A recurrence relation of Riccati-type differential equations known in
supersymmetric quantum mechanics is investigated to find exactly solvable
potentials. Taking some simple {\it ans\"atze}, we find new classes of solvable
potentials as well as reproducing the known shape-invariant ones.Comment: 14 pages, Late

Cho, Kyung Hyun

Kim, Sang Pyo

Soh, Dong Sup

English

arXiv

A recurrence relation of Riccati-type differential equations known in supersymmetric quantum mechanics is investigated to find exactly solvable potentials. Taking some simple {\it ans\"atze}, we find new classes of solvable potentials as well as reproducing the known shape-invariant ones

Soh, D S

Cho, K H

Kim, S P

CERN Document Server

KNUTH-31
Solvable Potentials from Supersymmetric Quantum Mechanics
Dong Sup Soh and Kyung-Hyun Cho
Department of Physics
Chonbuk National University
Chonju, 560-756, Korea
Sang Pyo Kim
Department of Physics
Kunsan National University
Kunsan, 573-701, Korea
(July 27, 1995)
Abstract
A recurrence relation of Riccati-type dierential equations known in super-
symmetric quantum mechanics is investigated to nd exactly solvable poten-
tials. Taking some simple ansa¨tze, we nd new classes of solvable potentials
as well as reproducing the known shape-invariant ones.
( Submitted to Journal of Korean Physical Society )
1
It is generally dicult to solve exactly the eigenvalue problem of a (time-independent)
hamiltonian in quantum mechanics. Among the various methods developed for this pur-
pose, a remarkably simple but powerful one is a ladder operator technique, whose typical
example is a simple harmonic oscillator. If a hamiltonian has a discrete eigenvalue spectrum
bounded from below, the energy eigenstates should be labeled by integers and formal rasing
and lowering operators can be written in this basis. However, it does not provide a way to
nd the ladder operators explicitly for a given hamiltonian. A practical method of getting
ladder operators has been studied by several authors( [3]- [8]) using ideas of supersymmetric
quantum mechanics( [1], [2]) and a concept of shape-invariant potentials [3]. This approach
has (re)produced many exactly solvable potentials. In this paper, we investigate the recur-
rence relation of Riccati-type dierential equations arising in the supersymmetric quantum
mechanics directly taking some ansa¨tze. In addition to reproducing known shape-invariant
potentials concisely, new classes of solvable potentials are found.
Let us begin by summarizing some results of supersymmetric quantum mechanics which
are used for the present work(for a review, see [2] for example). Consider a family of one
dimensional hamiltonians
Hk =
p2
2
+ Vk(x) ; k = 0; 1; 2;    ; (1)
where p = −i @
@x
; which are connected by
Hk+1 = BkB
y
k − Ek(1) = B
y
k+1Bk+1 ; H0 = B
y
0B0 : (2)
Here Bk and B
y
k are dened using superpotentials k(x)
Bk =
1
p
2
f
@
@x
+ k(x)g ; B
y
k =
1
p
2
f−
@
@x
+ k(x)g ; (3)
where k(x) is a real function such that the ground state wavefunction which is proportional
to exp(−
R
k(x)dx) should be normalizable. The nth eigenvalue of Hk is denoted by Ek(n)
n = 0; 1; 2;    ; and a relation between eigenvalues is given by
Ek+1(n) = Ek(n+ 1)− Ek(1) = E0(n+ k + 1) −E0(k + 1) : (4)
2
Note that all the ground state energies are dened to be zero. The Ek(1) is assumed to be
positive and is also denoted by sk, standing for a level spacing between the kth and (k+1)th
eigenstate of the starting hamiltonian H0. It is worth noting that E0(n+ 1) =
Pn
k=0 sk ; n =
0; 1; 2;    and E0(0) = 0 : Denoting the nth eigenfunction of Hk as  k(n)(x), important
relations between the eigenfunctions follow from (2) as
 k+1(n)(x) = fEk(n+ 1)g
−1=2 Bk k(n+1)(x) ;
Byk  k+1(n)(x) = fEk(n+ 1)g
1=2 k(n+1)(x) : (5)
Note that Bk  k(0)(x) = 0 : A repeated use of this relation gives
By0B
y
1   B
y
k  k+1(n)(x) = fE0(n+ k + 1)E1(n+ k)    Ek(n+ 1)g
1=2  0(n+k+1)(x) : (6)
Putting n = 0, this equation can be used to get the excited-state wavefunctions of H0 from
the knowledge of the ground states of systems related as superpartners.
Now, putting Bk and B
y
k in (3) into (2), a recurrence relation of Riccati-type dierential
equations is obtained as
2Vk+1(x) = 
2
k+1(x)− 
0
k+1(x) = 
2
k(x) + 
0
k(x)− 2sk ; (7)
where the prime means a derivative with respect to x : To search for the possible family of
k(x) and sk satisfying the dierence dierential equation (7), it is convenient to put k(x)
and k+1(x) as
k(x) =
1
2f 0k(x)
f2skfk(x) + f
00
k (x)g ; k+1(x) =
1
2f 0k(x)
f2skfk(x)− f
00
k (x)g ; (8)
which solve (7) identically. It holds for non-zero sk only, and in general, there can be added a
free parameter k, say, inside the curly brackets in (8). Note that fk(x) is proportional to the
rst excited-state wavefunction divided by that of the ground state of the kth hamiltonian.
The consistency in the two equations of (8) requires
