We investigate two methods of obtaining exactly solvable potentials with
analytic forms.Comment: 13 pages, Latex, to appear in Chineses Journal of Physic

Chang, Darwin

Chang, We-Fu

English

arXiv

We investigate two methods of obtaining exactly solvable potentials with analytic forms

Chang, D

Chang, W F

CERN Document Server

NHCU{HEP{95{07
On Exactly Solvable Potentials
Darwin Chang(1;2) and We-Fu Chang(1)
(1)Physics Department, National Tsing-Hua University, Hsinchu, Taiwan
(2)Institute of Physics, Academia Sinica, Taipei, Taiwan
Abstract
We investigate two methods of obtaining exactly solvable potentials with
analytic forms.
PACS numbers:03.65.Ge,11.30.Pb
There are two methods of obtaining exactly solvable potentials in quantum me-
chanics. The rst method was developed by applying the technique of supersymmetry
(SUSY) to the Schro¨dinger equation and obtain two potentials with almost identical
spectra. The two potentials can be considered to be superpartners of each other.
It has been shown [1, 2] that if the two partners happen to be related by a simple
relationship called shape invariance, the energy eigenvalues of the potential can be
solved exactly. All the known solvable potentials with closed analytic forms can be
shown to be shape invariant. In the literature, there are other solvable potentials
that have not been shown to be shape invariant, however, they exist only complex
numerical forms which we shall not consider in this article.
A second interesting method of obtaining solvable potential was proposed by
Klein and Li[3] based on some special quantum commutation relationships. Li[4] has
recently worked out the most general potential that can be obtained this way.
In this paper, we rst investigate the relationship between the two approaches.
We will show that the general solutions obtained by Li are special cases of the general
solutions that can be obtained by solving shape invariance condition. Therefore, the
solving shape invariance condition remains the most general method of obtaining
exactly solvable potentials which can be given in analytic form. Unfortunately, there
is no general method for getting analytic solutions of shape invariance condition.
The best one can do seems to be starting from a guessing ansatz with an unknown
function. The shape invariance is then enforced by demanding that the function
satises an ordinary dierential equation. The ansatz allows one to turn the dicult
shape invariance condition into a problem of solving dierential equation. It also
allows one to associate each ansatz as dening a particular class of solutions. While
the solutions obtained by Li correspond to those dened by a particular ansatz,
in the literature, the largest classes of analytic solutions of the shape invariance
condition was provided by Gendenshtein [1]. He proposed three ansatzs which dene
three classes of solutions which seem to cover all the known analytic exactly solvable
potentials. We, therefore, proceed to solve the dierential equations corresponding
to these ansatzs and the energy eigenvalues of these three classes of solutions of
shape invariance condition. In the process, we also demonstrate that all the known
2
solutions in the literature are special cases of the one provided by Gendenshtein.
The three ansatzs of Gendenshtein and the forth one corresponding to Li’s so-
lutions also demonstrated that each potential can be simultaneously represented in
many classes. It is also clear that one can in principle continue to invent more ansatz
without obtaining any new potentials. Therefore, one is immediately faced with the
intriguing question of what is the better way to classify these solvable potentials
than using the ansatzs. One method seems to be starting from the n dependence
of its spectrum. It is noted that only a few simple general forms of n dependence
are allowed for the known exactly solvable potentials. We shall make some com-
ments in this direction later and mention some interesting unsolved problems in the
conclusion.
We rst briefly review the procedure of constructing creation and annihilation op-
erators of simple harmonic oscillator, now we can apply this method to the Schro¨dinger
equation with an arbitrary potential V (x).
(−
1
2
d2
dx2
+ V (x)) = E (1)
Assuming that we have already shifted the potential by a constant so that the
ground state energy becomes zero, we shall denote this Hamiltonian and its potential
with (−) subscript and denote  (−)n as its eigenfunctions. That is, V−(x)  V (x),
 
(−)
0 (x)   0(x), E
(−)
0 = 0, with
H− 
(−)
0 = (−
1
2
d2
dx2
+ V−(x)) 
(−)
0 = 0: (2)
The above equation, Eq.(2) is identical to
H− =
1
2
(−
d2
dx2
+
 
(−)00
0
 
(−)
0
): (3)
Then, the general creation and annihilation operators are:
A+ =
1
p
2
(−
d
dx
−
 
