bounded energy eigenvalues for certain domain of underlying parameters. Therefore, it is possible to investigate the number of new non-hermitian Hamiltonian systems satisfying the P T -symmetric condition. The type of hermiticity discussed here is different from that of given by Bender et al. [2–4]. But both approaches deal with different types of non-hermitian Hamiltonians, the non-hermiticity (P T -symmetry) arising in the Bender’s approach is due to the complexity of the potential parameters whereas in the present study, in addition to the potential parameters, the underlying phase space is also taken as complex. Thus P T -symmetry discussed here is of generalized nature, which in certain limits reduces to the conventional P T -symmetry [2,13]. There are various methods available in the literature for complexifying a given Hamiltonian system [14,15], but here we use the scheme given by Xavier and de-Aguiar [11] to transform the potential in extended complex phase space approach (ECPSA) characterized by x = x1 + ip4 ,
y = x2 + ip5 ,
px = p1 + ix4 ,
z = x3 + ip6
py = p2 + ix5 ,
pz = p3 + ix6 .
(2)
The presence of variables (x4 , x5 , x6 , p1 , p2 , p3 ) in the above transformation represents coordinate-momentum interactions of a dynamical system. Note that in this complexifying scheme, the degrees of freedom of the underlying system just become double and the variables (x1 , p1 ), (x2 , p2 ), (x3 , p3 ), (x4 , p4 ), (x5 , p5 ), (x6 , p6 ) turn to be canonical pairs. Similar transformations (2) have also been used in the study of nonlinear evolution equations in the context of amplitude-modulated nonlinear Langmuir waves in plasma [12]. In the past, considerable progress has been made on the front of complex Hamiltonian up to two-dimensional systems [14,19–21] but the same effort has not been acceded for three-dimensional coupled systems. However, some attempts have been made to obtain the solution of the SE for a coupled harmonic potential in real domain [20–23] but no step has been taken towards complex domain in higher dimensions. The study of complex coupled harmonic potential has become of considerable interest due to the peculiar nature of its eigenvalue spectrum. With this motivation and for gaining the further insight into the nature of the eigenspectra of a three-dimensional complex coupled harmonic potential, we investigate the quasi-exact solutions of the SE in an extended complex phase space. The paper is structured as follows: in the next section, we demonstrate the ansatz method to enable its use in the subsequent sections. Using the same mathematical prescriptions, the explicit expressions for the energy eigenvalues and eigenfunctions of a coupled harmonic potential and its variant are presented in Section 3. Finally, concluding remarks are discussed in Section 4. 2. Ansatz method The SE (for h¯ = m = 1) for a three-dimensional system is written as
ˆ (x, y, z , px , py , pz )ψ(x, y, z ) = E ψ(x, y, z ), H
(3)
where
ˆ (x, y, z , px , py , pz ) = − H
1 d2 2 dx2
+
d2 dy2
+
d2 dz 2
+ V (x, y, z ).
(4)
Transformation (2), implies that [20]
∂ d 1 ∂ ∂ , = −i , dx 2 ∂ x1 ∂ p4 dpx 2 ∂ p1 ∂ x4 1 ∂ ∂ d 1 ∂ ∂ d = −i , = −i , dy 2 ∂ x2 ∂ p5 dpy 2 ∂ p2 ∂ x5 d 1 ∂ ∂ d 1 ∂ ∂ = −i , = −i . dz 2 ∂ x3 ∂ p6 dpz 2 ∂ p3 ∂ x6 On expressing V (x, y, z ), ψ(x, y, z ) and E in terms of real and imaginary components, we have d
=
1 ∂
−i
(5)
V (x, y, z ) = Vr (x1 , p4 , x2 , p5 , x3 , p6 ) + iVi (x1 , p4 , x2 , p5 , x3 , p6 ),
ψ(x, y, z ) = ψr (x1 , p4 , x2 , p5 , x3 , p6 ) + iψi (x1 , p4 , x2 , p5 , x3 , p6 ) E = Er + iEi ,
(6)
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by employing Eqs. (4)–(6) in Eq. (3) and then separating real and imaginary parts, one gets 1
ψr ,x1 x1 − ψr ,p4 p4 + 2ψi,x1 p4 + ψr ,x2 x2 − ψr ,p5 p5 + 2ψi,x2 p5 + ψr ,x3 x3 − ψr ,p6 p6 + 2ψi,x3 p6 + Vr ψr − Vi ψi = Er ψr − Ei ψi , 1 ψi,x1 x1 − ψi,p4 p4 − 2ψr ,x1 p4 + ψi,x2 x2 − ψi,p5 p5 − 2ψr ,x2 p5 8 + ψi,x3 x3 − ψi,p6 p6 − 2ψr ,x3 p6 + Vr ψi + Vi ψr = Er ψi + Ei ψr
−
8
(7a)
(7b)
where subscripts r and i denote the real and imaginary parts of the corresponding quantities and other subscripts to these quantities separated by comma denote the partial derivatives of the quantity concerned. For the wavefunction ψ(x, y, z ), Cauchy–Riemann analyticity condition implies
ψr ,x1 = ψi,p4 , ψr ,x2 = ψi,p5 ,
ψr ,p4 = −ψi,x1 , ψr ,p5 = −ψi,x2 ,
ψr ,x3 = ψi,p6 ,
ψr ,p6 = −ψi,x3 .
(8)
Under the analyticity condition (8), Eqs. (7a) and (7b) reduces to
− −
1 2 1 2
ψr ,x1 x1 + ψr ,x2 x2 + ψr ,x3 x3 + Vr ψr − Vi ψi = Er ψr − Ei ψi ,
(9a)
ψi,x1 x1 + ψi,x2 x2 + ψi,x3 x3 + Vr ψi + Vi ψr = Er ψi + Ei ψr .
(9b)
Ansatz for the eigenfunction ψ(x, y, z ) is assumed as [20,22]
ψ(x, y, z ) = φ(x, y, z ) exp g (x, y, z )
(10)
where, φ = φr + iφi and g = gr + igi . Thus real and imaginary parts of the wavefunction ψ(x, y, z ) are given by
ψr = egr (φr cos gi − φi sin gi ),
(11a)
ψi = e (φi cos gi + φr sin gi ).
(11b)
gr
Again, analyticity conditions for the functions gr and gi turn out to be gr ,x1 = gi,p4 ,
gi,x1 = −gr ,p4 ,
gr ,x2 = gi,p5 , gr ,x3 = gi,p6 ,
gi,x2 = −gr ,p5 , gi,x3 = −gr ,p6 .
(12)
After employing Eqs. (11a)–(12) in Eqs. (9a) and (9b), we have gr ,x1 x1 + gr ,x2 x2 + gr ,x3 x3 + (gr ,x1 )2 + (gr ,x2 )2 + (gr ,x3 )2 − (gi,x1 )2
− (gi,x2 )2 − (gi,x3 )2 +
1
φr (φr ,x1 x1 + φr ,x2 x2 + φr ,x3 x3
(φr2 + φi2 ) + 2φr ,x1 gr ,x1 + 2φr ,x2 gr ,x2 + 2φr ,x3 gr ,x3 − 2φi,x1 gi,x1 − 2φi,x2 gi,x2 − 2φi,x3 gi,x3 ) + φi (φi,x1 x1 + φi,x2 x2 + φi,x3 x3 + 2φr ,x1 gi,x1 + 2φr ,x2 gi,x2 + 2φr ,x3 gi,x3 + 2φi,x1 gr ,x1 + 2φi,x2 gr ,x2 + 2φi,x3 gr ,x3 ) + 2(Er − Vr ) = 0,
(13a)
gi,x1 x1 + gi,x2 x2 + gi,x3 x3 + 2gr ,x1 gi,x1 + 2gr ,x2 gi,x2 + 2gr ,x3 gi,x3
+
1
φr (φi,x1 x1 + φi,x2 x2 + φi,x3 x3 + 2φr ,x1 gi,x1 + 2φr ,x2 gi,x2
(φr2 + φi2 ) + 2φr ,x3 gi,x3 + 2φi,x1 gi,x1 + 2φi,x2 gi,x2 + 2φi,x3 gi,x3 ) + φi (−φr ,x1 x1 − φr ,x2 x2
− φr ,x3 x3 + 2φi,x1 gi,x1 + 2φi,x2 gi,x2 + 2φi,x3 gi,x3 − 2φr ,x1 gi,x1 − 2φr ,x2 gi,x2 − 2φr ,x3 gi,x3 ) + 2(Ei − Vi ) = 0.
(13b)
For the given potential V (x, y, z ), by considering the appropriate polynomial form of φ(x, y, z ) and selecting functional forms of gr and gi consistent with the potential,then rationalization of the resultant expressions ((13a) and (13b)) yields the ground state and excited state solutions of the SE.
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3. Examples In this section, we demonstrate the viability of the ansatz method as discussed in Section 2 for obtaining ground state as well as excited state eigenspectra of a coupled harmonic potential and its variant. 3.1. Coupled harmonic potential Consider a three-dimensional coupled harmonic potential as [23] V (x, y, z ) = ax2 + by2 + cz 2 + dxy + eyz + fzx
(14)
where a, b, c, d, e and f are coupling parameters and taken as complex. Under the transformation (2), the real and imaginary parts of the potential (14) are given by Vr = ar (x21 − p24 ) − 2ai x1 p4 + br (x22 − p25 ) − 2bi x2 p5 + cr (x23 − p26 ) − 2ci x3 p6 + dr (x1 x2 − p4 p5 ) − di (x1 p5 + x2 p4 ) + er (x2 x3 − p5 p6 ) − ei (x2 p6 + x3 p5 ) + fr (x1 x3 − p4 p6 ) − fi (x1 p6 + x3 p4 ),
(15a)
Vi = ai (x21 − p24 ) + 2ar x1 p4 + bi (x22 − p25 ) + 2br x2 p5 + ci (x23 − p26 ) + 2cr x3 p6 + di (x1 x2 − p4 p5 ) + dr (x1 p5 + x2 p4 ) + ei (x2 x3 − p5 p6 ) + er (x2 p6 + x3 p5 ) + fi (x1 x3 − p4 p6 ) + fr (x1 p6 + x3 p4 ).