ff 0k(x)f
0
k+1(x)g
0 = 2fskfk(x)f
0
k+1(x) − sk+1f
0
k(x)fk+1(x)g : (9)
3
We further rewrite equations (8), (9) by putting fk(x) as following :
fk(x) = ~gk(x) exp (
Z
~hk(x)dx ) : (10)
Note that there is an ambiguity in the choice of ~hk(x), or equivalently in ~gk(x) : Adopting
gk(x) = ~gk(x) exp (
R
k(x)dx ) ; and hk(x) = ~hk(x)−k(x) ; instead of tilded functions, fk(x)
remains the same for arbitrary k(x). We remove this ambiguity by choosing k(x) such
that h0(x) = 0 ; and
fk(x) = gk(x) exp (
Z
hk(x)dx ) ; f
0
k(x) = f
0
0(x) exp (
Z
hk(x)dx ) : (11)
The consistency in this procedure requires for all k = 0; 1; 2;    ;
f 00(x) = g
0
k(x) + gk(x) hk(x) ; (12)
then the dierence dierential equation (7) is satised. Now, equation (9) becomes
f 000 (x) +
1
2
f 00(x) (hk(x) + hk+1(x) ) = skgk(x) − sk+1gk+1(x) : (13)
Solving this equation for gk(x) and inserting it into (12) yields
kf 0000 (x) + pk(x)f
00
0 (x) + ( qk(x) + sk − s0 ) f
0
0(x) − s0hk(x)f0(x) = 0 ; (14)
where
pk(x) = khk(x) + k(x) ; qk(x) = 
0
k(x) + hk(x)k(x) ; (15)
and
k(x) =
1
2
k−1X
j=0
( hj(x) + hj+1(x) ) ; 0(x) = 0 : (16)
Since (14) is identically satised for k = 0, it is equivalent to the dierence of (k+ 1)th and
kth expressions in (14). Thus, with the dierence denoted by yk  yk+1 − yk, we nally
obtain
f 0000 (x) + pk(x)f
00
0 (x) + ( qk(x) + sk ) f
0
0(x) − s0hk(x)f0(x) = 0 : (17)
4
It is equivalent to the dierence dierential equation (7), and we will try to nd f0(x) and
sk consistently by making simple ansa¨tze on hk(x) below. The superpotential in (8) is now
given as
k(x) =
1
f 00(x)
f s0f0(x) − ( k −
1
2
) f 000 (x) g +
1
2
hk(x) − k(x) : (18)
Note that an overall constant factor of f0(x) does not aect the superpotential.
Let us solve (17). The simplest ansatz on hk(x) may be putting hk(x) = 0 for all
k = 0; 1; 2;    : In this case, (17) becomes
f 0000 (x) + sk f
0
0(x) = 0 : (19)
This is solved consistently by putting
sk = s ; (20)
where s is a constant parameter. It is easy to nd f 00(x) in (19), and the superpotential is
given in (18) with hk(x) = 0 ; k(x) = 0 in this ansatz, that is,
k(x) =
1
f 00(x)
f s0f0(x) − ( k −
1
2
) f 000 (x) g : (21)
The eigenvalues are given by the common formula
Ek(n) = s0n+
1
2
sn(n − 1 + 2k) ; (22)
referring to (20) and (4). In the following, we list the resulting superpotentials, while the
potentials can be written down explicitly by using 2Vk(x) = 2k(x)− 
0
k(x) in (7). With
constant parameters b; d ; and putting
 =
q
jsj ; ak =
2( s0 + ks )− s
2
;
(1) s > 0 ; f 00(x) = sin (x+ d)
k(x) =
8><>>:
−ak cot (x+ d) + bsin (x+d) Eckart potential
ak+b
2
tan 
2
(x+ d) − ak−b
2
cot 
2
(x+ d) Po¨schl-Teller potential I
(23)
5
(2) s < 0 ; f 00(x) = c1 sinh (x+ d) + c2 cosh (x+ d) with constant parameters c1; c2 :
Thus, the superpotential in (21) is
k(x) =
ak ( c1 + c2 tanh (x+ d) )
c2 + c1 tanh (x+ d)
+
b
c1 sinh (x+ d) + c2 cosh (x + d)
: (24)
Some well-known potentials in the literature come out :
a. f 00(x) = sinh (x+ d) ( c1 = 1 ; c2 = 0 )
k(x) =
8><>>:
ak coth (x+ d) +
b
sinh (x+d)
Rosen-Morse potential
ak−b
2
tanh 
2
(x+ d) + ak+b
2
coth 
2
(x+ d) Po¨schl-Teller potential II
(25)
b. Morse potential ; f 00(x) = cosh (x+ d) ( c1 = 0 ; c2 = 1 )
k(x) = ak tanh (x+ d) +
b
cosh (x + d)
: (26)
c. Morse potential ; f 00(x) = exp (x+ d) ; ( c1 = c2 =
1
2
) [ c1 = −c2 = −
1
2
also gives a
similar potential.]
k(x) = ak + b exp (−(x+ d) ) : (27)
(3) s = 0 ; two cases arise as following.
a. Harmonic oscillator potential ; f 00(x) = 1
k(x) = s0 (x+ d ) : (28)
b. Harmonic plus inverse-square potential ; f 00(x) = x+ d
k(x) =
s0
2
(x+ d) −
k
x+ d
; k = k −
1
2
+ b : (29)
It should be noted that k has a maximum allowed value when s < 0 since sk = s0 + ks is
assumed to be positive. It is remarkable that a large part of the known exactly solvable
6
potentials given in refs. [4], [5] are reproduced here within a single ansatz. Further details
of the potentials and the wavefunctions can be found in refs. [2], [4], [5], [7] where a notion
of shape-invariance of potentials is utilized in the framework of supersymmetric quantum
mechanics.
Let us solve (17) with an ansatz which is slightly more general than assuming hk(x) = 0
for all k ;
hk(x) =
8><>>:
0 for k = even
h(x) for k = odd :
(30)
Using for k = even, odd integers respectively,
hk(x) = h(x) ; pk(x) = h(x) f1 (k +
1
2
) g ; qk(x) =
h0(x)
2
+
h2
4
f1 (2k + 1) g ;
the equation (17) reduces to
f 0000 (x) +
3
2
h(x)f 000 (x) + f
h0(x)
2
+
h2(x)
2
+ s0 g f
0
0(x) − s0h(x)f0(x) = 0 ;
f 0000 (x) −
1
2
h(x)f 000 (x) + f
h0(x)
2
−
h2(x)
2
+ s1 g f
0
0(x) + s0h(x)f0(x) = 0 ; (31)
for k = 0; 1 and
f 000 (x) + f
h(x)
2
+
w
2h(x)
g f 00(x) = 0 ; (32)
which results from the consistency of (17) for all k = 2; 3; 4; 5;    ; together with sk
determined with a constant w as
s2k = s0 + kw ; s2k+1 = s1 − kw : (33)
Combining equations (31) and (32), one nally arrives at
h0(x) +
1
2
h2(x) (
1

