The main idea of the factorization method is to replace a given Schroedinger equation, which is a second-order differential equation, by an equivalent pair of first-order equations. This enables us to find 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 find the energy spectrum and the associated normalized wave functions.
Consider the following Schroedinger equation
−1 2
d2ψ(x)
dx2 + [V(x, c)−E]ψ(x) = 0 (5.1) where we suppose that the potentialV(x, c) is given in terms of a set of parametersc. We can think ofcas being represented byc=c0+m, m = 0,1,2, . . . or by a scaling ci = qci−1,0 < q < 1, i = 0,1,2, . . . However, any specific form ofc will not concern us until later in the chapter.
The factorizability criterion implies that we can replace (5.1) by a set of first-order differential 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)
To avoid confusion we have displayed explicitly the coordinatexand the parameter c on the wave function ψ and also on the first-order operatorsA and A+ which are taken to be
A(x, c) = d
dx +W(x, c) A+(x, c) = − d
dx+W(x, c) (5.3)
In (5.2) g is some function of c while in (5.3) W is an arbitrary function ofx and c.
It is easy to convince oneself that if ψ(x, E, c) is a solution of (5.1) then the two functions defined byψ(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 fixed 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 differ in the presence of the parameters. Moreover the operatorsAandA+ are mutually self-adjoint due toabφ(A+f)dx= b
a(Aφ)fdx, f being arbitrary subject to the continuity of the inte- grands and vanishing ofφf at the end-points of (a, b).
The necessary and sufficient conditions which the functionW(x, c) ought to satisfy for (5.1) to be consistent with the pair (5.2) are
W2(x, c+ 1) +W(x, c+ 1) = V(x, c)−g(c+ 1)
W2(x, c)−W(x, c) = V(x, c)−g(c) (5.4) Subtraction yields
W2(x, c+ 1) +W(x, c+ 1)
−W2(x, c)−W(x, c)=h(c) (5.5) whereh(c) =g(c)−g(c+ 1). Eq. (5.5) can also be recast in the form
V−(x, c+ 1) =V+(x, c) + 1
2h(c) (5.6)
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.
IH noted that in order for the factorization method to work the quantity g(c) should be independent ofx. Taking as a trial solution
W(x, c) =W0+cW1 (5.7)
the following constraints emerge from (5.5) a= 0 :W12+W1 = −a2 W0+W0W1 = −ka,
g(c) = a2c2+ 2kca2 (5.8) a= 0 :W1 = (x+d)−1
W0+W0W1 = b1
g(c) = −2bc (5.9)
where a, b, dand kare constants.
The solution (5.7) alongwith (5.8) lead to various types of fac- torizations
a= 0 Type A:
W1 = acota(x+x0)
W0 = kacota(x+x0) + c
sina(x+x0) (5.10) Type B:
W1 = ia
W0 = iak+eexp(−iax) (5.11)
a= 0 Type C:
W1 = 1 x W0 = bx
2 + e
x (5.12)
Type D:
W1 = 0
W0 = bx+p (5.13)
wherex0, e, and p are contents.
A possible enlargement of the decomposition (5.7) can be made by including an additional term Wc2. This induces two more types of factorizations
Type E:
W1 = acota(x+x0) W0 = 0
W2 = q (5.14)
Type F:
W1 = 1 W0 = 0x
W2 = q (5.15)
where
W =W0+cW1+ W2
c (5.16)
andq is a constant.
Each type of factorization determinesW(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 F, g(c) can be determined to be a2c2−qc22 and −qc22, respectively. IH concluded that the above types of factorizations are exhaustive if and only if a finite number of negative powers ofc are considered in the expansion ofW(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 valuesEk(k= 0,1,2, . . .) ofE for class II functions.
Replacing ψin (5.2) by the formYkc we can express the normal- ized solutions as
Class I:
Ykc−1= [g(k+ 1)−g(c)]−12
W(x, c) + d dx
Ykc (5.17)
Class II:
Ykc+1= [g(k)−g(c+ 1)]−1/2
W(x, c+ 1)− d dx
Ykc (5.18) where
Ykk=Aexp W(x, k+ 1)
dx (5.19)
for class I and
Ykk=Bexp
− W(x, k)dx
(5.20) for Class II with A and B fixed fromab(Ykk)2dx= 1.
We do not go into the details of the evaluation of the normalized solutions. Suffice 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 unified 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.