(15b)
For the underlying system, the functional forms of gr and gi in conformity with Eq. (12), are considered as gr =
1 2
1
1
2
2
α1 (x21 − p24 ) + α2 (x22 − p25 ) + α3 (x23 − p26 ) + β1 x1 p4 + β2 x2 p5
+ β3 x3 p6 + γ12 (x1 x2 − p4 p5 ) − γ21 (x1 p5 + x2 p4 ) + γ23 (x2 x3 − p5 p6 ) − γ32 (x2 p6 + x3 p5 ) + γ31 (x1 x3 − p4 p6 ) − γ13 (x1 p6 + x3 p4 ) gi =
1 2
1
1
2
2
(16a)
β1 (x21 − p24 ) − β2 (x22 − p25 ) − β3 (x23 − p26 )α1 x1 p4 + α2 x2 p5
+ α3 x3 p6 + γ21 (x1 x2 − p4 p5 ) + γ12 (x1 p5 + x2 p4 ) + γ32 (x2 x3 − p5 p6 ) + γ23 (x2 p6 + x3 p5 ) + γ13 (x1 x3 − p4 p6 ) + γ31 (x1 p6 + x3 p4 ).
(16b)
By selecting a suitable form of φ(x, y, z ) and employing Eqs. (15a)–(16b) in Eqs. (13a) and (13b), the rationalization of resultant expressions ((13a) and (13b)) yields the ground state and excited state solutions of the SE asCASE-I (Ground state solutions): For the ground state solutions, the functional form of φ(x, y, z ) is taken as a polynomial of zero degree, (say constant) i.e.
φ(x, y, z ) = 1.
(17)
By, inserting Eqs. (15a)–(17) in (13a) and (13b) and then equating the coefficients of various variables to zero, we obtain the following set of 14-non-repeating equations 1 Er = − (α1 + α2 + α3 ), 2 Ei =
1 2
(18a)
(β1 + β2 + β3 ),
(18b)
2 2 2 2 α12 − β12 + γ12 − γ21 + γ31 − γ13 = 2ar ,
(18c)
α1 β1 − γ12 γ21 − γ13 γ31 = −ai ,
(18d)
2 2 2 2 α22 − β22 + γ12 − γ21 + γ23 − γ32 = 2br ,
(18e)
α2 β2 − γ12 γ21 − γ23 γ32 = −bi ,
(18f)
α −β +γ −γ +γ −γ 2 3
2 3
2 31
2 13
2 23
2 32
= 2cr ,
α3 β3 − γ31 γ13 − γ23 γ32 = −ci ,
(18g) (18h)
(α1 + α2 )γ12 + (β1 + β2 )γ21 + γ23 γ31 − γ13 γ32 = dr ,
(18i)
−(α1 + α2 )γ21 + (β1 + β2 )γ12 − γ23 γ13 − γ31 γ32 = di ,
(18j)
(α2 + α3 )γ23 + (β2 + β3 )γ32 + γ12 γ31 − γ13 γ21 = er ,
(18k)
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−(α2 + α3 )γ32 + (β2 + β3 )γ23 − γ12 γ13 − γ31 γ21 = ei ,
(18l)
(α1 + α3 )γ31 + (β1 + β3 )γ13 + γ12 γ23 − γ32 γ21 = fr ,
(18m)
−(α1 + α3 )γ13 + (β1 + β3 )γ31 − γ12 γ32 − γ23 γ21 = fi .
(18n)
As such solutions of the various ansatz parameters α ’s, β ’s and γij ’s from the above equations seem to be very difficult. For this purpose, one can make a number of choices among the ansatz parameters but one should make some plausible choices among these potential parameters so that there is no conflict between the general solutions and the well established results (SHO). In this view, we choose
γ12 = γ21 = γ23 = γ32 = γ31 = γ13 α1 β1 = α2 β2 = α3 β3 = −γ12 γ21 = −γ23 γ32 = −γ31 γ13 . Under the above choices, Eqs. (18c)–(18h) immediately lead to
α1 = −a+ , β1 = a− , γ12 = γ21
α2 = −b+ , α3 = −c+ , β2 = b− , β3 = c− . ai , = γ23 = γ32 = γ31 = γ13 =
(19b) (19c)
3
where a± =
(19a)
±a r +
a2r + a2i /9, b± =
±b r +
b2r + b2i /9, c± =
±cr +
cr2 + ci2 /9, and ai = bi = ci .
With the above choice, Eqs. (18i)–(18n) yield the following six constraining relations
ai /3(−a+ − b+ + a− + b− ) = dr ,
(20a)
ai /3(a+ + b+ + a− + b− ) = di .
(20b)
ai /3(−b+ − c+ + b− + c− ) = er ,
(20c)
ai /3(b+ + c+ + b− + c− ) = ei .
(20d)
ai /3(−c+ − a+ + c− + a− ) = fr ,
(20e)
ai /3(c+ + a+ + c− + a− ) = fi .
(20f)
Now, substituting Eqs. (19a)–(19c) in Eqs. (18a) and (18b), the real and imaginary components of eigenvalue are written as Er(0) = (0)
Ei
=
1 2 1 2
a+ + b+ + c+ ,
(21a)
a− + b− + c− ,
(21b)
and the corresponding eigenfunction turns out to be
1 1 1 ai b+ + ib− y2 − c+ + ic− z 2 + (1 + i) xy + yz + zx . ψ (0) (x, y, z ) = N exp − a+ + ia− x2 − 2
2
2
3
(22)
P T -symmetric case: Under the P T -symmetric condition (1), the potential (14) yields ai = bi = ci = di = ei = fi = 0, and the ansatz parameters γ21 = γ32 = γ13 = β1 = β2 = β3 = 0. Then, under the same prescription as above, the rationalization of Eqs. (13a) and (13b) provides the following equations 1 Er = − (α1 + α2 + α3 ), 2 Ei = 0 ,
(23b)
2 2 α12 + γ12 + γ31 = 2ar ,
(23c)
α +γ +γ
= 2br ,
(23d)
α +γ +γ
= 2cr ,
(23e)
2 2
2 3
2 12
2 31
2 23
2 23
(23a)
(α1 + α2 )γ12 + γ23 γ31 = dr ,
(23f)
(α2 + α3 )γ23 + γ12 γ31 = er ,
(23g)
(α1 + α3 )γ31 + γ12 γ23 = fr ,
(23h)
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on selecting γ12 = γ23 = γ31 , Eqs. (23c)–(23e) lead to
√ α1 = − ar , γ12 = γ23 = γ31
α2 = − 2br − ar , ar = ,
α3 = − 2cr − ar , (24)
2
after implying Eq. (24) in (23a) and (23b), the eigenvalues are given by Er(0) P T =
1 √ [ ar + 2br − ar + 2cr − ar ], 2
(0)
Ei
PT
= 0,
(25)
and the eigenfunction reduces to the simpler form
1√ ar 1 1 2 2 2 (xy + yz + zx) , ψP T (x, y, z ) = N exp − ar x − 2br − ar y − 2cr − ar z − (0)
2
2
2
2
(26)
if a = b = c then Eqs. (25) and (26) reduces to Er(0) P T =
3√ ar ,
2
(0)
Ei
PT
= 0,
(27a)
1√ ψP(0T) (x, y, z ) = N exp − ar (x2 + y2 + z 2 ) −
2
ar 2
(xy + yz + zx) .
(27b)
CASE-II (Excited state solutions): Here, we concentrate ourselves to extract the excited state solutions of the SE for the coupled harmonic potential (14) with the same ansatz as represented by (16a)–(16b). Then functional form of φ(x, y, z ) is assumed as a polynomial of first degree [20,21] i.e.
φ(x, y, z ) = α x + β y + γ z + σ ,
(28)
where α , β , γ and σ are real constants. Under the transformation (2), the real and imaginary parts of (57) are written as
φr (x1 , p4 , x2 , p5 , x3 , p6 ) = α x1 + β x2 + γ x3 + σ ,
(29a)
φi (x1 , p4 , x2 , p5 , x3 , p6 ) = α p4 + β p5 + γ p6 ,
(29b)
where α , β , γ and σ are considered as real constants. On substituting Eqs. (15a)–(16b) and (58a)–(58b) in Eqs. (13a)–(13b) a complicated expression is obtained, which is very difficult to solve. To avoid this difficulty, we make the following restrictions on the ansatz parameters
α = β = γ, σ =0 α1 = α2 = α3 , β1 = β2 = β3 γ12 = γ21 , γ23 = γ32 , γ31 = γ13 α1 β1 = α2 β2 = α3 β3 = −γ12 γ21 = −γ23 γ32 = −γ31 γ13 . Under the above restrictions, rationalization of the resultant expression yields the following relations 5 Er = − α1 − 2γ12 , 2 5 Ei = β1 + 2γ21 , 2
(30a) (30b)
α12 − β12 = 2ar ,
(30c)
α1 β1 − 2γ12 γ21 = −ai , 2α1 γ12 + 2β1 γ21 = dr , −2α1 γ21 + 2β1 γ12 − 2γ23 γ13 = di ,
(30d) (30e) (30f)
on solving Eqs. (30c)–(30d), we have
α1 = α2 = α3 = −a+ , β1 = β2 = β3 = a− ,
(31a) (31b)
ai γ12 = γ21 = γ23 = γ32 = γ31 = γ13 = − , 3
(31c)
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by implying the above restrictions on Eqs. (30e) and (30f), one gets the following constraining relations 2(−a+ + a− ) 2(a+ + a− )
ai
= dr
3 ai
2ai
−
3
(32a)
= di ,
3
(32b)
where, a = b = c and d = e = f . After employing the values of the ansatz parameters (31a)–(31c) in Eqs. (59a) and (59b), the real and imaginary components of the first excited state energy eigenvalue are written as (1)
Er
(1)
Ei
=
=
5 2 5 2
ai
a+ + 2
3
ai
a− − 2
3
,
(33a)
,
(33b)
finally,the eigenfunction becomes
1 ψ (1) (x, y, z ) = α(x + y + z ) exp − (a+ + ia− )(x2 + y2 + z 2 ) − (1 + i) ai /3(xy + yz + zx) .
