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