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Hamiltonian hierarchy of pseudo-potentials
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IOP PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. 76 (2007) 689–692
doi:10.1088/0031-8949/76/6/018
Hamiltonian hierarchy of pseudo-potentials Rita de Cassia dos Anjos1 , Elso Drigo Filho1,3 and Regina Maria Ricotta2 1
UNESP, Universidade Estadual Paulista—IBILCE Rua Cristovão Colombo, 2265—15054-000 São José do Rio Preto-SP, Brazil 2 Faculdade de Tecnologia de São Paulo, FATEC/SP-CEETEPS-UNESP Praça Fernando Prestes, 30—01124-060 São Paulo-SP, Brazil E-mail:
[email protected] and
[email protected]
Received 2 July 2007 Accepted for publication 16 October 2007 Published 8 November 2007 Online at stacks.iop.org/PhysScr/76/689 Abstract The general structure of the Hamiltonian hierarchy of the pseudo-Coulomb and pseudo-Harmonic potentials is constructed by the factorization method within the supersymmetric quantum mechanics (SQMS) formalism. The excited states and spectra of eigenfunctions of the potentials are obtained through the generation of the members of the hierarchy. It is shown that the extra centrifugal term added to the Coulomb and Harmonic potentials maintain their exact solvability. PACS numbers: 3.65.Fd, 11.30.Pb
1. Introduction The Hamiltonian hierarchy of potentials has been extensively analysed since the pioneering work of Sukumar [1] on supersymmetric quantum mechanics [2]. It is an algebraic structure that introduces a formalism which allows us to solve the Schrödinger equation, not only for the exactly solvable potentials but for the partially solvable ones [3–7], closely related to the factorization method [8]. Recently, there has been a renewed interest and an expressive amount of work on pseudo-potentials [9–15]. Pseudo-potentials are derived from well-known potentials by the addition of an extra centrifugal term to the potential, revealing, for instance, a problem with an effective angular momentum. The modification of the original potential is such that the exact solvability is still maintained. In particular, the peculiarity stays in the fact that these new terms introduce new parameters in the theory, which are useful for applications in quantum chemistry and atomic and molecular physics [16–24]. In this paper, we evaluate the Hamiltonian hierarchy [1], for the pseudo-Coulomb potential, also known as Kratzer potential [25], and for the pseudo-harmonic potentials, in three dimensions. The first few members of the superfamily are explicitly evaluated and a general form for the superpotential is proposed by induction. We also evaluate 3
Work is partially supported by CNPq.
0031-8949/07/060689+04$30.00
the first excited states of the original potentials and consider a special case where we recover the results known in the literature [15, 23].
2. Pseudo-Coulomb potential—general case The pseudo-Coulomb potential is given, in atomic units, as follows b a2 VPC = c − + 2 (1) r r and the factorized radial Hamiltonian is given by (1) H1 = A+1 A− 1 + E0 = −
d2 + V1 (r ) dr 2
(2)
2
h¯ written in 2m units. V1 (r ) is the potential of the first Hamiltonian of the hierarchy, V1 (r ) = VPC + l(l + 1)/r 2 . The term proportional to 1/r 2 is the potential barrier term. As a result of the factorization of the Hamiltonian, the bosonic operators are given in terms of the superpotential,
A± 1 =∓
d + W1 (r ), dr
(3)
which solves the Riccati equation W12 (r ) −
d W1 (r ) = V1 (r ) − E 0(1) , dr
(4)
where E 0(1) is the ground state energy. © 2007 The Royal Swedish Academy of Sciences
Printed in the UK
689
Dr R C Anjos et al
In order to simplify our notation, we rescale the angular momentum l to a new constant γ , which satisfies the equation a 2 + l(l + 1) = γ (γ + 1),
Again, substituting this superpotential into the related Riccati equation we arrive at the following constraints for α1 and β1 , α1 = γ + 2,
(5)
whose solution for γ is (6)
E 0(2) = c − β12 = c −
Writing the superpotential as W1 (r ) = β −
α r
(7)
α = γ + 1,
W2 (r ) =
β=
b . 2(γ + 1)
(8)
E 0(1) = c − β 2 = c −
b2 . 4(γ + 1)2
0
Thus, with the superpotential given by (10), we obtain a convergent ground state wavefunction given by r
.
(12)
The supersymmetric partner of H1 is (1) + H2 = A− 1 A1 + E 0 = −
d2 + V2 (r ), dr 2
The ground state wavefunction is Z (2) 90 (r ) = N exp −
b (γ + 1)(γ + 2) + . r r2
W3 (r ) =
.
