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2r2
r2 3
+ в€’ l+j+ +2 (6.44)
2 2
A particular case of (6.43) and (6.44) when j = 0 corresponds
to the previous combinations (6.8) and (6.17). Equations (6.43) and
(6.44) are however more general and correctly predicting the energy
diп¬Ђerence of two units.
Thus using the ladder operator techniques one can successfully
give a supersymmetric interpretation to both the Coulomb and the
isotropic oscillator problems. We should stress that in getting these
results we did not require any speciп¬Ѓc form of the superpotential.
The mere existence of the operators A and A+ was enough to set up
a supersymmetric connection.

В© 2001 by Chapman & Hall/CRC
6.3 Isotropic Oscillator and Spin-orbit Cou-
pling
Dynamical SUSY can be identiп¬Ѓed [11,13] for the isotropic oscillator
problems involving a constant spin-orbit coupling or when inп¬‚uenced
by a particular spin-orbit coupling and an additional potential. Here
we discuss the case of spin-orbit coupling only .
The underlying Hamiltonian may be taken as
d2
1 3
в†’в†’
в€’ 2 + r2 + О» Пѓ . L +
H= (6.45)
2 dr 2
having energy levels
3 3 1
О»
Enlj = 2N + l + +О» l+ for j = l +
2 2 2
3 1 1
= 2N + l + + О» в€’l + for j = l в€’ (6.46)
2 2 2
where N as usual denotes the radial quantum number (N = 0, 1, 2, . . .)
for a п¬Ѓxed j and О» is a coupling constant. Equation (6.46) reп¬‚ects
an obvious degeneracy for О» = в€’1, l = j + 1 and О» = 1, l = j в€’ 1 .
2 2
In both cases EN turns out to be
EN = 2N + 2j + 2 (6.47)
To show that a dynamical SUSY is associated with the Hamilto-
nian we consider п¬Ѓrst an SU (1, 1) algebra in terms of the operators
KВ± and K0
[KВ± , K0 ] = в€“KВ±
[K+ , Kв€’ ] = в€’2K0 (6.48)
A convenient set of representations may be adopted for (6.48) which
is
3
1 ++
K+ = ОІi ОІi
2 i=1
3
1
Kв€’ = ОІi ОІi
2 i=1
3
1 1
+
K0 = ОІi ОІi + (6.49)
2 2
i=1

В© 2001 by Chapman & Hall/CRC
+
where ОІi and ОІi satisfy the bosonic commutation relations
+
ОІi , ОІj = Оґij , i, j = 1, 2, 3 (6.50)

Actually one can even enlarge the algebra (6.48) by introducing
a set of operators F+ and Fв€’ which are
3
1 0 1
+
F+ = Пѓi ОІi
1 0
2 i=1
3
1 0 1
Fв€’ = Пѓi ОІi (6.51)
1 0
2 i=1

Then the following commutation relations enlarge
1
[FВ± , K0 ] = в€“ FВ±
2
[KВ± , Fв€“ ] = в€“FВ±
[K+ , F+ ] = 0
[Kв€’ , Fв€’ ] = 0 (6.52)
along with
{FВ± , FВ± } = KВ±
{F+ , Fв€’ } = K0 (6.53)
Equations (6.48), (6.52), and (6.53) constitute the noncompact Osp( 1 )
2
superalgebra.
The quadratic Casimir operator of Osp ( 1 ) and SU (1, 1) can be
2
deп¬Ѓned in terms of the commutation of F+ and Fв€’ which we call C.
Thus
1 1
= C2 + C
C2 Osp( (6.54)
2 2
C2 [SU (1, 1)] = C 2 + C (6.55)
where
C = [F+ , Fв€’ ] (6.56)
Now corresponding to (6.51) C can be found to be
1 +
C= ОІi ОІj (Пѓi Пѓj в€’ Пѓj Пѓi ) в€’ 3 (6.57)
4 i j

В© 2001 by Chapman & Hall/CRC
which implies
1 в†’в†’ 3
Пѓ .L в€’
C=в€’ (6.58)
2 4
в†’ + +
where note that L = i(ОІ2 ОІ3 в€’ ОІ2 ОІ3 , cyclic).
Using (6.58), the Casimirs for Osp ( 1 ) and SU (1, 1) become
2