+ 1 ) +  = 0 ; 2f 00(x) = h(x)f0(x) ; w = 2 ; (34)
where ;  are
 = s0 + s1 ;  =
s0 −s1 − w
4s0
: (35)
7
We consider  6= 0, since otherwise h(x) = 0 in (34), which has been already covered. The
energy level spacings are now given as
s2k = s0 + k ; s2k+1 = ( 1 + 2 ) f s0 + (k +
1
2
)g : (36)
Here,  > −1=2 for positive level spacings. From this and (4), the energy levels are given as
Ek(n) = (1 + )n f s0 +

4
(n− 1 + 2k) g
− (−1)k  f
1− (−1)n
2
( s0 +

4
(2k − 1) ) −
n
4
(−1)n g : (37)
This formula reduces to (22) as 0 ; by noting in (36), sk = =2 in this case. The
superpotential in (18) is given nally using (34)
k(x) =
2
h(x)
( s0 −

4
+
k
2
) − (−1)k
h(x)
4
: (38)
Now, we only need to nd h(x) satisfying (34). The resulting superpotentials are listed
in the following with a constant d ; and putting
 =
s
jj ( 1 +  )
2
; bk = ( 1 +  ) ( s0 −

4
+
k
2
) ; k =
 (−1)k
2( 1 +  )
:
(1)  > 0 ; h(x) = 2
1+
cot  (x+ d)
k(x) = bk
tan  (x+ d)

− k  cot  (x+ d) : (39)
(2)  < 0 ; h(x) = 2
1+
coth  (x+ d)
k(x) = bk
tanh  (x+ d)

