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A possible enlargement of the decomposition (5.7) can be made
by including an additional term W2 . This induces two more types of
c
factorizations

Type E:
W1 = a cot a(x + x0 )
W0 = 0
W2 = q (5.14)
Type F:
1
W1 =
x
W0 =0
W2 = q (5.15)
where
W2
W = W0 + cW1 + (5.16)
c
and q is a constant.
Each type of factorization determines W (x, c) from the solutions
of W0 and W1 given above. For Types A-D factorizations, g(c) is
obtained from its expression in (5.8) whereas for the cases E and
2 2
F, g(c) can be determined to be a2 c2 ’ q2 and ’ q2 , respectively. IH
c c
concluded that the above types of factorizations are exhaustive if and
only if a ¬nite number of negative powers of c are considered in the
expansion of W (x, c).
Concerning the normalizability of eigenfunctions we note that
g(c) could be an increasing (class I) or a decreasing (class II) function
of the parameter c. So we can set c = 0, 1, 2, . . . k for each of a
discrete set of values Ek (k = 0, 1, 2, . . .) of E for class I and c =
k, k +1, k +2, . . . for each of a discrete set of values Ek (k = 0, 1, 2, . . .)
of E for class II functions.
Replacing ψ in (5.2) by the form Ykc we can express the normal-
ized solutions as

Class I:
d
1
Ykc’1 = [g(k + 1) ’ g(c)]’ 2 W (x, c) + Ykc (5.17)
dx


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Class II:
d
Ykc+1 = [g(k) ’ g(c + 1)]’1/2 W (x, c + 1) ’ Ykc (5.18)
dx
where
Ykk = A exp W (x, k + 1) dx (5.19)

for class I and
Ykk = B exp ’ W (x, k)dx (5.20)
b
for Class II with A and B ¬xed from a (Ykk )2 dx = 1.
We do not go into the details of the evaluation of the normalized
solutions. Su¬ce it to note that some of the representative potentials
for Types A ’ G are respectively those of Poschi Teller, Morse, a
system of identical oscillators, harmonic oscillator, Rosen-Morse, and
generalized Kepler problems. In the next section we shall return to
these potentials while addressing the question of SI in SUSYQM.
To summarize, the technique of the factorization method lays
down a procedure by which many physical problems can be solved
in a uni¬ed manner. We now turn to the SI condition which has
proved to be a useful concept in tackling the problem of solvability
of quantum mechanical systems.


5.3 Shape Invariance Condition
The SI condition was ¬rst utilized by Gendenshtein [4] to study the
properties of partner potentials in SUSYQM. Taking a cue from the
IH result (5.6), we can de¬ne SI as follows. If the pro¬les of V+ (x)
and V’ (x) are such that they satisfy the relationship
V’ (x, c0 ) = V+ (x, c1 ) + R(c1 ) (5.21)
where the parameter c1 is some function of c0 , say given by c1 =
f (c0 ), the potentials V± are said to be SI. In other words, to be SI
the potentials V± while sharing a similar coordinate dependence can
at most di¬er in the presence of some parameters. Note that (5.5) is
an equivalent condition to (5.21).
An example will make the de¬nition of SI clear. Let us take
W (x) = c0 tanh x : W (∞) = ’W (’∞) = c0 (5.22)


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Then
c2
1
V± (x, c0 ) = ’ c0 (c0 ± 1)sech x + 0
2
(5.23)
2 2
But these can also be expressed as

V’ (x, c0 ) = V+ (x, c1 ) + R(c1 ), c1 = c0 ’ 1 (5.24)

where
12
c0 ’ c2
R(c1 ) = (5.25)
1
2
So the potentials V± are SI in accordance with the de¬nition (5.21).
To exploit the SI condition let us assume that (5.21) holds for a
sequence of parameters {ck }, k = 0, 1, 2, . . . where ck = f f . . . k times
(c0 ) = f k (c0 ). Then

H’ (x, ck ) = H+ (x, ck+1 ) + R(ck ) (5.26)

where k = 0, 1, 2, . . . and we call H (0) = H+ (x, c0 ), H (1) = H’ (x, c0 ).
Writing H (m) as
m
1 d2
(m)
H =’ + V+ (x, cm ) + R(ck )
2 dx2 k=1
m
= H+ (x, cm ) + R(ck ) (5.27)
k=1

it follows on using (5.26) that
m
(m+1)
H = H’ (x, cm ) + R(ck ) (5.28)
k=1

Thus we are able to set up a hierarchy of Hamiltonians H (m)
for various m values. Now according to the principles of SUSYQM
highlighted in Chapter 2, H+ contains the lowest state with a zero-
energy eigenvalue. It then transpires from (5.27) that the lowest
energy level of H (m) has the value
m
(m)
E0 = R(ck ) (5.29)
k=1