(21)
(14)
(15)
b γ +3 − , 2(γ + 3) r
0
= N e−[b/2(γ +3)]r r γ +3 .
where E 0(2) is its ground state energy. In analogy to the first member of the hierarchy, the new superpotential W2 can be written as α1 W2 (r ) = β1 − . (17) r
(24)
From the results obtained so far, we can conceive by induction of the following form for the superpotential Wn+1 =
γ +n+1 b − , 2(γ + n + 1) r
(25)
or given in terms of the potentials Vn+1 = c −
b (γ + n)(γ + n + 1) + , r r2
(26)
with ground state wavefunctions 90(n+1) (r ) = N e−[b/2(γ +n+1)]r r γ +n+1 .
(27)
The corresponding ground state energy E 0(n+1) = c −
(16)
(23)
for the superpotential. The ground state wavefunction is Z r (3) 90 (r ) = N exp − W3 (¯r ) d¯r
The factorized radial Hamiltonian is given by (2) H2 = A+2 A− 2 + E0 ,
r
for the energy and
Substituting the superpotential W1 , we obtain the potential V2 V2 = c −
r
(20)
Following the same line of reasoning, we evaluate the next member of the hierarchy, H3 . The results obtained for the energy are b2 E 0(3) = c − (22) 4(γ + 3)2
(13)
where V2 (r ) is given by the supersymmetric quantum mechanics (SQM) results, d W1 (r ) = V2 (r ) − E 0(1) . dr
b γ +2 − . 2(γ + 2) r
=Ne
From the well-known results of supersymmetric quantum mechanics, we can evaluate a ground state wavefunction from the following equation Z r (1) 90 (r ) = N exp − W1 (¯r ) d¯r . (11)
= Ne
(19)
(9)
In terms of the parameters, the superpotential (7) is rewritten as b γ +1 W1 (r ) = − . (10) 2(γ + 1) r
−[b/2(γ +1)]r γ +1
b2 4(γ + 2)2
W2 (¯r ) d¯r 0 −[b/2(γ +2)]r γ +2
As a result, the energy is given by
690
(18)
and the superpotential, in terms of these constants becomes
and substituting it into equation (4), we arrive at
W12 (r ) +
b . 2(γ + 2)
The energy is then given by
p γ = − 12 + 12 1 + 4(a 2 + l(l + 1)).
90(1) (r )
β1 =
b2 . 4(γ + n + 1)2
(28)
The superalgebra also allows us to obtain the spectra H1 E n(1) = E 0(n+1) ,
9n(1) (r ) = A+1 A+2 . . . A+n 90(n+1) (r ), (29)
where A± n =∓
d + Wn (r ). dr
(30)
Hamiltonian hierarchy of pseudo-potentials
Thus, the first excited state is d 91(1) (r ) = A+1 90(2) (r ) = − + W1 (r ) e−[b/2(γ +2)]r r γ +2 dr 1 b = (2γ + 3) − + e−[b/2(γ +2)]r r γ +2 . r 2(γ + 1)(γ + 2)
The other excited states are calculated analogously to the previous case. The results quoted above are in perfect agreement with others found in the literature [15, 23].
4. The pseudo-Harmonic oscillator (31)
a 2 a 2 VPH = br + = 2 + 2ab + b2r 2 . r r
The second excited state then is given by 92(1) (r )
The potential for this case is given by
A+1 A+2 90(3) (r )
= d d = − + W1 (r ) − + W2 (r ) e−[b/2(γ +3)]r r γ +3 dr dr 1 1 b = − 2 + (2γ + 4) − + r r 2(γ + 2)(γ + 3) b 1 (2γ + 5)e−[b/2(γ +3)]r r γ +3 . × − + r 2(γ + 1)(γ + 3) (32)
The other excited states are straightforwardly calculated, through the systematic use of equation (29).
The factorized radial Hamiltonian H1 , first member of the hierarchy, is given by (1) H1 = A+1 A− 1 + E0 = −
d2 + V1 (r ), dr 2
At this point it is interesting to consider the usual form of the pseudo-Coulomb potential [15, 23]. This case is a particular case of potential (1) and is obtained when we set c → b2 ,
b → 2ab,
(33)
for which the pseudo-Coulomb potential becomes, in atomic units, a 2 2ab a 2 VPC = b − = b2 − + 2. (34) r r r It can be readily checked that the hierarchy of the general case, equation (25), reduces to Wn+1 =
ab γ +n+1 − , γ +n+1 r
90(n+1) (r ) = N e−[ab/γ +n+1]r r γ +n+1 , with ground state energies given by a2 E 0(n+1) = b2 1 − . (γ + n + 1)2
(35) (36)
(37)
The spectra of H1 are evaluated by successive use of equation (29). Thus the first excited state is d 91(1) (r ) = A+1 90(2) (r ) = − + W1 (r ) e−[ab/γ +2]r r γ +2 dr 1 ab = (2γ + 3) − + e−[ab/γ +2]r r γ +2 . (38) r (γ + 1)(γ + 2) The second excited state then is given by 92(1) (r )
=
A+1 A+2 90(3) (r )
d d − + W2 (r ) e−[ab/γ +3]r r γ +3 = − + W1 (r ) dr dr 1 ab = (2γ + 5) − + e−[ab/γ +3]r r γ +3 . r (γ + 1)(γ + 2) (39)
(41)
where V1 (r ) incorporates the potential barrier term, V1 (r ) = V P H + l(l + 1)/r 2 . Once again, in order to simplify our notation, we rescale the angular momentum l to a new constant γ , as in (5), whose solution for γ is exactly like (6). Writing the superpotential as α r and substituting it into equation (4), we arrive at W1 (r ) = βr −
3. Pseudo-Coulomb potential—the particular case
(40)
α = γ + 1,
(42)
β = b.