(−)0
0
 
(−)
0
);
A =
1
p
2
(
d
dx
−
 
(−)0
0
 
(−)
0
): (4)
The Hamiltonian can be written as H− = A+A. Now we dene a new Hamiltonian
H+ = AA+. which can be written as H+  −12
d2
dx2
+ V+(x): The corresponding
3
potential in the new Hamiltonian is:
V+ = −V− + (
 
(−)0
0
 
(−)
0
)2 = V− −
d2
dx2
ln (−)0 : (5)
The V+, V− are called supersymmetric partner potentials.
One can dene a new function, W (x), called superpotential,
W (x) = −
1
p
2
 
(−)0
0
 
(−)
0
: (6)
Solving the dierential equation we get
 
(−)
0 (x) = exp(−
p
2
Z x
W (x0)dx0); (7)
and A+ = − 1p
2
d
dx
+W (x), A = 1p
2
d
dx
+W (x). The two partner potentials are made
from superpotential W (x),
V = W
2 
1
p
2
d
dx
W (x): (8)
A+ and A do not commute with each other and satisfy [A;A+] =
p
2W (x)
0
. To
compare the spectra of these two partners, denote  (−) as the eigenfunction of H−
and  (+) as that of H+. Then, A (−) is an eigenfunction of H+ because
H+(A 
(−)
n ) = AA
+A (−)n = AH− 
(−)
n = E
(−)
n (A 
(−)
n ): (9)
Similarly, A+ (+) is an eigenfunction of H−, H−(A+ (+)n ) = E
(+)
n (A
+ (−)n ). There-
fore A, A+ can transform an eigenfunction of one potential into a eigenfunction of
it’s partner with the same energy. However, note that the ground state of V− is
annihilated by A and have no partner state of V+. Therefore the ground state of V−
has no superpartner.
Relations of energy eigenvalues and wavefunctions are:
E(+)n = E
(−)
n+1;
 (+)n =
1q
E
(−)
n+1
A 
(−)
n+1, n = 0; 1; 2; : : : ;
 
(−)
n+1 =
1q
E
(+)
n
A+ (+)n , n = 0; 1; 2; : : : : (10)
In 1983, Gendenshtein [1] suggested that all exactly solvable potentials are \shape
invariant" ( he used \form invariant" instead ). Shape invariance means that the
4
superpartner potentials have the same x−dependence modulo the changes in val-
ues of a set of parameters that dene the potential. Mathematically, it means the
superpartner of V−(x; a0), V+(x; a0) can be written as
V+(x; a0) = V−(x; a1) +R(a1) (11)
and a1 = f(a0), where a0 are the set of parameters in V+, and f is the transformation
function that maps a0 into a1. The remainder R(a1) is independent of x.
To show that the eigenvalues can be obtained easily from the above condition, we
construct a series of Hamiltonian H(k), k = 0; 1; 2; : : :, with H(0)  H−, H(1)  H+,
H(k) = −
1
2
d2
dx2
+ V−(x; ak) +
kX
s=1
R(as); (12)
where as = fs(a0), i.e., f mapped s times. Furthermore, we have From Eq.(12), we
see that the ground state of H(k) has the energy:
E
(k)
0 =
kX
s=1
R(as): (13)
Since (n + 1)th energy eigenvalue of H(0) (=H−), whose ground state energy is
zero, is coincident with the ground state energy of Hamiltonian H(n), the complete
eigenvalues of H− are:
E(−)n =
nX
k=1
R(ak) , E
(−)
0 = 0: (14)
If a potential is shape invariant, we can also get the bound state wavefunctions
 (−)n easily. This is because A and A
+ can link up the wavefunctions of the super-
partners with the same energy. Starting from H(−)n , it’s ground state  
(−)
0 (x; an)
corresponds to the rst excited state  (−)1 (x; an−1) of H
(−)
n−1. In the same manner,
eventually, it will correspond to the nth state of H
(−)
0 . Recalling Eq.(10), one ob-
tains
 (−)n / A
+(x; a0)A
+(x; a1) : : : A
+(x; an−1) 
(−)
0 (x; an): (15)
To compare the supersymmetry approach with the second approach proposed by
Klein and Li[3, 4], we shall rst give a brief review. Consider an one dimensional
quantum system with the Hamiltonian
H = −
1
2
d2
dx2
+ V (x): (16)
5
For an arbitrary function of position, f = f(x), one can easily derive
[[f;H]; H] = −(f 00H +Hf 00) + 2f 00V + f 0V 0 −
f (4)
4
: (17)
To obtain solvable potential Klein and Li proposed to impose a so-called linear double
commutator relation demanding that for some f and V , the right hand side become
a linear functional of f . In order to make the operator equation linear in f , one
imposes
2f 00V + f 0V 0 = f + ; (18)
and
f 00 = f + : (19)
Eq.(18) relates V to f and can be solved to give the potential V (x)
V (x) =
f2 + 2f + γ
2(f 0)2
: (20)
Eq.(19) can be solved for f as
f = ax2 + bx+ c (  = 0 ), (21)
or
f = Ae
p
x +Be−
p
x + C (  6= 0 ). (22)
For  = 0, the resulting potentials are just those for one- or three-dimensional
harmonic oscillator problems. For  6= 0, one can get the general potential
V (x) =
(Ae
p
x +Be−
p
x) + γ
2
(Ae
p
x − Be−
p
x)2
; (23)
which are special cases of Morse and P}oschl-Teller potentials. Li also solved the
energy eigenvalues by the method of Heisenberg’s matrix mechanics[4] under some
general assumptions. The eigenvalue is proportional to n2, where n is a quantum
number. Also, from Eqs. (19) and (20), the corresponding superpotential, with
linear double commutator relation, can be written as[4]
W (x) =
af(x) + b
f 0(x)
: (24)
Here we shall rst show that the superpotential is indeed shape invariance and then
solve the eigenvalues using the supersymmetric methods outlined earlier.
6
Given, Eqs.(24) and (19), one can easily check the shape invariance condition by
working out the potentials V as
V =
a(a p
2
)f2 + (a(b p
2
) + b(a p
2
))f + b(b p
2
)
(f 0)2

