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2r2
r2 3
+ ’ 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
di¬erence 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 speci¬c form of the superpotential.
The mere existence of the operators A and A+ was enough to set up
a supersymmetric connection.


© 2001 by Chapman & Hall/CRC
6.3 Isotropic Oscillator and Spin-orbit Cou-
pling
Dynamical SUSY can be identi¬ed [11,13] for the isotropic oscillator
problems involving a constant spin-orbit coupling or when in¬‚uenced
by a particular spin-orbit coupling and an additional potential. Here
we discuss the case of spin-orbit coupling only [12].
The underlying Hamiltonian may be taken as
d2
1 3
’’
’ 2 + r2 + » σ . L +
H= (6.45)
2 dr 2
having energy levels
3 3 1
»
Enlj = 2N + l + +» l+ for j = l +
2 2 2
3 1 1
= 2N + l + + » ’l + for j = l ’ (6.46)
2 2 2
where N as usual denotes the radial quantum number (N = 0, 1, 2, . . .)
for a ¬xed j and » is a coupling constant. Equation (6.46) re¬‚ects
an obvious degeneracy for » = ’1, l = j + 1 and » = 1, l = j ’ 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 ¬rst an SU (1, 1) algebra in terms of the operators
K± and K0
[K± , K0 ] = “K±
[K+ , K’ ] = ’2K0 (6.48)
A convenient set of representations may be adopted for (6.48) which
is
3
1 ++
K+ = βi βi
2 i=1
3
1
K’ = βi βi
2 i=1
3
1 1
+
K0 = βi βi + (6.49)
2 2
i=1



© 2001 by Chapman & Hall/CRC
+
where βi and βi satisfy the bosonic commutation relations
+
βi , βj = δij , i, j = 1, 2, 3 (6.50)

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

Then the following commutation relations enlarge
1
[F± , K0 ] = “ F±
2
[K± , F“ ] = “F±
[K+ , F+ ] = 0
[K’ , F’ ] = 0 (6.52)
along with
{F± , F± } = K±
{F+ , F’ } = 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
de¬ned in terms of the commutation of F+ and F’ 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+ , F’ ] (6.56)
Now corresponding to (6.51) C can be found to be
1 +
C= βi βj (σi σj ’ σj σi ) ’ 3 (6.57)
4 i j



© 2001 by Chapman & Hall/CRC
which implies
1 ’’ 3
σ .L ’
C=’ (6.58)
2 4
’ + +
where note that L = i(β2 β3 ’ β2 β3 , cyclic).
Using (6.58), the Casimirs for Osp ( 1 ) and SU (1, 1) become
2

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

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

Equation (6.62) re¬‚ects a dynamical SUSY corresponding to the su-
pergroup embedding Osp ( 1 ) ⊃ SU (1, 1) ⊃ SO(2).
2
To explicitly bring out the connection of the Hamiltonian (6.45)
to SUSYQM one has to take recourse to de¬ning some additional
operators. These are the U ™s, W ™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 σi βi

U+ =
0 0
2
1 0 σi βi

U’ =
0 0
2
1 0 0

W+ = +
σi βi 0
2
1 0 0

W’ =
σi βi 0
2
13 1 0
’’
+ σ .L
Y = (6.63)
0 ’1
22


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

1
[U± , K0 ] = “ U±
2
1
[W± , K0 ] = “ W±
2
[K± , U± ] = 0,
[K± , W± ] = 0
[K± , U“ ] = “U±
[K± , W“ ] = “W±
1
[Y, U± ] = U±
2
1
[Y, W± ] = ’ W±
2
[K± , Y ] = 0
[K0 , Y ] = 0
{U± , U± } = 0
{W± , W± } = 0
{U+ , U’ } = 0
{W+ , W’ } = 0
{U± , W± } = K±
{U“ , W± } = K0 ± Y (6.64)

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

H = 2(K0 + Y ) (6.65)

The degneracy indicated by (6.47) can be understood [12] from the
fact that both the Hamiltonians belong to the same representation
of Osp ( 2 ) ∼ SU (1, 1/1). That (6.65) can be put in the super-

2
+
symmetric form follows from the identi¬cations A ∼ 2 σi βi
√ +
and A+ ∼ 2 σi βi . In our notations of Chapter 2 these mean
√ √
U’ = 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+ }.


© 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 D’1 d l(l + D ’ 2)
’ ’ + + 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 ¬rst order derivative term
i
i=1
D’1
can be removed by employing the transformation R ’ r’ χ(r).
2

We then have the form
1 d2 ±l
’ + 2 + V (r) χ = Eχ (6.67)
2 dr2 2r
where
1
±l =
(D ’ 1) (D ’ 3) + l (l + D ’ 2) (6.68)
4
We now consider the following cases

(a) Coulomb potential

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