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broken.
The general structure of V± (x) in (2.29) is indicative of the pos-
sibility that we can replace the coordinate x in (2.27) by an arbitrary


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function W (x). Indeed the forms (2.29) of V± reside in the following
general expression of the supersymmetric Hamiltonian
12 1
p + W2 •
Hs = + σ3 W (2.30)
2 2
W (x) is normally taken to be a real, continuously di¬erentiable func-
tion in . However, should we run into a singular W (x), the necessity
of imposing additional conditions on the wave functions in the given
space becomes important [10].
Corresponding to Hs , the associated supercharges can be written
in analogy with (2.21) as
1 0 W + ip

Q=
0 0
2
1 0 0
Q+ = √ (2.31)
W ’ ip 0
2
As in (2.22), here too Q and Q+ may be combined to obtain

Hs = Q, Q+ (2.32)

Furthermore, Hs commutes with both Q and Q+

[Q, Hs ] = 0
Q+ , Hs =0 (2.33)

Relations (2.30) - (2.33) provide a general nonrelativistic basis
from which it follows that Hs satis¬es all the criterion of a formal
supersymmetric Hamiltonian. It is obvious that these relations allow
us to touch upon a wide variety of physical systems [12-53] including
approximate formulations [54-63].
In the presence of the superpotential W (x), the bosonic opera-
tors b and b+ go over to more generalized forms, namely
√ d
2ωb ’ A = W (x) +
dx
√ d
2ωb+ ’ A+ = W (x) ’ (2.34)
dx
In terms of A and A+ the Hamiltonian Hs reads
1 1
A, A+ •
+ σ3 A, A+
2Hs = (2.35)
2 2


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Expressed in a matrix structure Hs is diagonal

Hs ≡ diag (H’ , H+ )
1
diag AA+ , A+ A
= (2.36)
2
Note that Hs as in (2.30) is just a manifestation of (2.34). In the
literature it is customery to refer to H+ and H’ as “bosonic” and
“fermionic” hands of Hs , respectively.
The components H± , however, are deceptively nonlinear since
any one of them, say H’ , can always be brought to a linear form by
the transformation W = u /u. Thus for a suitable u, W (x) may be
determined which in turn sheds light on the structure of the other
component.
It is worth noting that both H± may be handled together by
taking recourse to the change of variables W = gu /u where, g,
which may be positive or negative, is an arbitrary parameter. We
see that H± acquire the forms
2
d2 u u
2H± = ’ 2 + g 2 ± g “g (2.37)
dx u u

It is clear that the parameter g e¬ects an interchange between the
“bosonic” and “fermionic” sectors : g ’ ’g, H+ ” H’ . To show
how this procedure works in practice we take for illustration [64] the
superpotential conforming to supersymmetric Liouville system [24]

2g
described by the superpotential W (x) = a exp ax , g and a are
√2
parameters. Then u is given by u(x) = exp 2 2 exp ax /a2 . 2
2
d
The Hamiltonian H+ satis¬es ’ dx2 + W 2 ’ W ψ+ = 2E+ ψ+ .

42
g exp ax
Transforming y = , the Schroedinger equation for H+
a2 2
becomes

d2 1d 1 1 8E+
ψ+ + ψ+ ’ ψ+ + 2 2 ψ+ = 0 (2.38)
dy 2 y dy 2g 4 ay

The Schroedinger equation for H’ can be at once ascertained from
(2.38) by replacing g ’ ’g which means transforming y ’ ’y. The
relevant eigenfunctions turn out to be given by con¬‚uent hypogeo-
metric function.


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The construction of the SUSYQM scheme presented in (2.30) -
(2.33) remains incomplete until we have made a connection to the
Schroedinger Hamiltonian H. This is what we™ll do now.
Pursuing the analogy with the harmonic oscillator problem, specif-
ically (2.27a), we adopt for V the form V = 1 W 2 ’ W + » in-
2
Wwhich the constant » can be adjusted to coincide with the ground-
state energy E0 oh H+ . In other words we write
1
W2 ’ W
V (x) ’ E0 = (2.39)
2
indicating that V and V+ can di¬er only by the amount of the ground-
state energy value E0 of H.
If W0 (x) is a particular solution, the general solution of (2.39) is
given by
exp [2 x W0 („ )d„ ]
W (x) = W0 (x) + , β∈R (2.40)
β ’ x exp [2 y W0 („ )d„ ] dy
On the other hand, the Schroedinger equation

