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H+ ПҲ 0 = (2.76)
2
corresponding to the solution given in (2.73).
All this can be generalized by rewriting the previous steps as
follows. We search for a potential V1 that satisп¬Ғes the Schroedinger
equation
1 d2
+ V1 ПҲ1 = вҲ’ПҮ2 ПҲ1
вҲ’ (2.77)
1
2
2 dx
with V1 +ПҮ2 signifying a zero-energy bound state. The relation (2.74)
1
is re-expressed as
W1 вҲ’ W1 = V! + ПҮ2
2
(2.78)
1

with ПҮ1 obtained from W1 +W1 = ПҮ2 and V1 identiп¬Ғed with вҲ’2ПҮ2 sech2
2
1 1
ПҮ1 (xвҲ’x0 ). Note that V1 (В±вҲһ) = 0. Further requring V f 1 to be sym-
metric means that W1 must be an odd function.
For an arbitrary n levels, we look for a chain of connections

Wn + Wn = VnвҲ’1 + ПҮ2
2
(2.79)
n

with VnвҲ’1 assumed to be known (notethat V0 = 0). Then Vn is
obtained from
Wn вҲ’ Wn = Vn + ПҮ2
2
(2.80)
n

where Wn (0) is taken to be vanishing.
Linearization of (2.79) is accomplished by the substition Wn =
gn /gn yielding
вҲ’ gn + VnвҲ’1 gn = вҲ’ПҮ2 gn (2.81)
n

В© 2001 by Chapman & Hall/CRC
1
As with (2.75), has also Un = satisп¬Ғes
gn

вҲ’ Un + (Wn вҲ’ Wn )un = вҲ’un + (Vn + ПҮ2 )Un = 0
2
(2.82)
n

That is
вҲ’ Un + Vn Un = вҲ’ПҮ2 Un (2.83)
n

which may be looked upon as a generalization of (2.76) to n-levels.
In this way one arrves at a form of the Schroedinger equation which
has n distinct eigenvalues. Evidently V1 = вҲ’2ПҮ2 sech2 ПҮ1 (x вҲ’ x0 ) is
1
reп¬‚ectionless.
In the study of nonlinear systems, V1 can be regarded as an in-
stantaneous frozen one-soliton solution of the KdV equation ut =
вҲ’uxxx + 6uux . The n-soliton solution of the KdV, similarly, also
emerges [79-85] as families of reп¬‚ectionless potentials. It may be
remarked that if we solve (2.81) and use gn (x) = gn (вҲ’x) then we
uniquely determine Wn (x). For a further discussion of the construc-
tion of reп¬‚ectionless potentials supporting a prescribed spectra of
bound states we refer to the work of Schonfeld et al. .

(c) SUSY and derivation of a hierarchy of Hamiltonians

The ideas of SUSYQM can also be used to derive a chain of
Hamiltonians having the properties that the adjacent members of
the hierarchy are SUSY partners. To look into this we п¬Ғrst note that
an important consequence of the representations (2.29) is that the
partner potentials VВ± are related through

d2 +
V+ (x) = VвҲ’ (x) + 2 log ПҲ0 (x) (2.84)
dx
where we have used (2.56). The above equation implies that once
the properties of VвҲ’ (x) are given, those of V+ (x) become immedi-
ately known. Actually in our discussion of reп¬‚ectionless potentials
we exploited this feature.
We now proceed to generate a sequence of Hamiltonians employ-
ing the preceding results of SUSY. Sukumar  pointed out that if a
certain one-dimensional Hamiltonian having a potential V1 (x) allows
for M bound states and has the ground-state eigenvalue and eigen-
(i) (i)
function as E0 and ПҲ0 , respectively, one can express this Hamilto-

В© 2001 by Chapman & Hall/CRC
nian in a similar form as H+
1 d2
H1 =вҲ’ + V1 (x)
2 dx2
1+ (i)
= A1 A1 + E0 (2.85)
2
(1)
where A1 and A+ are deп¬Ғned in terms of ПҲ0 . Using (2.34) and
1
+
(2.56) A1 and A1 can be expressed as
d (1) (1)
A1 = вҲ’ ПҲ0 /ПҲ0
dx
d (1) (1)
A+ = вҲ’ вҲ’ ПҲ0 /ПҲ0 (2.86)
1
dx
where a prime denotes a derivative with respect to x.
The supersymmetric partner to H1 is obtained simply by inter-
changing the operators A1 and A+ 1

1 d2 1 (1)
+ V2 (x) = A1 A+ + E0
H2 = вҲ’ (2.87)
1
2 dx2 2
where the correlation between V1 and V2 is provided by (2.84)
d2 (1)
V2 (x) = V1 (x) вҲ’ 2 lnПҲ0 = V1 (x) + A1 , A+ (2.88)
1
dx
From (2.59) we can relate the eigenvalues and eigenfunctions of
H1 and H2 as
(1) (2)
En+1 = En
(1) вҲ’1/2
(1) (1)
(2)
ПҲn = 2En+1 вҲ’ 2E0 A1 ПҲn+1 (2.89)