a
p
2
: (25)
Therefore, V are related by shape invariance condition
V+(a; b; x) = V−(a−

p
2
; b−

p
2
; x) +
p
2a−

2
: (26)
The energy eigenvalues can be straight-forwardly worked out to be
En =
p
2na−
n2
2
; (27)
where the ground state energy has been shifted to zero. SUSY method and Li’s
method gives the same results as expected.
In the SUSY approach, shape invariance requires that the superpotential W (x)
satises the functional dierential equation:
W 2(a; x) +
1
p
2
W 0(a; x) = W 2(a1; x)−
1
p
2
W 0(a1; x) +R(a1); (28)
where a0 represents a set of parameters, called the shape invariance relation(SIR).
The values of the set, a1, on the right-hand side, depend on the value of a0, that is,
a1 = f(a0) for some function f . The only known way of solving SIR is to impose an
educated ansatz to turn it into a dierential equation. In this direction, Gendenshtein
[1] had proposed three ansatzs which provide some general classes of solutions. There
is no claim that the three classes should encompass the most general solutions of SIR.
However, it is interesting to note that all the known solutions can be shown to be
special cases of one of the three classes as we shall demonstrate later. Also, the three
classes are not mutually exclusive. That is, some solutions can be represented in
more than one classes.
We shall discuss the three classes in order, solve the corresponding nonlinear
dierential equations and work out the potentials and energy eigenvalues for each
class. The three classes can be described as follow:
(I).Class one:
The ansatz for the superpotential is of the form
W = af1 + b: (29)
7
The SIR then requires f1 to satisfy
f 01 = pf
2
1 + qf1 + r: (30)
V can be worked out to be
V = a(a
p
p
2
)f21 + 2a(b
q
2
p
2
)f1 + b
2 
ar
p
2
; (31)
with the parameters transforming as
an = a+
np
p
2
;
bn =
ab
a+ npp
2
+
na qp
2
+ n
2pq
4
(a+ npp
2
)
; (32)
R = b20 − b
2
1 +
r
p
2
(a0 + a1):
For the case when p 6= 0, the eigenvalues can be worked out to be
En =
nX
k=1
R(ak; bk)
=
nX
k=1
[(b2k−1 − b
2
k) +
r
p
2
(ak−1 + ak)]
= b2 − [
ab
a+ npp
2
+
na qp
2
+ n
2pq
4
(a+ npp
2
)
]2 +
p
2arn+
pr
2
n2: (33)
Ordering the terms by power of n, En can be written as
En = (b−
aq
2p
)2 − (b−
aq
2p
)2
a2
(a+ n pp
2
)2
+
p
2(ar −
aq2
4p
)n+ (
pr
2
−
q2
8
)n2: (34)
For the case when p = 0, the transformation of parameter and the eigenvalues can
be obtained as
bn = b+
nq
p
2
; (35)
R = b20 − b
2
1 +
p
2ar;
En =
nX
k=1
Rk =
p
