2

ω2

0 00

If HΞ is rede¬ned slightly to have a change in the zero-point energy

so that

12

ωj bj , b+

HΞ = j

2 j=1

+ diag(E1 ’ E3 , 2E2 ’ E1 ’ E3 , E3 ’ E1 ) (8.106)

where the energy-frequency relationships are provided by (8.102), it

follows that HΞ along with Q± and Q± obey the relations

1 2

(Q± )2 = 0 i = 1, 2

i

H, Q± = 0 i = 1, 2

i

Q+ Q’ Q+ + Q+ Q’ = Q+ H

1 11 22 1

Q’ Q+ + Q+ Q’ Q+ = Q+ H (8.107)

11 22 2 2

along with their hermitean-conjugated counterparts.

We are thus led to a generalized scheme in which superpotentials

are introduced, in place of bosonic operators in (8.105). Thus we

© 2001 by Chapman & Hall/CRC

de¬ne in two dimensions

«

0 A+ (x) 0

1

1

Q+ = 0

√ 0 0

1

20 0 0

«

00 0

1

Q+ = √ 0 0 A+ (y) (8.108)

2 2

200 0

and read o¬ form (8.107)

H = diag(H1 , H2 , H3 ) (8.109)

where

1+

A1 (x)A’ (x) + A+ (y)A’ (y)

H1 = 1 2 2

2

d2 d2

1 2 2

= ’ 2 ’ 2 + W1 (x) + W2 (y)

2 dx dy

+W1 (x) + W2 (y) (8.110)

1’

A1 (x)A+ (x) + A+ (y)A’ (y)

H2 = 1 2 2

2

d2 d2

1 2 2

= ’ 2 ’ 2 + W1 (x) + W2 (y)

2 dx dy

’W1 (x) + W2 (y) (8.111)

1’

A1 (x)A+ (x) + A’ (y)A+ (y)

H3 = 1 2 2

2

d2 d2

1 2 2

= ’ 2 ’ 2 + W1 (x) + W2 (y)

2 dx dy

’W1 (x) ’ W2 (y) (8.112)

To derive H1 , H2 , H3 a constraint like (8.16) was not needed. These

components of the Hamiltonian H together with Q+ and Q+ de¬ned

1 2

by (8.108) o¬er a two-dimensional generalization of the conventional

PSUSY schemes. Note that the underlying algebra is provided by

(8.107).

It is pointless to mention that (8.110)-(8.112) are consistent with

(8.106) when W1 (x) = ω1 x and W2 (y) = ω2 y. We should also point

out that if we de¬ne Q and Q+ according to (8.24) using (8.108),

© 2001 by Chapman & Hall/CRC

then while Q and Q+ obey Q3 = (Q+ )3 = 0, Q+ and Q+ play the

1 2

role of supercharges.

This concludes our discussion on the generalized parasuperalge-

bras associated with the three-level system of Ξ type. The transition

operators, and consequently the supercharges for the systems V and

§ can be similarly built, and generalized schemes such as the one

described for the Ξ type can be set up.

8.6 References

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© 2001 by Chapman & Hall/CRC

Appendix A

The D-dimensional Schroedinger Equation in

a Spherically Symmetric Potential V (r)

In Cartesian coordinates, the Schroedinger equation under the in¬‚u-

ence of a potential V (r) reads

¯2 2 ’

h ’ ’

∇D ψ( r ) + V (r)ψ( r ) = Eψ( r )

’ (A1)

2m

where ’ ’

r = (x1 , x2 , . . . xD ), r = | r |

(A2)

‚‚

∇2 =

D

‚xi ‚xi

Our task is to transform (A1) to D-dimensional polar coordi-

nates. The latter are related to the Cartesian coordinates by

x1 = r cos θ1 sin θ2 sin θ3 . . . . . . sin θD’1

x2 = r sin θ1 sin θ2 sin θ3 . . . . . . sin θD’1

x3 = r cos θ2 sin θ3 sin θ4 . . . . . . sin θD’1

(A3)

x4 = r cos θ3 sin θ4 sin θ5 . . . . . . sin θD’1

.

.