(34)
2
P T -symmetric case: By imposing the P T -symmetric condition (1) on the potential (14), the first excited state eigenspectra turns out to be 5√ ar , 2
Er(1) (P T ) =
(1)
Ei (P T ) = 0,
(35)
√ √a ar r (x2 + y2 + z 2 ) − (xy + yz + zx) . ψ((P1)T ) (x, y, z ) = α(x + y + z ) exp − 2
(36)
2
3.2. Variant of coupled harmonic potential Here, we consider the variant of a coupled harmonic potential (14) by including inverse harmonic and cross terms as [23] x
V (x, y, z ) = ax2 + by2 + cz 2 + dxy + eyz + fzx + A1
y
y
z
z
x
A3
x
z
y
x
z
x2
+ A2 + B1 + B2 + C1 + C2 +
y
+
B3 y2
+
C3 z2
,
(37)
where, the coupling parameters a, b, c , d, e, f and Ai , Bi , Ci with (i = 1, 2, 3) are complex constants. Under the transformation (1), the real and imaginary parts of the potential (37) are considered as Vr = Vr1 +
+ − + −
−
(
x21
+
(
x23
)
+
+
)
p26 2
(
B1r (x2 x3 + p5 p6 )
(x23 + p26 ) B2i (x3 p5 − x2 p6 )
+
(x23 + p26 )2 (
+
p25
)
+
+
B3r (x22 − p25 )
(x22 + p25 )2
B1i (x3 p5 − x2 p6 )
(x23 + p26 ) C1r (x1 x3 + p4 p6 )
(x21 + p24 ) 2A3r x1 p4
(x23 + p26 )2
(
x21
)
p24 2
+
2C3r x3 p6
+
+
+
(
+
− + −
A2i (x2 p4 − x1 p5 )
(x21 + p24 ) B2r (x2 x3 + p5 p6 )
(x22 + p25 ) C1i (x1 p6 − x3 p4 )
(x21 + p24 )
B3i (x22 − p25 )
(
x22
+
)
p25 2
−
(x22 + p25 )2
(x22 + p25 ) p24
)
−
+
2B3r x2 p5
A1i (x1 x2 + p4 p5 )
A2i (x1 x2 + p4 p5 ) x21
2B3i x2 p5
(x22 + p25 )2
(x22 + p25 )
(x21 + p24 )
−
+
A1r (x1 x2 + p4 p5 )
A2r (x1 x2 + p4 p5 )
−
)
)
p24 2
p26 2
)
A1r (x2 p4 − x1 p5 ) x22
+
−
p24 2
C3i (x23 − p26 )
+
+
(x22 + p25 ) (
x23
+
(x22 + p25 )
x21
(
x21
2C3i x3 p6
A1i (x2 p4 − x1 p5 )
A3i (x21 − p24 )
2A3i x1 p4
+
p24 2
C3r (x23 − p26 )
Vi = Vi1 +
+
A3r (x21 − p24 )
A2r (x2 p4 − x1 p5 )
(x21 + p24 )
C2r (x1 x3 + p4 p6 )
(x23 + p26 )
−
C2i (x1 p6 − x3 p4 )
(x23 + p26 )
,
(38)
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
+ −
B1i (x2 x3 + p5 p6 )
(
x23
+
p26
)
B2r (x3 p5 − x2 p6 )
(x22 + p25 )
− +
B1r (x3 p5 − x2 p6 )
(
x23
+
p26
)
C1i (x1 x3 + p4 p6 )
(x21 + p24 )
+ −
2075
B2i (x2 x3 + p5 p6 )
(x22 + p25 ) C1r (x1 p6 − x3 p4 )
(x21 + p24 )
+
C2i (x1 x3 + p4 p6 )
(x23 + p26 )
−
C2r (x1 p6 − x3 p4 )
(x23 + p26 )
,
(39)
where, Vr1 and Vi1 are same as given by Eqs. (15a) and (15b). The functional forms of gr and gi consistent with Eqs. (38)–(39) and satisfying condition (12) are written as gr = gr1 + δ1 tan−1 gi = gi1 + γ1 tan−1
x1
+ δ2 tan−1
p4 x1 p4
+ γ2 tan−1
x2 p5 x2 p5
+ δ3 tan−1 + γ3 tan−1
x3 p6 x3 p6
−
γ1
+
δ1
2 2
ln(x21 + p24 ) − ln(x21 + p24 ) +
γ2 2
δ2 2
ln(x22 + p25 ) − ln(x22 + p25 ) +
γ3 2
δ3 2
ln(x23 + p26 )
(40a)
ln(x23 + p26 )
(40b)
where gr1 and gi1 are same as given by Eqs. (16a) and (16b). On employing Eqs. (38)–(40b) in Eqs. (13a) and (13b), then rationalization of the resultant expression yields the following equations in addition to Eqs. (18c)–(18n) as 1
Er = − Ei =
2
1 2
α1 + α2 + α3 + 2(δ1 β1 + δ2 β2 + δ3 β3 − γ1 α1 − γ2 α2 − γ3 α3 ) ,
β1 + β2 + β3 + 2(δ1 α1 + α2 δ2 + δ3 α3 − γ1 β1 − γ2 β2 − γ3 β3 ) ,
(41a) (41b)
γ12 − δ12 + γ1 = 2A3r ,
(41c)
2γ1 δ1 + δ1 = −2A3i ,
(41d)
γ − δ + γ2 = 2B3r ,
(41e)
2γ2 δ2 + δ2 = −2B3i ,
(41f)
γ − δ + γ3 = 2C3r ,
(41g)
2γ3 δ3 + δ3 = −2C3i ,
(41h)
2 2
2 3
2 2
2 3
γ12 γ2 + γ21 δ2 γ21 γ2 − γ12 δ2 γ12 γ1 + γ21 δ1 γ21 γ1 − γ12 δ1
= −A1r , = −A1i , = −A2r , = −A2i ,
γ23 γ3 + γ32 δ3 γ32 γ3 − γ23 δ3 γ23 γ2 + γ32 δ2 γ32 γ2 − γ23 δ2 γ31 γ1 + γ13 δ1 γ13 γ1 − γ31 δ1 γ31 γ3 + γ13 δ3 γ13 γ3 − γ31 δ3
= −B1r , = −B1i , = −B2r , = −B2i , = −C1r , = −C1i , = −C2r , = −C2i .
(41i) (41j) (41k) (41l) (42a) (42b) (42c) (42d) (42e) (42f) (42g) (42h)
The ansatz parameters αi , βi , γij are same as shown by Eq. (24), whereas δi ’s γi ’s are obtained by solving Eqs. (41c)–(41h) as
δ1 = −
4A3i
δ2 = −
4B3i
δ3 = −
4C3i
A1 B1 C1
1
a1
2
4
1
b1
2
4
1
c1
2
4
,
γ1 = − +
,
γ2 = − +
,
γ3 = − +
where
a1 =
b1 =
c1 =
2 + 16A3r + 2 1 + 16(A3r + 4|A3 |2 )
2 + 16B3r + 2 1 + 16(B3r + 4|B3 |2 )
2 + 16C3r + 2 1 + 16(C3r + 4|C3 |2 ).
(43a) (43b) (43c)
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Further, Eqs. (41i)–(42h) lead the following constraining relations
4B3i 1 ai b1 − − + = −A1r 3
b1
ai 4B3i 3
b1
2
−
1 2
+
(44a)
4
b1 4
= −A1i
(44b)
ai 4A3i 1 a1 − − + = −A2r 3
a1
ai 4A3i 3
a1
2
−
1 2
+
(44c)
4
a1 4
= −A2i
(44d)
ai 4C3i 1 c1 − − + = −B1r 3
c1
ai 4C3i 3
c1
2
−
A1 = B2 ,
1 2
+
(44e)
4
c1 4
= −B1i
A 2 = C1 ,
(44f)
B1 = C2 .
(44g)
After inserting the values of various ansatz parameters in Eqs. (41a) and (41b), the real and imaginary components of the energy eigenvalues are written as
b1 c1 4A3i 4B3i 4C3i + b+ 1 − + c+ 1 − + a− + b− + c−
(45a)
a1 4B3i 4C3i 4A3i b1 c1 = a− 1 − a+ − b+ − c+ + b− 1 − + c− 1 − −
(45b)
Er(0) = a+ 1 − (0)
Ei
a1 4
4
4
4
4
A1
4
B1
A1
C1
B1
C1
and the corresponding eigenfunction turns out to be i
i
i
ψ (0) (x, y, z ) = (x21 + p24 ) 2 (δ1 +iγ1 ) (x22 + p25 ) 2 (δ2 +iγ2 ) (x23 + p26 ) 2 (δ3 +iγ3 ) 1 1 1 b+ + ib− y2 − c+ + ic− z 2 × exp − a+ + ia− x2 − 2 2 2 ai x1 + (1 + i) xy + yz + zx + (γ1 + iδ1 ) tan−1 3
p4
x2 x3 + (γ2 + iδ2 ) tan−1 + (γ3 + iδ3 ) tan−1 . p5 p6
(46)
P T -symmetric case: Under the P T -symmetric condition (1), the potential (37), yields ai = bi = ci = di = ei = fi = Ani = Bni = Cni = δn = βn = 0 where n = 1, 2, 3, after adopting the same procedure as above, one gets the following set of equations in addition to Eqs. (23c)–(23h)
Er = α1 γ1 −
1 2
1 1 + α2 γ2 − + α3 γ3 − , 2
(47a)
2
Ei = 0 ,
(47b)
γ + γ1 = 2A3r ,
(47c)
γ + γ2 = 2B3r ,
(47d)
γ32 + γ3 = 2C3r ,
(47e)
γ12 γ2 γ12 γ1 γ23 γ3 γ23 γ2 γ31 γ1 γ31 γ3
(47f)
2 1
2 2
= −A1r , = −A2r , = −B1r , = −B2r , = −C1r , = −C2r .
(47g) (47h) (47i) (47j) (47k)
After solving the Eqs. (47c)–(47e), we have
√
√ γ1 =
−1 +
1 + 8A3r 2
,
γ2 =
−1 +
1 + 8B3r 2
√ ,
γ3 =
−1 +
1 + 8C3r 2
.