2(ar − bq)n−
q2
2
n2:
The known exactly solvable potentials in this class and their corresponding param-
eters are listed in Table 1.
8
Table 1: Class One W = af(x) + b ; f
0
= pf2 + qf + r
Potential W (x) f(x) a b p q r
Shifted Oscillator
q
1
2
!x− b x
q
1
2
! −b 0 0 1
Coulomb
q
1
2
e2
l+1
− l+1p
2r
r−1 − l+1p
2
q
1
2
e2
l+1
−1 0 0
Morse A− Be−x e−x −B A 0 − 0
Rosen-Morse A tanhx+ B
A
tanhx A B
A
− 0 
Eckart −A cothr + B
A
cothr −A B
A
− 0 
(B > A2)
The most general solutions for function f in this case can be summarized as
f = rx+ c ; (p = q = 0); (36)
f = −
1
px+ c
; (p 6= 0; q = r = 0); (37)
f = ceqx −
r
q
; (p = 0; q 6= 0); (38)
f =
p
4pr − q2
2p
tan(
p
4pr − q2
2
x+ c)−
q
2p
; (p 6= 0; q or r 6= 0); (39)
where c is an integration constant.
(II). Class two:
The superpotential in this ansatz is assumed to be of the form
W = af2 +
b
f2
;
f 02 = pf
2
2 + q: (40)
V can be worked out to be
V = a(a
p
p
2
)f22 + b(b
q
p
2
)f−22 + 2ab
1
p
2
(aq − pb); (41)
9
with the parameters transforming as
an = a+
np
p
2
;
bn = b−
nq
p
2
;
Rn = 2
p
2(aq − bp) + 2(2n − 1)pq: (42)
The eigenvalues are
En = 2
p
2(aq − bp)n+ 2pqn2: (43)
The well-known examples in the class and their corresponding parameters are listed
in Table 2 for illustration.
Table 2: Class Two W = af + b
f
; f
0
= pf2 + q
Potential W (x) f(x) a b p q
3-D Oscillator
q
1
2
!r − l+1p
2r
r
q
1
2
! − l+1p
2
0 1
Po¨schl-Teller I A tanx− B cotx tanx A −B  
(0 < x < 
2
)
Po¨schl-Teller II A tanhr − B cothr tanhr A −B − 
(B < A)
The general solutions for function f in this case can be written as
f = qx+ c (p = 0); (44)
f =
s
q
p
tan(
p
pqx+ c) (p 6= 0; q 6= 0); (45)
f = −
1
px+ c
(p 6= 0; q = 0); (46)
where c is an integration constant.
(III).Class three:
The superpotential is assumed to be of the form
W =
a+ b
q
pf23 + q
f3
;
f 03 =
q
pf23 + q: (47)
10
V can be worked out to be
V =
1
f23
[a2 + bq(b
s
1
2
) + 2a(b
1
2
p
2
)
q
pf23 + q] + b
2p; (48)
with the parameters transforming as
bn = b−
n
p
2
;
Rn = p(
p
2b+
1
2
)− pn: (49)
The eigenvalues are
En =
p
2npb−
pn2
2
: (50)
The well-known exactly solvable potentials in this class are listed in Table 3.
The general solutions for function f are
f =
p
qx+ c ; (p = 0); (51)
f = ce
p
px ; (p 6= 0; q = 0); (52)
f =
1
2
s
q
p
(e
p
p(x+c) − e−
p
p(x+c)) ; (p; q 6= 0); (53)
c is an integration constant.
Table 3: Class Three W = (a+ b
p
pf2 + q)=f ,f
0
=
p
pf2 + q
Potential W (x) f(x) a b p q
Morse A−Be−x ex −B A

