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© 2001 by Chapman & Hall/CRC

CHAPTER 5

Factorization Method,

Shape Invariance

5.1 Preliminary Remarks

As we already know modelling of SUSY in quantum mechanical sys-

tems rests in the possibility of factorizing the Schroedinger Hamil-

tonian. In e¬ect this amounts to solving a nonlinear di¬erential

equation for the superpotential that belongs to the Riccati class [see

(2.39)]. Not all forms of the Schroedinger equation however meet the

solvability criterion, only a handful of potentials exist which may be

termed as exactly solvable.

Tracking down solvable potentials is an interesting problem by

itself in quantum mechanics [1]. Those which possess normalizable

wavefunctions and yield a spectra of energy-levels include the har-

monic oscillator, Coulomb, isotropic oscillator, Morse, P¨schl-Teller,

o

Rosen-Morse, and sech2 potentials. The forms of these potentials

are generally expressible in terms of known functions of algebraic

polynomials, exponentials, or trigonometric quanties. Importance of

searching for solvable potentials stems from the fact that they very

often serve as a springboard for undertaking calculations of more

complicated systems. SUSY o¬ers a clue [2,3] to the general nature

of solvability in that most of the partner potentials derived from the

pair of isospectral Hamiltonians satisfy the condition of shape simi-

larity. In other words the functional forms of the partner potentials

© 2001 by Chapman & Hall/CRC

are similar except for the presence of the governing parameters in

the respective potentials. By imposing the so-called “shape invari-

ance” (SI) or “form invariance” condition [4,5] de¬nite expression

for the energy levels can be arrived at in closed forms. Although

su¬cient, the SI condition is not necessary for the solvability of the

Schroedinger equation [6]. However, a number of attempts have been

made to look for them by employing the SI condition. Before we take

up the SI condition let us review brie¬‚y the underlying ideas of the

factorization method in quantum mechanics [7-23].

5.2 Factorization Method of Infeld and Hull

The main idea of the factorization method is to replace a given

Schroedinger equation, which is a second-order di¬erential equation,

by an equivalent pair of ¬rst-order equations. This enables us to ¬nd

the eigenvalues and the normalized eigenfunctions in a far easier man-

ner than solving the original Schroedinger equation directly. Indeed

the factorization technique has proven to be a powerful tool in quan-

tum mechanics. The factorization method has a long history dating

back to the old papers of Schroedinger [17-19], Weyl [20], Dirac [21],

Stevenson [22], and Infeld and Hull (IH) [7,8]. IH showed that, for

a wide class of potentials, the factorization method enables one to

immediately ¬nd the energy spectrum and the associated normalized

wave functions.

Consider the following Schroedinger equation

1 d2 ψ(x)

’ + [V (x, c) ’ E] ψ(x) = 0 (5.1)

2 dx2

where we suppose that the potential V (x, c) is given in terms of a set

of parameters c. We can think of c as being represented by c = c0 +m,

m = 0, 1, 2, . . . or by a scaling ci = qci’1 , 0 < q < 1, i = 0, 1, 2, . . .

However, any speci¬c form of c will not concern us until later in the

chapter.

The factorizability criterion implies that we can replace (5.1) by

a set of ¬rst-order di¬erential operators A and A+ such that

A(x, c + 1)A+ (x, c + 1)ψ(x, E, c) = ’ [E + g(c + 1)] ψ(x, E, c)

A+ (x, c)A(x, c)ψ(x, E, c) = ’ [E + g(c)] ψ(x, E, c) (5.2a, b)

© 2001 by Chapman & Hall/CRC

To avoid confusion we have displayed explicitly the coordinate x and

the parameter c on the wave function ψ and also on the ¬rst-order

operators A and A+ which are taken to be

d

A(x, c) = + W (x, c)

dx

d

A+ (x, c) = ’ + W (x, c) (5.3)

dx

In (5.2) g is some function of c while in (5.3) W is an arbitrary

function of x and c.

It is easy to convince oneself that if ψ(x, E, c) is a solution of

(5.1) then the two functions de¬ned by ψ(x, E, c + 1) = A+ (x, c + 1)

ψ(x, E, c) and ψ(x, E, c ’ 1) = A(x, c)ψ(x, E, c) are also solutions

of the same equation for some ¬xed value of E. This follows in

a straightforward way by left multiplying (5.2a) and (5.2b) by the

operators A+ (x, c + 1) and A(x, c), respectively. As our notations

make the point clear, the solutions have the same coordinate depen-

dence but di¬er in the presence of the parameters. Moreover the

b

operators A and A+ are mutually self-adjoint due to a φ(A+ f )dx =

b

a (Aφ)f dx, f being arbitrary subject to the continuity of the inte-

grands and vanishing of φf at the end-points of (a, b).

The necessary and su¬cient conditions which the function W (x, c)

ought to satisfy for (5.1) to be consistent with the pair (5.2) are

W 2 (x, c + 1) + W (x, c + 1) = V (x, c) ’ g(c + 1)

W 2 (x, c) ’ W (x, c) = V (x, c) ’ g(c) (5.4)

Subtraction yields

W 2 (x, c + 1) + W (x, c + 1)

’ W 2 (x, c) ’ W (x, c) = h(c) (5.5)

where h(c) = g(c) ’ g(c + 1). Eq. (5.5) can also be recast in the form

1

V’ (x, c + 1) = V+ (x, c) + h(c) (5.6)

2

where V± can be recognized to be the partner components of the

supersymmetric Hamiltonian [see (2.29)]. So the function W (x) in

(5.3) essentially plays the role of the superpotential.

© 2001 by Chapman & Hall/CRC

IH noted that in order for the factorization method to work the

quantity g(c) should be independent of x. Taking as a trial solution

W (x, c) = W0 + cW1 (5.7)

the following constraints emerge from (5.5)

= 0W1 + W1 = ’a2

:2

a

W0 + W0 W1 = ’ka,

g(c) = a2 c2 + 2kca2 (5.8)

a = 0 : W1 = (x + d)’1

W 0 + W 0 W 1 = b1

g(c) = ’2bc (5.9)

where a, b, d and k are constants.

The solution (5.7) alongwith (5.8) lead to various types of fac-

torizations

a =0

Type A:

W1 = a cot a(x + x0 )

c

W0 = ka cot a(x + x0 ) + (5.10)

sin a(x + x0 )

Type B:

W1 = ia

W0 = iak + e exp(’iax) (5.11)

a=0

Type C:

1

W1 =

x

bx e

W0 = + (5.12)

2 x

Type D:

W1 = 0

W0 = bx + p (5.13)

© 2001 by Chapman & Hall/CRC

where x0 , e, and p are contents.