Download The Hierarchy of Hamiltonians for a Restricted Class of Natanzon

334
Brazilian Journal of Physics, vol. 31, no. 2, June, 2001
The Hierarchy of Hamiltonians for a Restricted
Class of Natanzon Potentials
Elso Drigo Filhoa and Regina Maria Ricotta b
a Instituto de Bioci^
encias, Letras e Ci^encias Exatas, IBILCE-UNESP
Rua Cristov~ao Colombo, 2265, 15054-000 S~ao Jose do Rio Preto, SP, Brazil
b Faculdade de Tecnologia de S~
ao Paulo, FATEC/SP-CEETPS-UNESP
Praca Fernando Prestes, 30, 01124-060, S~ao Paulo, SP, Brazil
Received on 10 November, 2000. Revised version received on 10 January, 2001
The restricted class of Natanzon potentials with two free parameters is studied within the context
of Supersymmetric Quantum Mechanics. The hierarchy of Hamiltonians and a general form for the
superpotential is presented. The rst members of the superfamily are explicitly evaluated.
I Introduction
The classes of Natanzon potentials, namely, the hypergeometric and the conuent, reer to potentials whose
Schrodinger equation is analytically and exactly solvable by means of hypergeometric functions. They have
motivated several works concerning the mathematical
and algebraic aspects of their structure and solutions
and have numerous applications in several branches of
physics, [1]-[7].
In particular, there have been studies within Supersymmetric Quantum Mechanics formalism. Cooper et
al, [6], for instance, investigated the relationship between shape invariance and exactly analytical solvable
potentials and showed that the Natanzon potential is
not shape invariant although it has analytical solutions
for the associated Schrodinger equation. Levai et al,
[7], have determined phase-equivalent potentials for a
class of Natanzon potentials employing the formalism
of supersymmetry.
However, the hierarchy of the Hamiltonians corresponding to Natanzon potentials has not been determined yet. In ref. [6], Cooper et al have sketched the
rst few potentials of the hierarchy from the knowledge
of their asymptotic behaviour from the series approximation. In this paper we construct the hierarchy of
Hamiltonians of the restricted class of Natanzon potentials, (Ginocchio class), with two free parameters.
The rst few members of the superfamily are explicitly
evaluated and a general form for the superportential is
Work
supported in part by CNPq
proposed by induction.
II
Supersymmetric Quantum
Mechanics Formalism
In the formalism of Supersymmetric Quantum Mechanics there are two operators Q and Q , that satisfy the
algebra
+
fQ; Qg = fQ
+
+
;Q
g = 0; fQ; Q g = HSS
+
(1)
where HSS is the supersymmetric Hamiltonian. The
usual realisation of the operators Q and Q is
0
0
Q=a =
A
0
+
1
= 00 A0
(2)
where are written in terms of the Pauli matrices
and A are bosonic operators. With this realisation
the supersymmetric Hamiltonian HSS is given by
A A
0
H
0
HSS =
0 A A = 0 H : (3)
where H are supersymmetric partner Hamiltonians
and share the same spectra, apart from the nondegenerate ground state. Using the super-algebra a given
+
Q
=a
+ +
1 +
+
+
+
335
Elso Drigo Filho e Regina Maria Ricotta
Hamiltonian can be factorized in terms of the bosonic