(48)
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
2077
On inserting the ansatz parameters (24) and (48) in Eqs. (47a)–(47b), the real and imaginary parts of the energy eigenvalue are written as Er(0) (P T ) =
√
ar 1 −
1 + 8A3r
+
2br − ar 1 −
1 + 8B3r
+
2cr − ar 1 −
1 + 8C3r ,
(0)
Ei (P T ) = 0,
(49) (50)
and eigenfunction is given by
ψ((P0)T ) (x, y, z ) = (x21 + p24 )
−γ1 2
(x22 + p25 )
−γ2 2
(x23 + p26 )
−γ3 2
1
a+ + ia− x2
exp −
2 1 1 ai 2 2 − b+ + ib− y − c+ + ic− z + (1 + i) xy + yz + zx 2 2 3 −1 x 1 −1 x2 −1 x 3 + iγ1 tan + iγ2 tan + iγ3 tan . p4 p5 p6
(51)
Special case: If inverse harmonic terms are absent in the potential (37) i.e. A3 = B3 = C3 = 0, then potential (37) reduces to V (x, y) = ax2 + by2 + cz 2 + dxy + eyz + fzx + A1
x
y
x
y
x
y
x
y
x
y
x
+ A2 + B1 + B2 + C1 + C2 .
y
(52)
Then adopting the same procedure as above, the energy eigenvalues are given by Er(0) =
(0)
Ei
=
1
(a+ + b+ + c+ ) + 3[(a+ + a− )A2r + (a+ − a− )A2i + (b+ + b− )B2r √ + (b+ − b− )B2i + (c+ + c− )C2r + (c+ − c− )C2i ]/2 ai , 2
(53a)
1
(a− + b− + c− ) + 3[(a− − a+ )A2r + (a+ + a− )A2i + (b− − b+ )B2r √ + (b− + b+ )B2i + (c− − c+ )C2r + (c− + c+ )C2i ]/2 ai , 2
(53b)
and finally eigenfunction is given by
ψ (0) (x, y, z ) = (x21 + p24 )
3A2 (1−i) √ 4 ai
3B2 (1−i) √ 4 ai
(x22 + p25 )
(x23 + p26 )
3C2 (1−i) √ 4 ai
1
exp −
2
a+ + ia− x2
a
1 1 − b+ + ib− y2 − c+ + ic− z 2 + (1 + i) 2 2 −
3A2 (1 − i)
√
4 ai
tan−1
x1 p4
−
3B2 (1 − i)
tan−1
√
4 ai
x2 p5
i
xy + yz + zx
3
−
3C2 (1 − i)
√
4 ai
tan−1
x3 p6
.
(54)
P T -symmetric case: Under the P T -symmetric condition (1), the eigenspectra for the potential (37) turns out to be Er(0) (P T ) =
√ 1 ar
2
−
2 ar
A2r
+
2br − ar
1 2
−
2 ar
B2r
+
2cr − ar
1 2
−
2 ar
(0)
Ei (P T ) = 0,
C2r ,
(55a) (55b)
ψ((P0)T ) (x, y, z ) = (x21 + p24 )
−γ1 2
(x22 + p25 )
−γ2 2
(x23 + p26 )
−γ3 2
1
exp −
a+ + ia− x2
2 1 1 ai 2 2 − b+ + ib− y − c+ + ic− z + (1 + i) xy + yz + zx 2 2 3 −1 x1 −1 x 2 −1 x 3 + iγ1 tan + iγ2 tan + iγ3 tan . p4 p5 p6
(56)
4. Conclusions In the present work, we have elaborated a general procedure to obtain the quasi-exact solutions of the SE for a threedimensional coupled harmonic potential and its variant using the extended complex phase space approach. For this purpose ansatz method is employed and besides the complexity of the phase space, complexity of the potential parameters is also taken into account [13,19,20]. It is found that imaginary component of the energy eigenvalue exists for complex coupling
2078
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
parameters, whereas it is reduced to zero for real coupling parameters. However, in case of complex coupling parameters the imaginary part of the energy eigenvalue can be made zero by imposing P T -symmetric condition. The interesting feature of the method is that it provides additional flexibility for obtaining real eigenspectra of a non-hermitian Hamiltonian system. It is also emphasized that quasi-exact solutions of the SE are obtained in the presence of some constraining relations among the potential parameters, which give rise to bound state energies of a system [21,22]. The number of constraints depends on the functional form of the ansatz. The constraining relations immediately help in identifying the usable domain for the parameters in the potential, which in turn suggests the desired features in the eigenspectrum. A quite general and visible sense is that analyticity property of the eigenfunction simplifies the underlying computation in determining the nature of the eigenspectra and extension of the parameters from real to complex domain lead to the complex eigenspectrum. Thus the present method suggests another degree of freedom to obtain the real eigenspectra for a non-hermitian operator. But for more involved three-dimensional coupled complex systems, it is too much tedious to obtain the solutions of the SE due to expansion of algebra and problem in choosing appropriate form of φ, gr and gi in the ansatz function. Acknowledgments The authors expresses their gratitude to Dr. R.S. Kaushal, Department of Physics and Astro physics, University of Delhi, Delhi (India), for his valuable suggestions regarding the manuscript. We are also thankful to the referees for several useful comments which helped to considerable improvement and fine-tuning of some of the ideas in the original version of the paper. Appendix As there are so many consistency conditions/constraints that the parameters have to satisfy in the present work. The values of the potential parameters (a, b, c ) can be chosen arbitrarily with the condition that ar ≤ 2br , ar ≤ 2cr as seen from Eqs. (24)–(25) and the values of the remaining coupling parameters (say d, e, f ) in the potential must be calculated from the constraining relations (20a)–(20f). In addition to reducing the result to some simple cases or some well known results, the set of parameters for the concrete calculation is chosen asa = b = c = 1 + 4i,
d = e = f = −1.1886 + 5.657i.
(57)
On using the Eq. (24) in Section 3, one finds the ground state as well as excited state solutions for a coupled harmonic potential and its variant. Coupled harmonic potential: After employing Eq. (57) in Eqs. (21a)–(22) and (59c)–(60a), the ground state and excited state solutions reduce to (0)
Er(0) = 2.450,
Ei
= 1.224,
(58a)
ψ (0) (x, y, z ) = N exp −0.817(1 + i/2)(x2 + y2 + z 2 ) + 1.155(1 + i) xy + yz + zx , (1)
Er(1) = 6.392,
Ei
= −0.266,
(58b) (58c)
ψ (1) (x, y, z ) = N α(x + y + z ) exp −0.817(1 + i/2)(x2 + y2 + z 2 ) + 1.155(1 + i) xy + yz + zx ,
(58d)
if a, b, c, d, e and f are real (say a = b = c = 1 and d = e = f = −1.1886), then Eqs. (27a)–(27b) and (60b)–(61a) yield the ground state and excited state solutions as Er(0) = 1.5,
(0)
= 0, ψ (0) (x, y, z ) = N exp −0.5(x2 + y2 + z 2 ) + 0.707 xy + yz + zx , Er(1) = 2.5,
Ei
(1)
Ei
= 0,
(59a) (59b) (59c)
ψ (1) (x, y, z ) = N α(x + y + z ) exp −0.5 x2 + y2 + z 2 + xy + yz + zx .
(59d)
Variant of coupled harmonic potential: After employing Eq. (57) in Eqs. (45a)–(46), the ground state solutions reduce to Er(0) = −1.5345,
(0)
Ei
= −4.545,
(60a)
ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )]
i (−0.617+1.122i) 2
exp −0.817(1 + i/2) × (x2 + y2 + z 2 ) + 1.155(1 + i) xy + yz + zx + (1.122 − 0.617i) x2 x3 x1 , × tan−1 + tan−1 + tan−1 p4
p5
p6
(60b)
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
2079
if a, b, c, d, e and f are real, then Eqs. (49)–(51) yield the solutions Er(0) = −6,
(0)
= 0,
Ei
(61a) −1
1 1 b+ + ib− y2 exp − a+ + ia− x2 −
ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )] 2 2 2 1 x x x3 a 1 2 i − c+ + ic− z 2 + (1 + i) xy + yz + zx + i tan−1 + tan−1 + tan−1 . 2
3
p4
p5
p6
(61b)
Special case: If inverse harmonic term is absent in the potential (37) i.e. A3 = B3 = C3 = 0, then employing Eq. (57) in Eqs. (53a)–(54), the ground state solutions reduce to Er(0) = 2.441,
(0)
Ei
= 13.57,
(62a) (−0.583+1.739i) 4.619
exp −0.817(1 + i/2)(x2 + y2 + z 2 ) ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )] x2 x3 x1 +1.155(1 + i) xy + yz + zx + (−0.583 + 1.739i) tan−1 + tan−1 + tan−1 p4
p5
p6
(62b)
if a, b, c, d, e and f are real then Eqs. (55a)–(56) give rise to Er(0) = 3.972,
(0)
Ei
= 0,
(63a)
ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )](−0.5) exp −0.817(1 + i/2)(x2 + y2 + z 2 ) x2 x3 x1 + tan−1 + tan−1 . + 1.155(1 + i) xy + yz + zx + i tan−1 p4
p5
p6
(63b)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Eigenspectra of a complex coupled harmonic potential in three dimensions S.B. Bhardwaj a,b , Ram Mehar Singh a,∗ , S.C. Mishra c a
Department of Physics, Ch. Devi Lal University, Sirsa-125055, India
b
Department of Physics, Jai Parkash Mukand Lal Innovative Engineering and Technology Institute, Radaur (Haryana), India
c
Department of Physics, Kurukshetra University, Kurukshetra-136119, India
article
info
Article history: Received 30 July 2013 Received in revised form 17 July 2014 Accepted 8 September 2014 Available online 23 October 2014 Keywords: Ansatz Complex potential Eigenvalues and eigenfunctions
abstract Within the framework of extended complex phase space approach characterized by position and momentum coordinates, we investigate the quasi-exact solutions of the Schrödinger equation for a coupled harmonic potential and its variants in three dimensions. For this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone. The energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex. However, in case of complex coupling parameters, the imaginary component of energy eigenvalue reduces to zero if the P T -symmetric condition is satisfied. Thus a non-hermitian Hamiltonian possesses real eigenvalue if it is P T -symmetric. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the study of complex potentials has become more desirable [1–3] for the better theoretical understanding of several newly discovered phenomena in different contexts like optical model of a nucleus, delocalization transitions in condensed matter systems such as a vortex flux line depinning in type-II superconductors, population biology, Bose system of hard spheres, energy spectra of complex Toda lattice, quantum cosmology, quantum field theory and super symmetric quantum mechanics etc. [4–9]. As far as viability of the complex Hamiltonian [10,11] is concerned, it has been investigated in the context of quantum mechanics and semiclassical field theories. Complex Hamiltonians have also been studied in several other theoretical contexts—for example, the study of complex trajectories with regard to the calculation of a semiclassical coherent state propagator in the path integral method has attracted particular interest in the laser physics [11]. Besides some general studies of a complex Hamiltonian in a non-linear domain [1,12], some efforts have been made to study both classical [13,14] as well as quantum aspects [15,16] of a system. At the classical level, construction of exact invariants has been carried out using a more general transformation and the analyticity property of the Hamiltonian leads to a class of integrable systems. However, in the present study, we make use of such transformation to study the quantum aspect of a system and look for the analytic solutions of the Schrödinger equation (SE). It is to be noted that a complex Hamiltonian is no longer hermitian and ordinarily does not guarantee for real eigenvalues. However, in P T -symmetric form [3,4,17], the system is found to exhibit real eigenvalue spectrum [18]. The reality of the spectrum is a direct consequence of combined action of parity and time reversal invariance of Hamiltonian [13,14]. The parity (P ) and time reversal (T ) operators defined
∗
Corresponding author. E-mail address: [email protected] (R.M. Singh).
http://dx.doi.org/10.1016/j.camwa.2014.09.006 0898-1221/© 2014 Elsevier Ltd. All rights reserved.