1 d2
’ + V (x) ’ E0 ψ0 = 0 (2.41)
2 dx2

subject to (2.39) has the solution
x x
ψ0 (x) = A exp ’ W („ )d„ + B exp ’ W („ )d„
x y
exp 2 W („ )d„ dy (2.42)

where A, B, ∈ R and assuming ψ(x) ∈ L2 (’∞, ∞). If (2.40) is sub-
stituted in (2.42), the wave function is the same [65] whether a par-
ticular W0 (x) or a general solution to (2.39) is used in (2.42).
In N = 2 SUSYQM, in place of the supercharges Q and Q+ de-
¬ned in (2.31), we can also reformulate the algebra (2.32) - (2.35) by
introducing a set of hermitean operators Q1 and Q2 being expressed
as
Q = (Q1 + iQ2 ) /2, Q+ = (Q1 ’ iQ2 ) /2 (2.43)
While (2.32) is converted to Hs = Q2 = Q2 that is
1 2

{Qi , Qj } = 2δij Hs (2.44)


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(2.33) becomes
[Qi , Hs ] = 0, i = 1, 2 (2.45)
In terms of the superpotential W (x), Q1 and Q2 read
1 p
√ σ1 W ’ σ2 √
Q1 =
m
22
1 p
√ σ1 √ + σ2 W
Q2 = (2.46)
m
22

On account of (2.45), Q1 and Q2 are constants of motion: Q1 = 0

and Q2 = 0.
From (2.44) we learn that the energy of an arbitrary state is
strictly nonnegative. This is because [66]
Eψ = < ψ|Hs |ψ >
= < ψ|Q+ Q1 |ψ >
1
= < φ|φ >≥ 0 (2.47)
where |φ >= Q1 |ψ >, and we have used in the second step the
representation (2.44) of Hs .
For an exact SUSY
Q1 |0 > = 0
Q2 |0 > = 0 (2.48)
So |φ > = 0 would mean existence of degenerate vacuum states0 >|
and |0 > related by a supercharge signalling a spontaneous symmetry
breaking.
It is to be stressed that the vanishing vacuum energy is a typ-
ical feature of unbroken SUSY models. For the harmonic oscillator
whose Hamiltonian is given by (2.3) we can say that HB remains
invariant under the interchange of the operators b and b+ . However,
the same does not hold for its vacuum which satis¬es b|0 >. In the
case of unbroken SUSY both the Hamiltonian Hs and the vacuum
are invariant with respect to the interchange Q ” Q+ .


2.3 Physical Interpretation of Hs
As for the supersymmetric Hamiltonian in the oscillator case here
also we may wish to seek [66, 36] a physical interpretation of (2.30).


© 2001 by Chapman & Hall/CRC
To this end let us restore the mass parameter m in Hs which then
reads
1 p2 1W
+ W 2 • + σ3 √
Hs = (2.49)
2m 2 m
Comparing with the Schroedinger Hamiltonian for hhe electron (mass
m and charge ’e) subjected to an external magnetic ¬eld namely

p2 e2 ’2
1 ie e’’ |e| ’ ’

div A ’ A. p + σ .B
H= +A+ (2.50)
2 mm 2m m 2m
’ ’ ’
where A = 1 B — r is the vector potential, we ¬nd that (2.50) goes
2 √
’ m
over to (2.49) for the speci¬c case when A = 0. 2|e| W, 0 . The point
to observe is the importance of the electron magnetic moment term
in (2.50) without which it is not reducible to (2.49). We thus see
that a simple problem of an electron in the external magnetic ¬eld
exhibies SUSY.
Let us dwell on the Hamiltonian H a little more. If we assume

the magnetic ¬eld B to be constant and parallel to the Z axis so that

B = B k, it follows that
1
’’
A. p = BLz
2
’2 ’’ 2
= r2 B 2 ’ r .B
4A

x2 + y 2 B 2
= (2.51)

As a result H becomes
1 1
p2 + (p2 + p2 ) + mω 2 x2 + y 2 ’ ω (Lz ’ σ3 ) (2.50a)
H=
2m z x y
2
Apart from a free motion in the z direction, H describes two harmonic
oscillators in the xy-plane and also involves a coupling to the orbital
eB
and spin moments. In (2.50a) ω is the Larmor frequency: ω = 2m
’ ’
and S = 1 σ .
2
In the standard approach of quantization of oscillators the cou-
pling terms look like

ω (Lz ’ σ3 ) = ’iω b+ by ’ b+ bx + ωσ3 (2.52)
x y



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However, setting
1
B+ = √ b+ + ib+
x y

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