operators. In ~ = c = 1 units, it is given by
H1
d2
=
dr2
+ V (r) = A
+E
+
1 A1
1
The eigenfunction for the lowest state is related to the
superpotential W as
(4)
(1)
0
(r) = N exp(
(1)
0
where E is the lowest eigenvalue. The bosonic operators are dened by
(1)
0
=
drd + W (r)
A1
( )=
W1 r
where the superpotential W (r) satises the Riccati
equation
0
W
W = V (r)
E :
(6)
1
2
1
(7)
d
dr
( ):
(8)
(1)
0
ln
Now it is possible to construct the supersymmetric
partner Hamiltonian,
(1)
0
1
1
() )
W1 r dr
0
or conversely
(5)
1
Z r
c
H2
=A
1
+
A1
d2
+E =
(1)
0
dr2
+ (W + W 0 ) + E
2
1
1
(9)
(1)
0 :
If one factorizes H in terms of a new pair of bosonic operators, A one gets,
2
2
H2
=A
+
2 A2
d2
+E =
(2)
0
dr2
+ (W
2
2
0 ) + E (2)
W2
(10)
0
d
where E is the lowest eigenvalue of H and W satisfy
the Riccati equation,
0
W
W = V (r)
E :
(11)
Thus a whole hierarchy of Hamiltonians can be constructed , with simple relations connecting the eigenvalues and eigenfunctions of the n-members, [8]-[13]
n
Hn = An An + E
(12)
(2)
0
2
2
2
( )
0
+
=
An
drd + Wn (r)
+ +
1 A2 :::
(1)
(n+1)
0
(r) = N exp(
(1)
0
(2)
0
2
2
n = A
2
(1)
En
Z r
0
=E n
(14)
() )
(15)
( +1)
0
W1 r dr :
III Natanzon Potential and the
Hierarchy of Hamiltonians
The restricted class of Natanzon potentials having two
parameters and given in terms of the variable y(r) is,
(13)
c
( ) = f v(v + 1) + 1=4(1 )[5(1 )y (7 )y + 2]g(1 y ) ;
(16)
where the variable function y(r) satises dy=dr = (1 y )[1 (1 )y ]. The dimensionless free parameters v and
measure the depth and the shape of the potential, respectively.
The Schrodinger equation for this potential, [2], [3], in dimensionless units, is given by
V r
2
2
2
4
2
[
2
d =dr
2
2
2
2
2
2
+ V (r)]n (r) = n n (r)
(17)
where V (x) = v V (r), n = En =v and r = bx = (2mv =~ ) = x.
The analytic solutions for the energy eigenfunctions are given by,
0
0
n = (1
0
2
)n = [g(y)]
2
2 1 2
(2n +1)=4
Cn n
+1=2
(y=[g(y)] = )
1 2
(18)
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Brazilian Journal of Physics, vol. 31, no. 2, June, 2001
where g(y) = 1 (1 )y . The factor Cna (x) is a Gegenbauer polynomial when n is a non-negative integer,
which is our case. The corresponding energy eigenvalues are given by n = n ; n > 0, where
=
n = [ (v + 1=2) + (1 )(n + 1=2) ]
(n + 1=2):
(19)
Notice the relationship between the energy levels which will be extensively used in what follows,
(n n ) = ( 2n (2n + 1)n + (2n 1)n ):
(20)
2
( )
2
2
2
2
2
2
2
2
4
2 1 2
2
1
1
d
In order to construct the superfamily we rstly factorize the Natanzon potential, calling V (r) = V (r) =
V (r) + , [6]. The factorized Schr
odinger equation
is given by
H
= a a ; a = d=dr + W (r) (21)
where n = n . The superpotential W (r) is evaluated
from the knowledge of the ground state eigenfunction
of V (r) by using (8) with n = n , given by (18). It
satises the Riccati equation and it is given by
W (r) = (1 )y (y
1)=2 + y : (22)
(1)
0
(1)
0
1
The superpartner Hamiltonian satises the equation
1
+
1 1
1
1
(1)
H2
1
2
0
1
(23)
+
a1
2
2
W1
(1)
2
=a
which is written in terms of V (r) as
1
2
(1)
0
+ W 0 = V (r)
(24)
(1)
0
2
1