− k  coth  (x+ d) : (40)
(3)  = 0 ; h(x) = 2
1+
1
x+d
k(x) = s0 (1 + ) (x+ d) −
k
x+ d
: (41)
8
These potentials are similar but dierent from the ones obtained by our rst ansatz, that
is, Po¨schl-Teller potentials I, II given in (23), (25) respectively, and (29). This fact is clear
in view of their dierent energy-level structures and the wavefunctions. For example, the
potentials corresponding to the superpotentials in (29), (41) are given respectively (putting
d = 0 for simplicity)
Vk(x) =
s20
8
x2 +
2k − k
2x2
−
s0
2
(k +
1
2
) ; (42)
Vk(x) =
1
2
s20(1 + )
2 x2 +
2k − k
2x2
−
s0
2
( 1 +  + (−1)k ) : (43)
The energy level spacings are sk = s0 for all k, for the potentials (42), while sk = s0 ; s0 ( 1+
2 ) for k = even, odd integers respectively for (43). The dierence between the two systems
can be seen more clearly in their wavefunctions. The unnormalized wavefunctions for V0(x)
given in (42) are
 0(k)(x) = x
0 exp(−
1
4
s0x
2 )L
0−
1
2
k (
s0x
2
2
) ; (44)
while for V0(x) given in (43)
 0(2k)(x) = x
0exp(−
1
2
s0(1 + )x
2 )L
0−
1
2
k ( s0(1 + )x
2 ) ;
 0(2k+1)(x) = x
1−0 exp(−
1
2
s0(1 + )x
2 )L
1
2
−0
k ( s0(1 + )x
2 ) ; (45)
where Lan(z) is a generalized Laguerre polynomial. Note that as 0, the potentials in (43),
the wavefunctions in (45) and the energy levels all reduce to those of the simple harmonic
oscillator. In contrast, if the parameter is chosen such that the potentials in (42) become
quadratic, only ‘half’ of the energy levels and wavefunctions of a corresponding harmonic
oscillator are obtained as can be seen in (44). We remark that the wavefunctions in (45)
are normalizable as long as  > −1=2, although their rst derivatives are not continuous at
x = 0 : The point is that another class of potentials like in (43) is possible as well as the well-
known ones like in (42) if we seek supersymmetry-related potentials possessing normalizable
wavefunctions.
9
How can the (radial) Coulomb potential consisting of 1=x and 1=x2 terms be reproduced
in our approach? Since we need hk(x) as inputs for solving (17), we assume hk(x) to be that
of Coulomb potential
hk(x) = g (
1
k + a
−
1
k + 1 + a
) −
g
a(a+ 1)
; (46)
where g; a are constant parameters. The derivation of this will be described later together
with other general idea of getting superpotentials. Then (17) becomes for k = 0 ;
f 0000 (x) +
3h1
2
f 000 (x) + f ( +
g(2a+ 1)
a(a+ 1)
)h1 +
g2
a2(a+ 1)2
+ b g f 00(x) − s0h1 f0(x) = 0 ;
(47)
where h1 = −2g=f a(a+ 1)(a+ 2) g ; and
(2a+ 1) f 000 (x) − f 2 +
2g(2a+ 1)
a(a+ 1)
g f 00(x) + 2s0 f0(x) = 0 ; (48)
which results from the consistency of (17) for k = 1; 2; 3;    : Here, sk is determined as
sk =
g2
2
(
1
(k + a)2
−
1
(k + 1 + a)2
) + hk + bk + c ; (49)
with another constants ; b; c : Combining (47) and (48), interesting cases arise only when
 = 0 ; c = ( a+
1
2
) b ; a 6= −
1
2
; (50)
and the nal equation is
f 000 (x) − 2B f
0
0(x) + (B
2 + b ) f0(x) = 0 ; (51)
where
B =
g
a(a+ 1)
:
Now, energy levels are obtained from above and (4) as
Ek(n) =
g2
2
f
1
(k + a)2
−
1
(k + n+ a)2
g + b f
n2
2
+ (k + a)n g : (52)
The superpotentials are obtained using (18), (51)
10
k(x) = ( k + a )F (x) +
g
k + a
; F (x) = (B2 + b)
f0(x)
f 00(x)
− B ; (53)
where f0(x) is easily obtained in (51). We list the resulting superpotentials in the following
with a constant d (to be chosen suitably if necessary), and putting
 =
q
jbj :
(1) b > 0 ; f0(x) = sin (x+ d) exp (Bx )
k(x) = −(k + a)  cot x +
g
k + a
: (54)
(2) b < 0 ; f0(x) = ( c1 sinh x + c2 cosh x ) exp (Bx ) ; with appropriate choices of
c1; c2 ; there result two well-known potentials
a. Eckart potential
k(x) = −(k + a)  coth x +
g
k + a
: (55)
b. Rosen-Morse potential
k(x) = −(k + a)  tanh x +
g
k + a
: (56)
(3) b = 0 ; Coulomb potential ; f0(x) = (x+ d) exp (Bx )
k(x) = −
k + a
x
+
g