(43)
E 0(1) = 2ab + 2γ b + 3b.
(44)
As a result, the energy is given by
In terms of the parameters, the superpotential (42) is rewritten as γ +1 W1 (r ) = br − . (45) r With the above superpotential we can evaluate, through the equation (11), a convergent ground state wavefunction. It is given by 90(1) (r ) = N e−[ab/γ +1]r r γ +1 . (46) The supersymmetric partner of H1 is (1) + H2 = A− 1 A1 + E 0 = −
d2 + V2 (r ), dr 2
(47)
where V2 (r ), as in (13), satisfies W12 (r ) +
d W1 (r ) = V2 (r ) − E 0(1) . dr
(48)
Substituting the superpotential W1 , we obtain the potential V2 V2 = b2r 2 +
(γ + 1)(γ + 2) + 2b + 2ab. r2
(49)
The corresponding factorized Hamiltonian is given by (2) H2 = A+2 A− 2 + E0 ,
(50)
where E 0(2) is its ground state energy. Again, in analogy to the first member of the hierarchy, the new superpotential W2 can be written as α1 W2 (r ) = β1r − . (51) r 691
Dr R C Anjos et al
Thus, substituting back this superpotential into the related Riccati equation, we arrive at the following constraints for α1 and β1 , α1 = γ + 2, β1 = b. (52)
Again, other excited states are systematically calculated, using (29).
The energy is then given by
5. Conclusions
E 0(2)
= 2ab + 7b + 2bγ
(53)
(56)
In this work the Hamiltonian hierarchy of pseudo-Coulomb and pseudo-Harmonic potentials were evaluated. For both cases, the first few members of the superfamily were explicitly evaluated and a general form for the superpotential was proposed by induction. The introduction of a centrifugal term, important for studies of molecular systems, did not harm the exact solvability of the potential. Through the superalgebra we showed that the whole superfamily is a collection of exactly solvable Hamiltonians. The important point to remark on is the fact that the method allows us to evaluate the spectra of energy and eigenfunctions of the original potential in a systematic way, as we did for the first excited states of the pseudo-Coulomb and pseudo-Harmonic potentials, recovering already known results given by other methods [15, 23].
(57)
References
and the superpotential, in terms of these constants becomes W2 (r ) = br −
γ +2 . r
The ground state wavefunction is Z 90(2) (r ) = N exp −
r
W2 (¯r ) d¯r
(54)
0
= N e−(b/2)r r γ +2 . 2
(55)
Repeating the same procedure, we can evaluate the third member of the hierarchy and arrive at the following results for the energy, E 0(3) = 2ab + 11b + 2γ b for the superpotential, γ +3 , r and for the ground state wavefunction W3 (r ) = br −
90(3) (r )
= N e−(b/2)r r γ +3 . 2
(58)
From the above results a general form for the hierarchy of this potential can be conceived by induction. For the (n + 1)th member the superpotential is Wn+1 = br −
γ +n+1 , r
(59)
the ground state wavefunction is 90(n+1) (r ) = N e−(b/2)r r γ +n+1 , 2
(60)
and the ground state energy is E 0(n+1) = 2ab + 2γ b + (4n + 3)b,
(61)
where γ is defined by (6). From equations (29) and (59), we can calculate the spectra of H1 . Thus, the first excited state is 91(1) (r ) = A+1 90(2) (r ) d 2 = − + W1 (r ) e−(b/2)r r γ +2 dr (2γ + 3) −(b/2)r 2 γ +3 = 2b − e r . r2
(62)
The second excited state then is given by d 92(1) (r ) = A+1 A+2 90(3) (r ) = − + W1 (r ) dr d 2 × − + W2 (r ) e−(b/2)r r γ +3 dr (2γ + 5)(2γ + 3) −(b/2)r 2 γ +3 e r . = −2b(4γ + 10) + 4b2r 2 + r2 (63) 692
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