To generate a hierarchy of Hamiltonians we put H2 in place of
H1 and carry out a similar set of operations as we have just now
done. It turns out that H2 can be represented as
1 d2 1 (2)
+ V2 (x) = A+ A2 + E0
H2 = вҲ’ (2.90)
2
2
2 dx 2
with
d (2) (2)
A2 = вҲ’ ПҲ0 /ПҲ0
dx
d (2) (2)
A+ =вҲ’ вҲ’ ПҲ0 /ПҲ0 (2.91)
2
dx

В© 2001 by Chapman & Hall/CRC
H2 induces for itself a supersymmetric partner H3 which can be
obtained by reversing the order of the operators A+ and A2 . In
2
this way we run into H4 and build up a sequence of Hamiltonian
H4 , H5 , . . . etc. A typical Hn in this family reads
1 d2 1 (n) (nвҲ’1)
+Vn (x) = A+ An +E0 = AnвҲ’1 A+ +E0
Hn = вҲ’ (2.92)
n nвҲ’1
2
2 dx 2
with
d (n) (n)
An = вҲ’ ПҲ0 /ПҲ0
dx
d (n) (n)
A+ = вҲ’ вҲ’ ПҲ0 /ПҲ0 (2.93)
n
dx
and having the potential Vn
d2 (nвҲ’1)
Vn (x) = VnвҲ’1 (x) вҲ’ 1nПҲ0 , n = 2, 3, . . . M (2.94)
dx2
Further, the eigenvalues and eigenfunctions of Hn are given by
(nвҲ’1) (1)
(n)
Em = Em+1 = . . . = E(m+nвҲ’1) ,
m = 0, 1, 2, . . . M вҲ’ n, n = 2, 3, . . . M (2.95)
(1) (1) (1) (1)
(n)
ПҲm = 2Em+nвҲ’1 вҲ’ 2EnвҲ’2 2Em+nвҲ’1 вҲ’ 2EnвҲ’3 . . .
вҲ’1
(1) (1) (n+mвҲ’1)
2
2Em+nвҲ’1 вҲ’ 2E0 Г— AnвҲ’1 AnвҲ’2 . . . A1 ПҲ1 (2.96)
The following two illustrations will make clear the generation of
Hamiltonian hierarchy.

(1) Harmonic oscillator

Take V1 = 1 Пү 2 x2 . The ground-state wave function is known to
2
(0) 2
be ПҲ1 вҲј eвҲ’Пүx /2 . It follows from (2.84) that V2 (x) = V1 (x) + Пү,
V3 (x) = V2 (x) + Пү = V1 (x) + 2Пү etc. leading to Vk (x) = V1 (x) + (k вҲ’
1)Пү. This amounts to a shifting of the potential in units of Пү.

(2) Particle in a box problem

Here the relevant potential is given by
V1 = 0 |x| < a
= вҲһ |x| = a

В© 2001 by Chapman & Hall/CRC
The energy spectrum and ground-state wave function are well known

ПҖ2
(1)
(m + 1)2 , m = 0, 1, 2 . . .
Em = 2
8a ПҖx
(1)
ПҲ0 = A cos
2a
where A is a constant. From (2.89) we п¬Ғnd for Vn the result
ПҖ ПҖx
n(n вҲ’ 1) sec2
Vn (x) = V1 (x) + n = 1, 2, 3 . . .
8a 2 2a
ПҖ
(1)
(n)
= Em+nвҲ’1 = 2 (n + m)2 m = 0, 1, 2 . . .
Em
8a
We thus see that the вҖңparticle in the boxвҖқ problem generates a se-
ries of sec2 ПҖx potentials. The latter is a well-studied potential in
2a
quantum mechanics and represents an exactly solvable system.

(d) SUSY and the Fokker-Planck equation

As another example of SUSY in physical systems let us examine
its subtle role  on the evaluation of the small eigenvalue associated
with the вҖңapproach to equilibriumвҖқ problem in a bistable system. For
a dissipative system under a random force F (t) we have the Langevin
equation
вҲ‚U
x=вҲ’
Л™ + F (t) (2.97)
вҲ‚x
where U is an arbitrary function of x and F (t) depicts the noise term.
1
Assuming F (t) to have the вҖңwhite-noiseвҖқ correlation (ОІ = T )

F (t) = 0, F (t)F (t ) = 2ОІОҙ(t вҲ’ t ) (2.98)

the probability of п¬Ғnding F (t) becomes Gaussian

1
F 2 (t)dt
P [F (t)] = A exp вҲ’ (2.99)
2ОІ
1
F 2 (t)dt
вҲ’
where AвҲ’1 = D[F ]e 2ОІ .
The Fokker-Planck eqution for the probability distribution P is
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