k + a
: (57)
It is satisfying to see that a single ansatz reflecting the common structure of superpoten-
tials as in (53) produces a class of potentials in addition to the starting one, the Coulomb
potential in this case. The above potentials and the results of our rst ansatz given in (23)-
(29), constitute the whole shape-invariant potentials given in refs. [4], [5]. (There are some
discrepancies compared with the tables presented in these references.)
Finally, we want to mention another way of getting superpotentials as following. Since
it is generally expected that the ground state wavefunctions are products of two factors, or
11
equivalently k(x) is a sum of two parts, one may try to nd new solutions to the dierence
dierential equation (7) by adding an appropriately chosen second part to an old one which
is assumed to satisfy it. As a simple ansatz to implement this idea, we take
k(x) = ak F (x) + bk G(x) : (58)
Here, simple forms are taken for the two parts with constants ak; bk and k-independent
functions F (x); G(x) ; and akF (x) is assumed to solve (7). Then inserting (58) into (7)
results in
(ak + ak+1) F
0(x) + (a2k − a
2
k+1) F
2(x) = 2s(a)k ; (59)
(bk + bk+1) G
0(x) + 2 (akbk − ak+1bk+1) F (x)G(x) + (b
2
k − b
2
k+1) G
2(x) = 2s(b)k ; (60)
together with a condition
sk = s
(a)
k + s
(b)
k : (61)
Recall that sk should be positive denite, while s
(a)
k ; s
(b)
k need not be so. In fact, all the
results obtained so far can also be obtained within this simple ansatz as may be clear from
the forms of the obtained superpotentials, (21), (38), (53). We illustrate the procedure by
obtaining the Coulomb potential : A solution to (59) corresponding to s(a)k = 0 is directly
obtained to be proportional to 1=x, ak being determined correspondingly. Next, note that
(60) is consistently solved by taking akbk to be independent of k and G(x) to be a constant,
say g : Then one may get
k(x) = −
k + a
x
+
g
k + a
; Vk(x) = −
g
x
+
(k + a)(k + a− 1)
2x2
+
g2
2(k + a)2
:
One can nd hk(x) by taking the dierence of (18),
k(x) = −
f 000 (x)
f 00(x)
− hk(x) ; (62)
which can be used to get the f0(x) and hk(x) for the known superpotentials by recalling
h0(x) = 0 .
12
In summary, we have investigated the recurrence relation of Riccati-type dierential
equations in supersymmetric quantum mechanics. Recasting the problem into (17) and
taking some simple ansa¨tze on hk(x), we reproduced the known shape-invariant potentials
in refs. [4], [5], and obtained another classes of potentials which look similar but are dierent
from the known ones, as illustrated in (42)-(45). It should be stressed that dierent types
of potentials can be obtained altogether within a common ansatz on the superpotential as
shown in the text. By using more intricate ansa¨tze or introducing some new ideas into the
formalism, we expect that the method presented in this paper will be helpful for nding out
more solvable potentials, as well as reproducing the known ones, e.g., Natanzon potentials(
[9], [7], [2]).
ACKNOWLEDGMENTS
K. H. Cho is grateful to S. U. Park for helpful comments. The present studies were
supported in part by the Basic Science Research Institute Program, Ministry of Education,
1995, Project No BSRI-95-2434, and by the Korea Science and Engineering Foundation
under Grant No 951-0207-056-2.
13
REFERENCES
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[2] F Cooper F, A Khare A and U P Sukhatme, preprint LA-UR-94-569, hep-th/9405029, Super-
symmetry and Quantum Mechanics (1994)
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Solvable potentials from supersymmetric quantum mechanics

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.266.3893

Solvable Potentials from Supersymmetric Quantum Mechanics

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