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
2069
by the action of position and momentum operators are
Pˆ : (x, y, z , px , py , pz , i) → (−x, −y, −z , −px , −py , −pz , i), Tˆ : (x, y, z , px , py , pz , i) → (x, y, z , −px , −py , −pz , −i), whereas, the combined action of Pˆ Tˆ -operation is given by
Pˆ Tˆ : (x, y, z , px , py , pz , i) → (−x, −y, −z , px , py , pz , −i)
(1)
where Pˆ 2 Tˆ 2 = 1. Such studies reveal that a complex Hamiltonian can give real and bounded energy eigenvalues for certain domain of underlying parameters. Therefore, it is possible to investigate the number of new non-hermitian Hamiltonian systems satisfying the P T -symmetric condition. The type of hermiticity discussed here is different from that of given by Bender et al. [2–4]. But both approaches deal with different types of non-hermitian Hamiltonians, the non-hermiticity (P T -symmetry) arising in the Bender’s approach is due to the complexity of the potential parameters whereas in the present study, in addition to the potential parameters, the underlying phase space is also taken as complex. Thus P T -symmetry discussed here is of generalized nature, which in certain limits reduces to the conventional P T -symmetry [2,13]. There are various methods available in the literature for complexifying a given Hamiltonian system [14,15], but here we use the scheme given by Xavier and de-Aguiar [11] to transform the potential in extended complex phase space approach (ECPSA) characterized by x = x1 + ip4 ,
y = x2 + ip5 ,
px = p1 + ix4 ,
z = x3 + ip6
py = p2 + ix5 ,
pz = p3 + ix6 .
(2)
The presence of variables (x4 , x5 , x6 , p1 , p2 , p3 ) in the above transformation represents coordinate-momentum interactions of a dynamical system. Note that in this complexifying scheme, the degrees of freedom of the underlying system just become double and the variables (x1 , p1 ), (x2 , p2 ), (x3 , p3 ), (x4 , p4 ), (x5 , p5 ), (x6 , p6 ) turn to be canonical pairs. Similar transformations (2) have also been used in the study of nonlinear evolution equations in the context of amplitude-modulated nonlinear Langmuir waves in plasma [12]. In the past, considerable progress has been made on the front of complex Hamiltonian up to two-dimensional systems [14,19–21] but the same effort has not been acceded for three-dimensional coupled systems. However, some attempts have been made to obtain the solution of the SE for a coupled harmonic potential in real domain [20–23] but no step has been taken towards complex domain in higher dimensions. The study of complex coupled harmonic potential has become of considerable interest due to the peculiar nature of its eigenvalue spectrum. With this motivation and for gaining the further insight into the nature of the eigenspectra of a three-dimensional complex coupled harmonic potential, we investigate the quasi-exact solutions of the SE in an extended complex phase space. The paper is structured as follows: in the next section, we demonstrate the ansatz method to enable its use in the subsequent sections. Using the same mathematical prescriptions, the explicit expressions for the energy eigenvalues and eigenfunctions of a coupled harmonic potential and its variant are presented in Section 3. Finally, concluding remarks are discussed in Section 4. 2. Ansatz method The SE (for h¯ = m = 1) for a three-dimensional system is written as
ˆ (x, y, z , px , py , pz )ψ(x, y, z ) = E ψ(x, y, z ), H
(3)
where
ˆ (x, y, z , px , py , pz ) = − H
1 d2 2 dx2
+
d2 dy2
+
d2 dz 2
+ V (x, y, z ).
(4)
Transformation (2), implies that [20]
∂ d 1 ∂ ∂ , = −i , dx 2 ∂ x1 ∂ p4 dpx 2 ∂ p1 ∂ x4 1 ∂ ∂ d 1 ∂ ∂ d = −i , = −i , dy 2 ∂ x2 ∂ p5 dpy 2 ∂ p2 ∂ x5 d 1 ∂ ∂ d 1 ∂ ∂ = −i , = −i . dz 2 ∂ x3 ∂ p6 dpz 2 ∂ p3 ∂ x6 On expressing V (x, y, z ), ψ(x, y, z ) and E in terms of real and imaginary components, we have d
=
1 ∂
−i
(5)
V (x, y, z ) = Vr (x1 , p4 , x2 , p5 , x3 , p6 ) + iVi (x1 , p4 , x2 , p5 , x3 , p6 ),
ψ(x, y, z ) = ψr (x1 , p4 , x2 , p5 , x3 , p6 ) + iψi (x1 , p4 , x2 , p5 , x3 , p6 ) E = Er + iEi ,
(6)
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by employing Eqs. (4)–(6) in Eq. (3) and then separating real and imaginary parts, one gets 1
ψr ,x1 x1 − ψr ,p4 p4 + 2ψi,x1 p4 + ψr ,x2 x2 − ψr ,p5 p5 + 2ψi,x2 p5 + ψr ,x3 x3 − ψr ,p6 p6 + 2ψi,x3 p6 + Vr ψr − Vi ψi = Er ψr − Ei ψi , 1 ψi,x1 x1 − ψi,p4 p4 − 2ψr ,x1 p4 + ψi,x2 x2 − ψi,p5 p5 − 2ψr ,x2 p5 8 + ψi,x3 x3 − ψi,p6 p6 − 2ψr ,x3 p6 + Vr ψi + Vi ψr = Er ψi + Ei ψr
−
8
(7a)
(7b)
where subscripts r and i denote the real and imaginary parts of the corresponding quantities and other subscripts to these quantities separated by comma denote the partial derivatives of the quantity concerned. For the wavefunction ψ(x, y, z ), Cauchy–Riemann analyticity condition implies
ψr ,x1 = ψi,p4 , ψr ,x2 = ψi,p5 ,
ψr ,p4 = −ψi,x1 , ψr ,p5 = −ψi,x2 ,
ψr ,x3 = ψi,p6 ,
ψr ,p6 = −ψi,x3 .
(8)
Under the analyticity condition (8), Eqs. (7a) and (7b) reduces to
− −
1 2 1 2
ψr ,x1 x1 + ψr ,x2 x2 + ψr ,x3 x3 + Vr ψr − Vi ψi = Er ψr − Ei ψi ,
(9a)
ψi,x1 x1 + ψi,x2 x2 + ψi,x3 x3 + Vr ψi + Vi ψr = Er ψi + Ei ψr .
(9b)
Ansatz for the eigenfunction ψ(x, y, z ) is assumed as [20,22]
ψ(x, y, z ) = φ(x, y, z ) exp g (x, y, z )
(10)
where, φ = φr + iφi and g = gr + igi . Thus real and imaginary parts of the wavefunction ψ(x, y, z ) are given by
ψr = egr (φr cos gi − φi sin gi ),
(11a)
ψi = e (φi cos gi + φr sin gi ).
(11b)
gr
Again, analyticity conditions for the functions gr and gi turn out to be gr ,x1 = gi,p4 ,
gi,x1 = −gr ,p4 ,
gr ,x2 = gi,p5 , gr ,x3 = gi,p6 ,
gi,x2 = −gr ,p5 , gi,x3 = −gr ,p6 .
(12)
After employing Eqs. (11a)–(12) in Eqs. (9a) and (9b), we have gr ,x1 x1 + gr ,x2 x2 + gr ,x3 x3 + (gr ,x1 )2 + (gr ,x2 )2 + (gr ,x3 )2 − (gi,x1 )2
− (gi,x2 )2 − (gi,x3 )2 +
1
φr (φr ,x1 x1 + φr ,x2 x2 + φr ,x3 x3
(φr2 + φi2 ) + 2φr ,x1 gr ,x1 + 2φr ,x2 gr ,x2 + 2φr ,x3 gr ,x3 − 2φi,x1 gi,x1 − 2φi,x2 gi,x2 − 2φi,x3 gi,x3 ) + φi (φi,x1 x1 + φi,x2 x2 + φi,x3 x3 + 2φr ,x1 gi,x1 + 2φr ,x2 gi,x2 + 2φr ,x3 gi,x3 + 2φi,x1 gr ,x1 + 2φi,x2 gr ,x2 + 2φi,x3 gr ,x3 ) + 2(Er − Vr ) = 0,
(13a)
gi,x1 x1 + gi,x2 x2 + gi,x3 x3 + 2gr ,x1 gi,x1 + 2gr ,x2 gi,x2 + 2gr ,x3 gi,x3
+
1
φr (φi,x1 x1 + φi,x2 x2 + φi,x3 x3 + 2φr ,x1 gi,x1 + 2φr ,x2 gi,x2
(φr2 + φi2 ) + 2φr ,x3 gi,x3 + 2φi,x1 gi,x1 + 2φi,x2 gi,x2 + 2φi,x3 gi,x3 ) + φi (−φr ,x1 x1 − φr ,x2 x2
− φr ,x3 x3 + 2φi,x1 gi,x1 + 2φi,x2 gi,x2 + 2φi,x3 gi,x3 − 2φr ,x1 gi,x1 − 2φr ,x2 gi,x2 − 2φr ,x3 gi,x3 ) + 2(Ei − Vi ) = 0.
(13b)
For the given potential V (x, y, z ), by considering the appropriate polynomial form of φ(x, y, z ) and selecting functional forms of gr and gi consistent with the potential,then rationalization of the resultant expressions ((13a) and (13b)) yields the ground state and excited state solutions of the SE.