2 0
A tanhx+B sechx coshx B A

2 −2
Rosen-Morse A cothr − B cschr sinhr −B A

2 2
(A < B)
Eckart −A cotx+B cscx sinx B −A

−2 2
(0 < x < ;A > B)
11
The tables showed that all known exact solvable potentials with analytic forms
can be put into one of these three classes. Whatever has spectrum 1/n2 can be
classied as the rst case. However, only the class two solutions can produce three
dimensional oscillator, type-I P}oshl-Teller or type-II P}oshl-Teller potentials. But
two potentials, Morse and Eckart, can be considered both as case one and as case
three. As mentioned before, Li’s results are just the special case of shape invariance
solutions. However it cannot be so easily t into one of the three classes given by
Gendenshtein. In fact one can consider Eq.(19) and Eq.(24) to be the equations that
dene a fourth ansatz for solutions of SIR. The set of solutions overlaps with those of
the other three classes provided by Gendenshtein but does not generate a new one.
From this point of view, it is clear that the ansatz does not provide a very precise
classication of the solutions of shape invariance condition. It is not too hard to
propose new ansatz, however, it is much harder to generate new solutions. Typically
one can not be sure whether an ansatz generates any new solution or not until they
are solved completely. Therefore it seems that a better classifying solutions may be
to use the energy spectrum and its quantum number dependence instead.
In conclusion, we have discussed the two methods of obtaining exactly solvable
potentials in quantum mechanics. One of them requires the shape invariance be-
tween the superpartners of the potentials in the supersymmetric formulation. The
other one imposes a so-called \linear double commutator relations". We have shown
that the second case only produces solutions which are special solutions of the rst
approach. From this point of view, the shape invariance approach is still the most
general method of producing the analytic, exactly solvable potentials. We also ar-
gued that the n dependence of the energy spectrum may be a better way of telling
the dierence between dierent classes of potentials. In addition, we work out the
energy eigenvalues of the most general classes of potentials in the literature.
Unfortunately, it is still not possible to obtain the most general solutions to SIR.
From the table of [2], one observes that all the known shape invariant potentials
have basically only one parameter changing under the SUSY transformation. In
particular, for Po¨schl-Teller I potentials, only the combination A + B is changing.
The parameter A − B is invariant. For Po¨schl-Teller potentials II, on the other
12
hand, the combination A−B is changing and the parameter A+B is invariant. It is
not surprising because as long as the transformation property of the parameters are
linear for a proper choice of parameters, one can always make linear combinations
such that only one of them is changing during the transformation. Since, in general,
there is no reason for the transformation of the parameters to be linear, one would
expect to have a lot more interesting solutions of shape invariance condition waiting
to be discovered.
Another interesting observation is that all the shape invariant solutions has the
spectrum of one of the following the forms (modulo a constant that sets the ground
state energy to zero): (1) an, (a > 0); (2) − b
(n+a)2
, (b > 0); (3) (a+ bn)2 or their
linear combination. The harmonic oscillator is an example for the rst form. The
hydrogen atom is an example for the second form and, the square well is the simplest
example for the third form. Also note that the general spectra of all three classes of
solutions suggested by Gendenshtein are all linear combinations of the above three
forms. One may wonder if there is something fundamental about these kinds of
spectra that made them represent the spectra of all the shape invariant, exactly
solvable, potentials.
Finally, regarding the approach proposed by Klein and Li, since Li’s solutions in
[4] produce only part of the solutions of SIR, it suggests that it may be possible to
generalize their approach within its’ framework. We have made some attempts in
this direction, however, so far without success.
This work was supported in part by National Science Council of Republic of
China grants No. NSC 84-2112-M-007-042 and No. NSC 84-2112-M-007-016.
References
[1] L. E. Gendenshtein, LETP Lett 38, 356 (1984).
[2] R. Dutt, A. Khare, and U. Sukhatman, Am. J. Phys.56, 163 (1988).
[3] C. T. Li, A. Klein, and F. Krejs, Phys. Rev. D17, 2311 (1975).
[4] C.T. Li, Chin. J. Phys.32, 309 (1994).
13


On Exactly Solvable Potentials

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