where V (r), the potential for the second member of the
hierarchy, is given by
2
c
( )=f
2 4
V2 r
0 + + 1=4(1
2
0
2
)[ 7(1
2
)y + (9 3
4
8 )y
2
0
2
2]g(1
2
y
2
):
(25)
d
To construct the next member of the superfamily, we
factorize the Schrodinger equation for V . It gives
H
= a a ; a = d=dr + W (r) (26)
where W (r) satises the associated Riccati equation,
0
W
W = V (r)
:
(27)
is the energy ground state of the potential V (r)
and it is such that = . The superpotential W
can be computed from the ground state wave function
. It is given by W (r) = drd log( ), where
=a
, i.e.,
W (r) = 3=2(1 )y (y
1)+ y drd log(f ) (28)
where
( ) =1+a
2
(2)
0
2
+
2 2
2
2
2
2
2
(2)
0
(2)
0
(2)
0
(2)
0
2
(1)
1
2
1
(1)
1
2
2
2
1
2
1
2
(29)
;
11
a11
= (
)
1:
1 2
0
(30)
The new superpartner of H is given by
2
(2)
0
2
11 y
and the coeÆcient a is given by
(2)
0
2
2
f1 y
W22
+ W 0 = V (r )
3
2
(31)
(2)
0
where V (r), the potential for the third member of the
hierarchy, is given by
3
c
( ) = f + + 1=4(1 ) 27(1 )y + (33 15 )y 6 +
+ 2af (gy()y) f1 + ( 9 + 6 2 )y + 8(1 )y g + 8a y (1 y )( fg((yy)) ) g(1
2
1
V3 r
11
1
4
1
2
2
2
2
1
2
2
4
2
2
4
2
2
11
2
2
2
1
(32)
y
2
):
337
Elso Drigo Filho e Regina Maria Ricotta
and g(y) = 1 (1
) . Thus, factorizing the Hamiltonian for this potential we have
H
= a a ; a = d=dr + W (r)
(33)
where W (r) satises the Riccati equation,
W
W 0 = V (r )
:
(34)
is the energy ground state of the potential V (r), with = = . Again, the superpotential W can be
computed from the ground state wave function , dened by W (r) = drd log( ), with = a a
. It
is given by
W = 5=2(1
)y (y
1) + y + drd log(f ) drd log(f )
(35)
where
f (y ) = 1 + a y + a y
(36)
with coeÆcients are given by
a = 2(
)
2
2 y 2
(3)
0
3
+
3 3
3
3
3
2
3
(3)
0
3
2
3
(3)
0
3
3
(3)
0
(3)
0
(2)
1
(1)
2
(3)
0
3
2
2
2
2
2
21
21
1
1
3
(3)
0
2
2
(4)
0
0
1
(4)
0
3
W4
= 7=2(1
2
4
2
1
4
22
2
2
0
1
2
0
2
0
2
0
2
)y(y
1) + y + drd log(f ) + drd log(f )
2
2
3
1
d
2
dr
where f and f are evaluated in (29) and (36) and f is set to
f (y ) = 1 + a y + a y + a y ;
with the coeÆcients given by
a = 3(
) + 2(
)
5
1
1
1
2
(1)
3
2
(1)
2
2
= 1 + (2 3 + 6 )=3 + ( 4 + 6 2 5 + 3 + 3 )=3:
For the next member of the superfamily, we show the result of the evaluation of the superpotential,
d
. It is given by
dr log ( ) with = a a a
a22
1
( )
()=
W4 r
(37)
log f3
3
3
31
= 10 +
+
a32
2
31
2
2
3
4
32
0
(38)
6
33
2
1
(6 25 + 15 6 + 22 )=3 +
(6 + 2
18 6 + 18 + 13 3 6 11 3 + 8 )=3
2
0
4
0
1
1
2
0
2
2
1
3
0
3
0
1
3
1
2
3
0
2
2
3
148
484 206 84 +
( 6 + 13 3 12 4 ) + +
15 45
45
5
42
1
394
497
59
16
+ 45 + 2 + 3 45 + 45 5 3 +
116 24 48 + 12 + 8 + 82 + 83 + 328
45
45
5
5
5
3
9
14 6 + 2 8 58 + 182 2 9
15
45
45
5
26 + 8 + 44 4 5
9
9
Casting all the results we have so far for the hierarchy, the following nth-term for the superpotential is induced
a33
= 10 +
2
0
0
6
Wn+1
1
2
2
0
1
2
0
2
1
3
= (n + 1=2)(1
2
0
2
1
3
)(
2 y y 2
1
2
0
3
0
0
3
2
2
2
0
( )=
n
X
i=0
ani y 2i
; an0
1
2
0
3
Qn
where fn (y) is a 2n-order polynomial of the form
fn y
1
3
1
3
2
3
2
0
2
1
3
3
1) + yn + drd log(
2
3
1
0
0
2
0
0
3
3
0
2
4
3
1
2
0
0
2
0
1
= 1:
i=0 fi
fn
1
)
; f0
=f =1
1
(39)
(40)