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2071
3. Examples In this section, we demonstrate the viability of the ansatz method as discussed in Section 2 for obtaining ground state as well as excited state eigenspectra of a coupled harmonic potential and its variant. 3.1. Coupled harmonic potential Consider a three-dimensional coupled harmonic potential as [23] V (x, y, z ) = ax2 + by2 + cz 2 + dxy + eyz + fzx
(14)
where a, b, c, d, e and f are coupling parameters and taken as complex. Under the transformation (2), the real and imaginary parts of the potential (14) are given by Vr = ar (x21 − p24 ) − 2ai x1 p4 + br (x22 − p25 ) − 2bi x2 p5 + cr (x23 − p26 ) − 2ci x3 p6 + dr (x1 x2 − p4 p5 ) − di (x1 p5 + x2 p4 ) + er (x2 x3 − p5 p6 ) − ei (x2 p6 + x3 p5 ) + fr (x1 x3 − p4 p6 ) − fi (x1 p6 + x3 p4 ),
(15a)
Vi = ai (x21 − p24 ) + 2ar x1 p4 + bi (x22 − p25 ) + 2br x2 p5 + ci (x23 − p26 ) + 2cr x3 p6 + di (x1 x2 − p4 p5 ) + dr (x1 p5 + x2 p4 ) + ei (x2 x3 − p5 p6 ) + er (x2 p6 + x3 p5 ) + fi (x1 x3 − p4 p6 ) + fr (x1 p6 + x3 p4 ).
(15b)
For the underlying system, the functional forms of gr and gi in conformity with Eq. (12), are considered as gr =
1 2
1
1
2
2
α1 (x21 − p24 ) + α2 (x22 − p25 ) + α3 (x23 − p26 ) + β1 x1 p4 + β2 x2 p5
+ β3 x3 p6 + γ12 (x1 x2 − p4 p5 ) − γ21 (x1 p5 + x2 p4 ) + γ23 (x2 x3 − p5 p6 ) − γ32 (x2 p6 + x3 p5 ) + γ31 (x1 x3 − p4 p6 ) − γ13 (x1 p6 + x3 p4 ) gi =
1 2
1
1
2
2
(16a)
β1 (x21 − p24 ) − β2 (x22 − p25 ) − β3 (x23 − p26 )α1 x1 p4 + α2 x2 p5
+ α3 x3 p6 + γ21 (x1 x2 − p4 p5 ) + γ12 (x1 p5 + x2 p4 ) + γ32 (x2 x3 − p5 p6 ) + γ23 (x2 p6 + x3 p5 ) + γ13 (x1 x3 − p4 p6 ) + γ31 (x1 p6 + x3 p4 ).
(16b)
By selecting a suitable form of φ(x, y, z ) and employing Eqs. (15a)–(16b) in Eqs. (13a) and (13b), the rationalization of resultant expressions ((13a) and (13b)) yields the ground state and excited state solutions of the SE asCASE-I (Ground state solutions): For the ground state solutions, the functional form of φ(x, y, z ) is taken as a polynomial of zero degree, (say constant) i.e.
φ(x, y, z ) = 1.
(17)
By, inserting Eqs. (15a)–(17) in (13a) and (13b) and then equating the coefficients of various variables to zero, we obtain the following set of 14-non-repeating equations 1 Er = − (α1 + α2 + α3 ), 2 Ei =
1 2
(18a)
(β1 + β2 + β3 ),
(18b)
2 2 2 2 α12 − β12 + γ12 − γ21 + γ31 − γ13 = 2ar ,
(18c)
α1 β1 − γ12 γ21 − γ13 γ31 = −ai ,
(18d)
2 2 2 2 α22 − β22 + γ12 − γ21 + γ23 − γ32 = 2br ,
(18e)
α2 β2 − γ12 γ21 − γ23 γ32 = −bi ,
(18f)
α −β +γ −γ +γ −γ 2 3
2 3
2 31
2 13
2 23
2 32
= 2cr ,
α3 β3 − γ31 γ13 − γ23 γ32 = −ci ,
(18g) (18h)
(α1 + α2 )γ12 + (β1 + β2 )γ21 + γ23 γ31 − γ13 γ32 = dr ,
(18i)
−(α1 + α2 )γ21 + (β1 + β2 )γ12 − γ23 γ13 − γ31 γ32 = di ,
(18j)
(α2 + α3 )γ23 + (β2 + β3 )γ32 + γ12 γ31 − γ13 γ21 = er ,
(18k)
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−(α2 + α3 )γ32 + (β2 + β3 )γ23 − γ12 γ13 − γ31 γ21 = ei ,
(18l)
(α1 + α3 )γ31 + (β1 + β3 )γ13 + γ12 γ23 − γ32 γ21 = fr ,
(18m)
−(α1 + α3 )γ13 + (β1 + β3 )γ31 − γ12 γ32 − γ23 γ21 = fi .
(18n)
As such solutions of the various ansatz parameters α ’s, β ’s and γij ’s from the above equations seem to be very difficult. For this purpose, one can make a number of choices among the ansatz parameters but one should make some plausible choices among these potential parameters so that there is no conflict between the general solutions and the well established results (SHO). In this view, we choose
γ12 = γ21 = γ23 = γ32 = γ31 = γ13 α1 β1 = α2 β2 = α3 β3 = −γ12 γ21 = −γ23 γ32 = −γ31 γ13 . Under the above choices, Eqs. (18c)–(18h) immediately lead to
α1 = −a+ , β1 = a− , γ12 = γ21
α2 = −b+ , α3 = −c+ , β2 = b− , β3 = c− . ai , = γ23 = γ32 = γ31 = γ13 =
(19b) (19c)
3
where a± =
(19a)
±a r +
a2r + a2i /9, b± =
±b r +
b2r + b2i /9, c± =
±cr +
cr2 + ci2 /9, and ai = bi = ci .
With the above choice, Eqs. (18i)–(18n) yield the following six constraining relations
ai /3(−a+ − b+ + a− + b− ) = dr ,
(20a)
ai /3(a+ + b+ + a− + b− ) = di .
(20b)
ai /3(−b+ − c+ + b− + c− ) = er ,
(20c)
ai /3(b+ + c+ + b− + c− ) = ei .
(20d)
ai /3(−c+ − a+ + c− + a− ) = fr ,
(20e)
ai /3(c+ + a+ + c− + a− ) = fi .
(20f)
Now, substituting Eqs. (19a)–(19c) in Eqs. (18a) and (18b), the real and imaginary components of eigenvalue are written as Er(0) = (0)
Ei
=
1 2 1 2
a+ + b+ + c+ ,
(21a)
a− + b− + c− ,
(21b)
and the corresponding eigenfunction turns out to be
1 1 1 ai b+ + ib− y2 − c+ + ic− z 2 + (1 + i) xy + yz + zx . ψ (0) (x, y, z ) = N exp − a+ + ia− x2 − 2
2
2
3
(22)
P T -symmetric case: Under the P T -symmetric condition (1), the potential (14) yields ai = bi = ci = di = ei = fi = 0, and the ansatz parameters γ21 = γ32 = γ13 = β1 = β2 = β3 = 0. Then, under the same prescription as above, the rationalization of Eqs. (13a) and (13b) provides the following equations 1 Er = − (α1 + α2 + α3 ), 2 Ei = 0 ,
(23b)
2 2 α12 + γ12 + γ31 = 2ar ,
(23c)
α +γ +γ
= 2br ,
(23d)
α +γ +γ
= 2cr ,
(23e)
2 2
2 3
2 12
2 31
2 23
2 23
(23a)
(α1 + α2 )γ12 + γ23 γ31 = dr ,
(23f)
(α2 + α3 )γ23 + γ12 γ31 = er ,
(23g)
(α1 + α3 )γ31 + γ12 γ23 = fr ,
(23h)
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2073
on selecting γ12 = γ23 = γ31 , Eqs. (23c)–(23e) lead to
√ α1 = − ar , γ12 = γ23 = γ31
α2 = − 2br − ar , ar = ,
α3 = − 2cr − ar , (24)
2
after implying Eq. (24) in (23a) and (23b), the eigenvalues are given by Er(0) P T =
1 √ [ ar + 2br − ar + 2cr − ar ], 2
(0)
Ei
PT
= 0,
(25)
and the eigenfunction reduces to the simpler form
1√ ar 1 1 2 2 2 (xy + yz + zx) , ψP T (x, y, z ) = N exp − ar x − 2br − ar y − 2cr − ar z − (0)
2
2
2
2
(26)
if a = b = c then Eqs. (25) and (26) reduces to Er(0) P T =
3√ ar ,
2
(0)
Ei
PT
= 0,
(27a)
1√ ψP(0T) (x, y, z ) = N exp − ar (x2 + y2 + z 2 ) −
2
ar 2
(xy + yz + zx) .
(27b)
CASE-II (Excited state solutions): Here, we concentrate ourselves to extract the excited state solutions of the SE for the coupled harmonic potential (14) with the same ansatz as represented by (16a)–(16b). Then functional form of φ(x, y, z ) is assumed as a polynomial of first degree [20,21] i.e.
φ(x, y, z ) = α x + β y + γ z + σ ,
(28)
where α , β , γ and σ are real constants. Under the transformation (2), the real and imaginary parts of (57) are written as
φr (x1 , p4 , x2 , p5 , x3 , p6 ) = α x1 + β x2 + γ x3 + σ ,
(29a)
φi (x1 , p4 , x2 , p5 , x3 , p6 ) = α p4 + β p5 + γ p6 ,
(29b)
where α , β , γ and σ are considered as real constants. On substituting Eqs. (15a)–(16b) and (58a)–(58b) in Eqs. (13a)–(13b) a complicated expression is obtained, which is very difficult to solve. To avoid this difficulty, we make the following restrictions on the ansatz parameters
α = β = γ, σ =0 α1 = α2 = α3 , β1 = β2 = β3 γ12 = γ21 , γ23 = γ32 , γ31 = γ13 α1 β1 = α2 β2 = α3 β3 = −γ12 γ21 = −γ23 γ32 = −γ31 γ13 . Under the above restrictions, rationalization of the resultant expression yields the following relations 5 Er = − α1 − 2γ12 , 2 5 Ei = β1 + 2γ21 , 2
(30a) (30b)
α12 − β12 = 2ar ,
(30c)
α1 β1 − 2γ12 γ21 = −ai , 2α1 γ12 + 2β1 γ21 = dr , −2α1 γ21 + 2β1 γ12 − 2γ23 γ13 = di ,
(30d) (30e) (30f)
on solving Eqs. (30c)–(30d), we have
α1 = α2 = α3 = −a+ , β1 = β2 = β3 = a− ,
(31a) (31b)
ai γ12 = γ21 = γ23 = γ32 = γ31 = γ13 = − , 3
(31c)
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S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
by implying the above restrictions on Eqs. (30e) and (30f), one gets the following constraining relations 2(−a+ + a− ) 2(a+ + a− )
ai
= dr
3 ai
2ai
−
3
(32a)
= di ,
3
(32b)
where, a = b = c and d = e = f . After employing the values of the ansatz parameters (31a)–(31c) in Eqs. (59a) and (59b), the real and imaginary components of the first excited state energy eigenvalue are written as (1)
Er
(1)
Ei
=
=
5 2 5 2
ai
a+ + 2
3
ai
a− − 2
3
,
(33a)
,
(33b)
finally,the eigenfunction becomes
1 ψ (1) (x, y, z ) = α(x + y + z ) exp − (a+ + ia− )(x2 + y2 + z 2 ) − (1 + i) ai /3(xy + yz + zx) .