It is also not too di¬cult to realize [5] that because of the chain
H (m) ’ H (m’1) . . . ’ H (1) (≡ H’ ) ’ H (0) (≡ H+ ), the nth member


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in this sequence carries the nth level of the energy spectra of H (0)
(or H+ ), namely
n
(+)
(+)
En = R(ck ), E0 =0 (5.30)
k=1

(+)
Moreover if ψ0 (x, cm ) is to represent the ground-state wave
function for H (m) then the nth wave function for H+ (x, c0 ) can be
constructed from it by repeated applications of the operator A+ . To
1
+
establish this we note from (2.59) that ψn+1 = (2En )’ 2 A+ ψn and
’ ’
(’) +
that for SI potentials ψn (x, c0 ) = ψn (x, c1 ). So we can write
+
ψn+1 (x, c0 ) = (2En )’1/2 A+ (x, c0 )ψn (x, c1 )
’ +
(5.31)

In the presence of n parameters c0 , c1 , . . . cn , repeated use of (5.31)
gives the result
+
ψn (x, c0 ) = N A+ (x, c0 )A+ (x, c1 ) . . . A+ (x, cn’1 )ψ0 (x, cn )
+
(5.32)

where N is a constant. These correspond to the energy eigenfunctions
of H+ (x, c0 ).
Let us now return to the example (5.22). We rewrite (5.24) as
1 1
V’ (x, c0 ) = V+ (x, c0 ’ 1) + c2 ’ (c0 ’ 1)2 (5.33)
0
2 2
and note that we can generate ck from c0 as ck = c0 ’ k. Therefore
the levels of V+ (x, c0 ) are given by
n
+
En = R(ck )
k=1
n
1
c2 ’ c2
=
2 k=1 0 k

12
c0 ’ c2
= n
2
12
c0 ’ (c0 ’ n)2
= (5.34)
2
On the other hand, the ground state wave function for H+ (x, c0 )
may be obtained from (2.56b) using the form of W (x) in (5.22). It
turns out to be proportional to sechx.


© 2001 by Chapman & Hall/CRC
+
En for V+ being obtianed from (5.34) we can easily calculate the
energy levels En of the potential

V (x) = ’βsech2 x (5.35)

with β = 1 c0 (c0 + 1) derived from (5.23). We ¬nd
2

1 1
En = En ’ c2 = ’ (c0 ’ n)2
+
(5.36)
20 2

where c0 can be expressed in terms of the coe¬cient β of V (x).
In Table 5.1 we furnish a list of solvable potentials which are
SI in the sense of (5.26). It is worth noting that the well-known
potentials such as the Coulomb, the oscillator, Poschl-Teller, Eckart,
Rosen-Morse, and Morse, all satisfy the SI condition. The forms of
these potentials are also consistent with the following ansatz [6] for
the superpotential W (x, c)

q(x)
W (x, c) = (a + b)p(x) + + r(x) (5.37)
a+b

where c = f (a) and b is a constant. Substituting (5.37) into (5.21)
it follows that the case p(x) = q(x) = 0 leads to the one-dimensional
harmonic oscillator, the case q(x) = 0 leads to the three-dimensional
oscillator and the Morse while the case r(x) = 0 along with q(x)
= constant leads to the Rosen-Morse, the Coulomb, and the Eckart
potentials. Note that the Rosen-Morse potential includes as a par-
ticular case (B = 0) the Poschl-Teller potential.
The SI condition has yielded new potentials for a scaling ansatz
of the change of parameters as well [24]. With c1 = f (c0 ), let us
express (5.21) in terms of the superpotential W (x). We have the
form

W 2 (x, c0 ) + W (x, c0 ) = W 2 (x, c1 ) ’ W (x, c1 ) + R(c0 ) (5.38)

The scaling ansatz deals with the proposition

c1 = qc0 (5.39)


© 2001 by Chapman & Hall/CRC
Table 5.1
A list of SI potentials with the parameters explicitly displayed. In the presence of m and
¯ , V+ (x) is de¬ned as V+ (x) = 1 (W 2 ’ √m W ). The variables x and r run between ’∞ < x < ∞
¯
h
h 2
and 0 < r < ∞. The results in this table are consistent with the list provided in Ref. [4]. The

ground state wave function can be calculated using ψ0 (x) = exp ’ ¯m x W (y)dy .
h


Potential V+ (x) Shape-invariant Parameters
c0 c1 R(c1 )
2
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