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)

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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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