(34)
2
P T -symmetric case: By imposing the P T -symmetric condition (1) on the potential (14), the first excited state eigenspectra turns out to be 5√ ar , 2
Er(1) (P T ) =
(1)
Ei (P T ) = 0,
(35)
√ √a ar r (x2 + y2 + z 2 ) − (xy + yz + zx) . ψ((P1)T ) (x, y, z ) = α(x + y + z ) exp − 2
(36)
2
3.2. Variant of coupled harmonic potential Here, we consider the variant of a coupled harmonic potential (14) by including inverse harmonic and cross terms as [23] x
V (x, y, z ) = ax2 + by2 + cz 2 + dxy + eyz + fzx + A1
y
y
z
z
x
A3
x
z
y
x
z
x2
+ A2 + B1 + B2 + C1 + C2 +
y
+
B3 y2
+
C3 z2
,
(37)
where, the coupling parameters a, b, c , d, e, f and Ai , Bi , Ci with (i = 1, 2, 3) are complex constants. Under the transformation (1), the real and imaginary parts of the potential (37) are considered as Vr = Vr1 +
+ − + −
−
(
x21
+
(
x23
)
+
+
)
p26 2
(
B1r (x2 x3 + p5 p6 )
(x23 + p26 ) B2i (x3 p5 − x2 p6 )
+
(x23 + p26 )2 (
+
p25
)
+
+
B3r (x22 − p25 )
(x22 + p25 )2
B1i (x3 p5 − x2 p6 )
(x23 + p26 ) C1r (x1 x3 + p4 p6 )
(x21 + p24 ) 2A3r x1 p4
(x23 + p26 )2
(
x21
)
p24 2
+
2C3r x3 p6
+
+
+
(
+
− + −
A2i (x2 p4 − x1 p5 )
(x21 + p24 ) B2r (x2 x3 + p5 p6 )
(x22 + p25 ) C1i (x1 p6 − x3 p4 )
(x21 + p24 )
B3i (x22 − p25 )
(
x22
+
)
p25 2
−
(x22 + p25 )2
(x22 + p25 ) p24
)
−
+
2B3r x2 p5
A1i (x1 x2 + p4 p5 )
A2i (x1 x2 + p4 p5 ) x21
2B3i x2 p5
(x22 + p25 )2
(x22 + p25 )
(x21 + p24 )
−
+
A1r (x1 x2 + p4 p5 )
A2r (x1 x2 + p4 p5 )
−
)
)
p24 2
p26 2
)
A1r (x2 p4 − x1 p5 ) x22
+
−
p24 2
C3i (x23 − p26 )
+
+
(x22 + p25 ) (
x23
+
(x22 + p25 )
x21
(
x21
2C3i x3 p6
A1i (x2 p4 − x1 p5 )
A3i (x21 − p24 )
2A3i x1 p4
+
p24 2
C3r (x23 − p26 )
Vi = Vi1 +
+
A3r (x21 − p24 )
A2r (x2 p4 − x1 p5 )
(x21 + p24 )
C2r (x1 x3 + p4 p6 )
(x23 + p26 )
−
C2i (x1 p6 − x3 p4 )
(x23 + p26 )
,
(38)
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
+ −
B1i (x2 x3 + p5 p6 )
(
x23
+
p26
)
B2r (x3 p5 − x2 p6 )
(x22 + p25 )
− +
B1r (x3 p5 − x2 p6 )
(
x23
+
p26
)
C1i (x1 x3 + p4 p6 )
(x21 + p24 )
+ −
2075
B2i (x2 x3 + p5 p6 )
(x22 + p25 ) C1r (x1 p6 − x3 p4 )
(x21 + p24 )
+
C2i (x1 x3 + p4 p6 )
(x23 + p26 )
−
C2r (x1 p6 − x3 p4 )
(x23 + p26 )
,
(39)
where, Vr1 and Vi1 are same as given by Eqs. (15a) and (15b). The functional forms of gr and gi consistent with Eqs. (38)–(39) and satisfying condition (12) are written as gr = gr1 + δ1 tan−1 gi = gi1 + γ1 tan−1
x1
+ δ2 tan−1
p4 x1 p4
+ γ2 tan−1
x2 p5 x2 p5
+ δ3 tan−1 + γ3 tan−1
x3 p6 x3 p6
−
γ1
+
δ1
2 2
ln(x21 + p24 ) − ln(x21 + p24 ) +
γ2 2
δ2 2
ln(x22 + p25 ) − ln(x22 + p25 ) +
γ3 2
δ3 2
ln(x23 + p26 )
(40a)
ln(x23 + p26 )
(40b)
where gr1 and gi1 are same as given by Eqs. (16a) and (16b). On employing Eqs. (38)–(40b) in Eqs. (13a) and (13b), then rationalization of the resultant expression yields the following equations in addition to Eqs. (18c)–(18n) as 1
Er = − Ei =
2
1 2
α1 + α2 + α3 + 2(δ1 β1 + δ2 β2 + δ3 β3 − γ1 α1 − γ2 α2 − γ3 α3 ) ,
β1 + β2 + β3 + 2(δ1 α1 + α2 δ2 + δ3 α3 − γ1 β1 − γ2 β2 − γ3 β3 ) ,
(41a) (41b)
γ12 − δ12 + γ1 = 2A3r ,
(41c)
2γ1 δ1 + δ1 = −2A3i ,
(41d)
γ − δ + γ2 = 2B3r ,
(41e)
2γ2 δ2 + δ2 = −2B3i ,
(41f)
γ − δ + γ3 = 2C3r ,
(41g)
2γ3 δ3 + δ3 = −2C3i ,
(41h)
2 2
2 3
2 2
2 3
γ12 γ2 + γ21 δ2 γ21 γ2 − γ12 δ2 γ12 γ1 + γ21 δ1 γ21 γ1 − γ12 δ1
= −A1r , = −A1i , = −A2r , = −A2i ,
γ23 γ3 + γ32 δ3 γ32 γ3 − γ23 δ3 γ23 γ2 + γ32 δ2 γ32 γ2 − γ23 δ2 γ31 γ1 + γ13 δ1 γ13 γ1 − γ31 δ1 γ31 γ3 + γ13 δ3 γ13 γ3 − γ31 δ3
= −B1r , = −B1i , = −B2r , = −B2i , = −C1r , = −C1i , = −C2r , = −C2i .
(41i) (41j) (41k) (41l) (42a) (42b) (42c) (42d) (42e) (42f) (42g) (42h)
The ansatz parameters αi , βi , γij are same as shown by Eq. (24), whereas δi ’s γi ’s are obtained by solving Eqs. (41c)–(41h) as
δ1 = −
4A3i
δ2 = −
4B3i
δ3 = −
4C3i
A1 B1 C1
1
a1
2
4
1
b1
2
4
1
c1
2
4
,
γ1 = − +
,
γ2 = − +
,
γ3 = − +
where
a1 =
b1 =
c1 =
2 + 16A3r + 2 1 + 16(A3r + 4|A3 |2 )
2 + 16B3r + 2 1 + 16(B3r + 4|B3 |2 )
2 + 16C3r + 2 1 + 16(C3r + 4|C3 |2 ).
(43a) (43b) (43c)
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Further, Eqs. (41i)–(42h) lead the following constraining relations
4B3i 1 ai b1 − − + = −A1r 3
b1
ai 4B3i 3
b1
2
−
1 2
+
(44a)
4
b1 4
= −A1i
(44b)
ai 4A3i 1 a1 − − + = −A2r 3
a1
ai 4A3i 3
a1
2
−
1 2
+
(44c)
4
a1 4
= −A2i
(44d)
ai 4C3i 1 c1 − − + = −B1r 3
c1
ai 4C3i 3
c1
2
−
A1 = B2 ,
1 2
+
(44e)
4
c1 4
= −B1i
A 2 = C1 ,
(44f)
B1 = C2 .
(44g)
After inserting the values of various ansatz parameters in Eqs. (41a) and (41b), the real and imaginary components of the energy eigenvalues are written as
b1 c1 4A3i 4B3i 4C3i + b+ 1 − + c+ 1 − + a− + b− + c−
(45a)
a1 4B3i 4C3i 4A3i b1 c1 = a− 1 − a+ − b+ − c+ + b− 1 − + c− 1 − −
(45b)
Er(0) = a+ 1 − (0)
Ei
a1 4
4
4
4
4
A1
4
B1
A1
C1
B1
C1
and the corresponding eigenfunction turns out to be i
i
i
ψ (0) (x, y, z ) = (x21 + p24 ) 2 (δ1 +iγ1 ) (x22 + p25 ) 2 (δ2 +iγ2 ) (x23 + p26 ) 2 (δ3 +iγ3 ) 1 1 1 b+ + ib− y2 − c+ + ic− z 2 × exp − a+ + ia− x2 − 2 2 2 ai x1 + (1 + i) xy + yz + zx + (γ1 + iδ1 ) tan−1 3
p4
x2 x3 + (γ2 + iδ2 ) tan−1 + (γ3 + iδ3 ) tan−1 . p5 p6
(46)
P T -symmetric case: Under the P T -symmetric condition (1), the potential (37), yields ai = bi = ci = di = ei = fi = Ani = Bni = Cni = δn = βn = 0 where n = 1, 2, 3, after adopting the same procedure as above, one gets the following set of equations in addition to Eqs. (23c)–(23h)
Er = α1 γ1 −
1 2
1 1 + α2 γ2 − + α3 γ3 − , 2
(47a)
2
Ei = 0 ,
(47b)
γ + γ1 = 2A3r ,
(47c)
γ + γ2 = 2B3r ,
(47d)
γ32 + γ3 = 2C3r ,
(47e)
γ12 γ2 γ12 γ1 γ23 γ3 γ23 γ2 γ31 γ1 γ31 γ3
(47f)
2 1
2 2
= −A1r , = −A2r , = −B1r , = −B2r , = −C1r , = −C2r .
