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2
1
√ (bx ’ iby )
B= (2.53)
2
2
pz
we may diagonilize (2.50a) to obtain 1 H ’ 2m = ω B + B + 1
2 2
• + ω σ which is a look-alike of (2.28). Summarizing, the two-
23
dimensional Pauli equation (2.50) gives a simple illustration of how
SUSY can be realized in physical systems.


2.4 Properties of the Partner Hamiltonians
As interesting property of the supersymmetric Hamiltonian Hs is
that the partner components H+ and H’ are almost isopectral. In-
deed if we set
+ ++
H+ ψ n = En ψ n (2.54)
it is a simple exercise to work out
1
+
AA+ Aψn +
H’ Aψn =
2
1+ +
=A A Aψn
2
+ +
= En Aψn (2.55)
+
This clearly shows En to be the energy spectra of H’ also. However,
+ +
Aψ0 is trivially zero since ψ0 being the ground-state solution of H+
satis¬es
+ +
’(ψ0 ) + (W 2 ’ W )ψ0 = 0 (2.56a)
and so is constrained to be of the form
x
+
ψ0 = C exp ’ W (y)dy (2.56b)

C is a constant.
We conclude that the spectra of H+ and H’ are identical except
for the ground state (n = 0) which is nondegenerate and, in the
present setup, is with the H+ component of Hs . This is the case of


© 2001 by Chapman & Hall/CRC
unbroken SUSY (nondegenerate vacuum). However, if SUSY were
to be broken (spontaneously) then H+ along with H’ can not posses
any normalizable ground-state wave function and the spectra of H+
and H’ would be similar. In other words the nondegeneracy of the
ground-state will be lost.
For square-integrability of ψ0 in one-dimension we may require
from (2.56) that W (y)dy ’ ∞ as |x| ’ ∞. One way to realize
this condition is to have W (x) di¬ering in sign at x ’ ±∞. In other
words, W (x) should be an odd function. As an example we may
keep in mind the case W (x) = ωx. On the other hand, if W (x) is
an even function, that is it keeps the same sign at x ’ ±∞, the
square-integrability condition cannot be ful¬lled. A typical example
is W (x) = x2 .
From (2.54) and (2.55) we also see for the following general eigen-
value problems of H±
(+) (+) (+)
H+ ψn+1 = En+1 ψn+1
(’) (’) (’)
H’ ψ n = En ψ n (2.57a, b)
+ +
that if Aψ0 = 0 holds for a normalizable eigenstate ψ0 of H+ , then
+ +
since H+ ψ0 ≡ 1 A+ Aψ0 = 0, it follows that such a normalizable
2
+
eigenstate is also the ground-state of H+ with the eigenvalue E0 = 0.
Of course, because of the arguments presented earlier, H’ does not
possess any normalHzed eigenstate with zero-energy value.
To inquire how the spectra and wave functions of H+ and H’
are related we use the decompositions (2.36) to infer from (2.57) the
eigenvalue equations
1
H+ A+ ψn = A+ A A+ ψn = A+ H’ ψn = En A+ ψn (2.58a)
’ ’ ’ ’ ’
2
1
H’ Aψn = AA+ Aψn = AH+ ψn = En Aψn
+ + + + +
(2.58b)
2
It is now transparent that the spectra and wave functions of H+ and
H’ are related a la [52]
+ +

En = En+1 , n = 0, 1, 2, . . . ; E0 = 0 (2.59a)
’1
+ +

2En+1 2
ψn = Aψn+1 (2.59b)
1
+
ψn+1 = (2En )’ 2 A+ ψn
’ ’
(2.59c)
We now turn to some applications of the results obtained so far.


© 2001 by Chapman & Hall/CRC
2.5 Applications
(a) SUSY and the Dirac equation

One of the important aspects of SUSY is that it appears natu-
rally in the ¬rst quantized massless Dirac operator in even dimen-
sions. To examine this feature [47, 54-74] we consider the Dirac equa-
tion in (1+2) dimensions with minimal electromagnetic coupling

(iγ µ Dµ ’ m) ψ = 0 (2.60)

where Dµ = Dµ +iqAµ with q = ’|e|. The γ matrices may be realized
in terms of the Pauli matrices since in (1+2) dimensions (2.60) can
be expressed in a 2 — 2 matrix form: γ0 = σ3 , γ1 = iσ1 and γ2 = iσ2 .
Introducing covariant derivatives


D1 = ’ ieA1 ,
‚x

D2 = ’ ieA2 (2.61)
‚y

Then (2.60) translates, in the massless case, to

’ (σ1 D1 + σ2 D2 ) ψ = σ3 Eψ (2.60a)