2
1 1 в†’ 1в†’
L+ Пѓ
C2 Osp = (6.59)
2 4 2
1в†’ 3
2
C2 [SU (1, 1)] = Lв€’ (6.60)
2 16
while the Hamiltonian (6.45) takes the form

H = 2K0 в€’ 2О»C (6.61)

In fact H can be written in terms of the Casimir operators of
Osp ( 1 ) and SU (1, 1) by inserting (6.59) and (6.60) in (6.61).
2

1
H = 4О» C2 Osp в€’ C2 (SU (1, 1)) + 2K0 (6.62)
2

Equation (6.62) reп¬‚ects a dynamical SUSY corresponding to the su-
pergroup embedding Osp ( 1 ) вЉѓ SU (1, 1) вЉѓ SO(2).
2
To explicitly bring out the connection of the Hamiltonian (6.45)
to SUSYQM one has to take recourse to deп¬Ѓning some additional
operators. These are the U вЂ™s, W вЂ™s, and Y which along with (6.48),
(6.52) and (6.53) enlarge Osp ( 1 ) to the Osp ( 2 ) superalgebra. Their
2 2
representations can be taken to be
+
1 0 Пѓi ОІi
в€љ
U+ =
0 0
2
1 0 Пѓi ОІi
в€љ
Uв€’ =
0 0
2
1 0 0
в€љ
W+ = +
Пѓi ОІi 0
2
1 0 0
в€љ
Wв€’ =
Пѓi ОІi 0
2
13 1 0
в†’в†’
+ Пѓ .L
Y = (6.63)
0 в€’1
22

В© 2001 by Chapman & Hall/CRC
where a summation over the label i is suggested. One then п¬Ѓnds the
following relations to hold

1
[UВ± , K0 ] = в€“ UВ±
2
1
[WВ± , K0 ] = в€“ WВ±
2
[KВ± , UВ± ] = 0,
[KВ± , WВ± ] = 0
[KВ± , Uв€“ ] = в€“UВ±
[KВ± , Wв€“ ] = в€“WВ±
1
[Y, UВ± ] = UВ±
2
1
[Y, WВ± ] = в€’ WВ±
2
[KВ± , Y ] = 0
[K0 , Y ] = 0
{UВ± , UВ± } = 0
{WВ± , WВ± } = 0
{U+ , Uв€’ } = 0
{W+ , Wв€’ } = 0
{UВ± , WВ± } = KВ±
{Uв€“ , WВ± } = K0 В± Y (6.64)

Moreover corresponding to the two cases О» = В±1, l = j в€“ 1 , the
2
Hamiltonians (6.45) are expressible as

H = 2(K0 + Y ) (6.65)

The degneracy indicated by (6.47) can be understood  from the
fact that both the Hamiltonians belong to the same representation
of Osp ( 2 ) в€ј SU (1, 1/1). That (6.65) can be put in the super-
в€љ
2
+
symmetric form follows from the identiп¬Ѓcations A в€ј 2 Пѓi ОІi
в€љ +
and A+ в€ј 2 Пѓi ОІi . In our notations of Chapter 2 these mean
в€љ в€љ
Uв€’ = Q/ 2 and W+ = Q+ / 2, so that from the last equation of
(6.64) corresponding to the positive sign we have H = {Q, Q+ }.

В© 2001 by Chapman & Hall/CRC
6.4 SUSY in D Dimensions
The radial Schroedinger equation in D dimensions reads (see Ap-
pendix A for a detailed derivation)

1 d2 Dв€’1 d l(l + D в€’ 2)
в€’ в€’ + + V (r) R = ER (6.66)
2 dr2 2r2
2r dr
where r in terms of D cartesian coordinates xi is given by r =
1
D 2

x2 . As with (6.2), here, too, the п¬Ѓrst order derivative term
i
i=1
Dв€’1
can be removed by employing the transformation R в†’ rв€’ П‡(r).
2

We then have the form
1 d2 О±l
в€’ + 2 + V (r) П‡ = EП‡ (6.67)
2 dr2 2r
where
1
О±l =
(D в€’ 1) (D в€’ 3) + l (l + D в€’ 2) (6.68)
4
We now consider the following cases

(a) Coulomb potential
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