(47g) (47h) (47i) (47j) (47k)
After solving the Eqs. (47c)–(47e), we have
√
√ γ1 =
−1 +
1 + 8A3r 2
,
γ2 =
−1 +
1 + 8B3r 2
√ ,
γ3 =
−1 +
1 + 8C3r 2
.
(48)
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2077
On inserting the ansatz parameters (24) and (48) in Eqs. (47a)–(47b), the real and imaginary parts of the energy eigenvalue are written as Er(0) (P T ) =
√
ar 1 −
1 + 8A3r
+
2br − ar 1 −
1 + 8B3r
+
2cr − ar 1 −
1 + 8C3r ,
(0)
Ei (P T ) = 0,
(49) (50)
and eigenfunction is given by
ψ((P0)T ) (x, y, z ) = (x21 + p24 )
−γ1 2
(x22 + p25 )
−γ2 2
(x23 + p26 )
−γ3 2
1
a+ + ia− x2
exp −
2 1 1 ai 2 2 − b+ + ib− y − c+ + ic− z + (1 + i) xy + yz + zx 2 2 3 −1 x 1 −1 x2 −1 x 3 + iγ1 tan + iγ2 tan + iγ3 tan . p4 p5 p6
(51)
Special case: If inverse harmonic terms are absent in the potential (37) i.e. A3 = B3 = C3 = 0, then potential (37) reduces to V (x, y) = ax2 + by2 + cz 2 + dxy + eyz + fzx + A1
x
y
x
y
x
y
x
y
x
y
x
+ A2 + B1 + B2 + C1 + C2 .
y
(52)
Then adopting the same procedure as above, the energy eigenvalues are given by Er(0) =
(0)
Ei
=
1
(a+ + b+ + c+ ) + 3[(a+ + a− )A2r + (a+ − a− )A2i + (b+ + b− )B2r √ + (b+ − b− )B2i + (c+ + c− )C2r + (c+ − c− )C2i ]/2 ai , 2
(53a)
1
(a− + b− + c− ) + 3[(a− − a+ )A2r + (a+ + a− )A2i + (b− − b+ )B2r √ + (b− + b+ )B2i + (c− − c+ )C2r + (c− + c+ )C2i ]/2 ai , 2
(53b)
and finally eigenfunction is given by
ψ (0) (x, y, z ) = (x21 + p24 )
3A2 (1−i) √ 4 ai
3B2 (1−i) √ 4 ai
(x22 + p25 )
(x23 + p26 )
3C2 (1−i) √ 4 ai
1
exp −
2
a+ + ia− x2
a
1 1 − b+ + ib− y2 − c+ + ic− z 2 + (1 + i) 2 2 −
3A2 (1 − i)
√
4 ai
tan−1
x1 p4
−
3B2 (1 − i)
tan−1
√
4 ai
x2 p5
i
xy + yz + zx
3
−
3C2 (1 − i)
√
4 ai
tan−1
x3 p6
.
(54)
P T -symmetric case: Under the P T -symmetric condition (1), the eigenspectra for the potential (37) turns out to be Er(0) (P T ) =
√ 1 ar
2
−
2 ar
A2r
+
2br − ar
1 2
−
2 ar
B2r
+
2cr − ar
1 2
−
2 ar
(0)
Ei (P T ) = 0,
C2r ,
(55a) (55b)
ψ((P0)T ) (x, y, z ) = (x21 + p24 )
−γ1 2
(x22 + p25 )
−γ2 2
(x23 + p26 )
−γ3 2
1
exp −
a+ + ia− x2
2 1 1 ai 2 2 − b+ + ib− y − c+ + ic− z + (1 + i) xy + yz + zx 2 2 3 −1 x1 −1 x 2 −1 x 3 + iγ1 tan + iγ2 tan + iγ3 tan . p4 p5 p6
(56)
4. Conclusions In the present work, we have elaborated a general procedure to obtain the quasi-exact solutions of the SE for a threedimensional coupled harmonic potential and its variant using the extended complex phase space approach. For this purpose ansatz method is employed and besides the complexity of the phase space, complexity of the potential parameters is also taken into account [13,19,20]. It is found that imaginary component of the energy eigenvalue exists for complex coupling
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S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
parameters, whereas it is reduced to zero for real coupling parameters. However, in case of complex coupling parameters the imaginary part of the energy eigenvalue can be made zero by imposing P T -symmetric condition. The interesting feature of the method is that it provides additional flexibility for obtaining real eigenspectra of a non-hermitian Hamiltonian system. It is also emphasized that quasi-exact solutions of the SE are obtained in the presence of some constraining relations among the potential parameters, which give rise to bound state energies of a system [21,22]. The number of constraints depends on the functional form of the ansatz. The constraining relations immediately help in identifying the usable domain for the parameters in the potential, which in turn suggests the desired features in the eigenspectrum. A quite general and visible sense is that analyticity property of the eigenfunction simplifies the underlying computation in determining the nature of the eigenspectra and extension of the parameters from real to complex domain lead to the complex eigenspectrum. Thus the present method suggests another degree of freedom to obtain the real eigenspectra for a non-hermitian operator. But for more involved three-dimensional coupled complex systems, it is too much tedious to obtain the solutions of the SE due to expansion of algebra and problem in choosing appropriate form of φ, gr and gi in the ansatz function. Acknowledgments The authors expresses their gratitude to Dr. R.S. Kaushal, Department of Physics and Astro physics, University of Delhi, Delhi (India), for his valuable suggestions regarding the manuscript. We are also thankful to the referees for several useful comments which helped to considerable improvement and fine-tuning of some of the ideas in the original version of the paper. Appendix As there are so many consistency conditions/constraints that the parameters have to satisfy in the present work. The values of the potential parameters (a, b, c ) can be chosen arbitrarily with the condition that ar ≤ 2br , ar ≤ 2cr as seen from Eqs. (24)–(25) and the values of the remaining coupling parameters (say d, e, f ) in the potential must be calculated from the constraining relations (20a)–(20f). In addition to reducing the result to some simple cases or some well known results, the set of parameters for the concrete calculation is chosen asa = b = c = 1 + 4i,
d = e = f = −1.1886 + 5.657i.
(57)
On using the Eq. (24) in Section 3, one finds the ground state as well as excited state solutions for a coupled harmonic potential and its variant. Coupled harmonic potential: After employing Eq. (57) in Eqs. (21a)–(22) and (59c)–(60a), the ground state and excited state solutions reduce to (0)
Er(0) = 2.450,
Ei
= 1.224,
(58a)
ψ (0) (x, y, z ) = N exp −0.817(1 + i/2)(x2 + y2 + z 2 ) + 1.155(1 + i) xy + yz + zx , (1)
Er(1) = 6.392,
Ei
= −0.266,
(58b) (58c)
ψ (1) (x, y, z ) = N α(x + y + z ) exp −0.817(1 + i/2)(x2 + y2 + z 2 ) + 1.155(1 + i) xy + yz + zx ,
(58d)
if a, b, c, d, e and f are real (say a = b = c = 1 and d = e = f = −1.1886), then Eqs. (27a)–(27b) and (60b)–(61a) yield the ground state and excited state solutions as Er(0) = 1.5,
(0)
= 0, ψ (0) (x, y, z ) = N exp −0.5(x2 + y2 + z 2 ) + 0.707 xy + yz + zx , Er(1) = 2.5,
Ei
(1)
Ei
= 0,
(59a) (59b) (59c)
ψ (1) (x, y, z ) = N α(x + y + z ) exp −0.5 x2 + y2 + z 2 + xy + yz + zx .
(59d)
Variant of coupled harmonic potential: After employing Eq. (57) in Eqs. (45a)–(46), the ground state solutions reduce to Er(0) = −1.5345,
(0)
Ei
= −4.545,
(60a)
ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )]
i (−0.617+1.122i) 2
exp −0.817(1 + i/2) × (x2 + y2 + z 2 ) + 1.155(1 + i) xy + yz + zx + (1.122 − 0.617i) x2 x3 x1 , × tan−1 + tan−1 + tan−1 p4
p5
p6
(60b)
S.B. Bhardwaj et al. / Computers and Mathematics with Applications 68 (2014) 2068–2079
2079
if a, b, c, d, e and f are real, then Eqs. (49)–(51) yield the solutions Er(0) = −6,
(0)
= 0,
Ei
(61a) −1
1 1 b+ + ib− y2 exp − a+ + ia− x2 −
ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )] 2 2 2 1 x x x3 a 1 2 i − c+ + ic− z 2 + (1 + i) xy + yz + zx + i tan−1 + tan−1 + tan−1 . 2
3
p4
p5
p6
(61b)
Special case: If inverse harmonic term is absent in the potential (37) i.e. A3 = B3 = C3 = 0, then employing Eq. (57) in Eqs. (53a)–(54), the ground state solutions reduce to Er(0) = 2.441,
(0)
Ei
= 13.57,
(62a) (−0.583+1.739i) 4.619
exp −0.817(1 + i/2)(x2 + y2 + z 2 ) ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )] x2 x3 x1 +1.155(1 + i) xy + yz + zx + (−0.583 + 1.739i) tan−1 + tan−1 + tan−1 p4
p5
p6
(62b)
if a, b, c, d, e and f are real then Eqs. (55a)–(56) give rise to Er(0) = 3.972,
(0)
Ei
= 0,
(63a)
ψ (0) (x, y, z ) = [(x21 + p24 )(x22 + p25 )(x23 + p26 )](−0.5) exp −0.817(1 + i/2)(x2 + y2 + z 2 ) x2 x3 x1 + tan−1 + tan−1 . + 1.155(1 + i) xy + yz + zx + i tan−1 p4
p5
p6
(63b)
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