The above equation is also representative of

0 A
ψ = ’σ3 Eψ (2.62)
A+ 0

where A = D1 ’ iD2 and A+ = D1 + iD2 . From (2.35) and (2.36)
we therefore conclude
2Hs ψ = E 2 ψ (2.63)
The supersymmetric Hamiltonian thus gives the same eimen-
function and square of the energy of the original massless equation.
This also makes clear the original curiosity [75] of SUSY which was
to consider the “square root” of the Dirac operator in much the same
manner as the “square-root” of the Klein-Gordon operator was uti-
lized to arrive at the Dirac equation. In the case of a massive fermion
the eigenvalue in (2.63) gets replaced as E 2 ’ E 2 ’ m2 .


© 2001 by Chapman & Hall/CRC
In connection with the relation between chiral anomaly and
fermionic zero-modes, Jackiw [68] observed some years ago that the
Dirac Hamiltonian for (2.60), namely



H = ± . p + eA (2.64)

where ± = (’σ 2 , σ 1 ), displays a conjugation-symmetric spectrum
with zero-modes under certain conditions for the background ¬eld.
The symmetry, however, is broken by the appearance of a mass term.
Actually, in a uniform magnetic ¬eld the square of H coincides with
the Pauli Hamiltonian. As already noted by us the latter exhibits
SUSY which when exact possesses a zero-value nondegenerate vac-
uum.
Hughes, Kostelecky, and Nieto [69] have studied SUSY of mass-
less Dirac operator in some detail by focussing upon the role of
Foldy-Wouthusen (FW) transformations and have demonstrated the
relevance of SUSY in the ¬rst-order Dirac equation. To bring out
Dirac-FW equivalence let us follow the approach of Beckers and De-
bergh [71]. These authors have pointed out that since SUSYQM is
characterized by the algebra (2.32) and (2.33) involving odd super-
charges, it is logical to represent the Dirac Hamiltonian as a sum of
odd and even parts
HD = Q1 + βm (2.65)
where Q1 is odd and the mass term being even has an attached
multiplicative coe¬cient β that anticommutes with Q1

{Q1 , β} = 0 (2.66)

Squaring (2.65) at once yields

HD = Q2 + m2
2
1
= Hs + m2 (2.67)

from (2.44).
We an interpret (2.67) from the point of view of FW transfor-
mation which works as

= U HD U ’1
HF W
= β(Hs + m2 )1/2 (2.68)


© 2001 by Chapman & Hall/CRC
implying that the square of HF W is just proportional to the right-
hand side of (2.67). Note that U , which is unitary, is given by
S = S+ : U = exp(iS)
i
S = ’ βQ1 K ’1 θ
2
K
tanθ =
m
[θ, β] = 0
{HD , S} = 0 (2.69)

where K is even and stands for Hs with the positive sign. We can
also write
E + βQ1 + m
U= (2.70)
1/2
[2E(E + m)]
with E = (Hs + m2 )1/2 .
The SUSY of he massless Dirac operator links directly to two
very important ¬elds in quantum theory, namely index theorems and
anomalies. Indeed it is just the asymmetry of the Dirac ground state
that leads to these phenomena.

(b) SUSY and the construction of re¬‚ectionless potentials

In quantum mechanics it is well known that symmetric, re¬‚ec-
tionless potentials provide good approximations to con¬nement and
their constructions have always been welcome [48,76-78]. In the fol-
lowing we demonstrate [76-86] how the ideas of SUSYQM can be
exploited to derive the forms of such potentials.
Of the two potentials V± , let us impose upon V’ the criterion
that it possesses no bound state. So we take it to be a constant 1 χ2
2

1 1
W2 + W = χ2 > 0
V’ ≡ (2.71)
2 2
Equation (2.71) can be linearized by a substitution W = g /g which
converts it to the form
g
= χ2 (2.72)
g
The solution of (2.72) can be used to determine W (x) as
W (x) = χ tanh χ(x ’ x0 ) (2.73)


© 2001 by Chapman & Hall/CRC
Knowing W (x), V+ can be ascertained to be

1
W2 ’ W
V+ ≡
2
12
χ 1 ’ 2sech2 χ (x ’ x0 )
= (2.74)
2
One can check that H+ possesses a zero-energy bound state wave
function given by
1
ψ0 ∼ ∼ sechχ(x ’ x0 ) (2.75)
g

becausc
1
’ψ0 + W 2 ’ W ψ0 = 0

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