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Brazilian Journal of Physics, vol. 31, no. 2, June, 2001
We stress that since Wn is a superpotential it checks the Riccati equation,
+1
0
= Vn (r) n
where Vn (r) is the superpartner potential of Vn which satises
Wn + Wn0 = Vn (r)
n :
We have therefore a recursive relationship between Wn and Wn given by
2
Wn+1
Wn+1
(41)
( +1)
0
+1
+1
2
+1
(42)
0
+1
Wn0 +1
Wn2+1
where n =
0
= Wn + Wn0 + n
2
(43)
(n+1)
0
0
and n = n . After the substitutions we end up with the condition
2n(1 ) y (y 1) + (y 1) ((2n 1)n (2n + 1)n )(1 (1 )y ) 2n +
2n
1
( +1)
0
4
2 2 2
+
nX1
fi0
f
i=0 i
+fn0
n
nX1
2
2
fn0
1
n
1
4f
1
4
2
1
2
2
fn0
2 f + 2(1
n
2
1
2
)y(y
2
1) + 2y (n
2
2
fi00
f
i=0 i
2
1
1
1
+2
2
2
2
nX1
i=0
2
n
4n(1 )y(y 1) 2fn0 =fn + 2y (n + n )
(1 )y(y 1)(2n + 1) + 2y n + fn00 =fn
1 =fn
f 0 =fn
2
1
) +
(44)
1
2
0
( ffi ) + 2n(1
i
1 2
1
2
)(1 3y )
2
2
(n + n )
1
dy=dr
= 0:
where f 0 = df =dr and f 00 = d f =dr .
2
2
d
Therefore, fn can be determined from the knowledge of fn . In this way, the particular cases of n = 1,
n = 2 and n = 3 can be checked by inspection and the
resulting functions f , f and f perfectly agree with
equations (29) , (36) and (38) respectively. Notice that
the particular case when = 1 trivially reduces the nth term superpotential, equation (39) to Wn = yn ,
with n = v n, since all the f 's become 1 once all
the ain 's are checked to reduce to zero. The related
potentials of the hierarchy are then given by
Vn (r) = (v
n)(v
n + 1)=cosh r n = 0; 1; :::
(45)
This is known as the shape invariant Poschl-Teller (PT)
potential.
discussions about the exactly solvable potentials. In
ref. [8] there is a discussion about this subject which
has recently been extended in [14] concerning potentials
depending on n parameters . The Natanzon potential
is not shape invariant in the usual sense, as most of
the exactly solvable potentials are. However, for the
restricted class analised here, it was possible to obtain
a general form for the superpotential, as shown in the
previous section.
The Hulthen potential without the potential barrier
term is another example of an exactly solvable potential
which is not shape invariant, but for which it is possible
to determine a general expression for the superpotential
in the hierarchy, [13].
IV Conclusions
References
+1
1
2
3
+1
+1
2
The hierarchy of Hamiltonians is studied for the restricted class of Natanzon potentials, (Ginocchio class),
with two parameters and a general form for the superpotential is proposed. The superalgebra drives us to the
conclusion that the whole superfamily is a collection of
exactly solvable Hamiltonians. The case = 1 served
as a check of our formulae and was shown to reduce the
original potential to the Poschl-Teller (PT) potential,
known to be shape invariant.
As a nal remark, the shape invariance concept introduced by Gedenshtein, [12], has motivated several
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Codriansky, P. Cordero and Salamo , Nuovo Cimento
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Elso Drigo Filho e Regina Maria Ricotta
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