The general structure of V± (x) in (2.29) is indicative of the pos-

sibility that we can replace the coordinate x in (2.27) by an arbitrary

© 2001 by Chapman & Hall/CRC

function W (x). Indeed the forms (2.29) of V± reside in the following

general expression of the supersymmetric Hamiltonian

12 1

p + W2 •

Hs = + σ3 W (2.30)

2 2

W (x) is normally taken to be a real, continuously di¬erentiable func-

tion in . However, should we run into a singular W (x), the necessity

of imposing additional conditions on the wave functions in the given

space becomes important [10].

Corresponding to Hs , the associated supercharges can be written

in analogy with (2.21) as

1 0 W + ip

√

Q=

0 0

2

1 0 0

Q+ = √ (2.31)

W ’ ip 0

2

As in (2.22), here too Q and Q+ may be combined to obtain

Hs = Q, Q+ (2.32)

Furthermore, Hs commutes with both Q and Q+

[Q, Hs ] = 0

Q+ , Hs =0 (2.33)

Relations (2.30) - (2.33) provide a general nonrelativistic basis

from which it follows that Hs satis¬es all the criterion of a formal

supersymmetric Hamiltonian. It is obvious that these relations allow

us to touch upon a wide variety of physical systems [12-53] including

approximate formulations [54-63].

In the presence of the superpotential W (x), the bosonic opera-

tors b and b+ go over to more generalized forms, namely

√ d

2ωb ’ A = W (x) +

dx

√ d

2ωb+ ’ A+ = W (x) ’ (2.34)

dx

In terms of A and A+ the Hamiltonian Hs reads

1 1

A, A+ •

+ σ3 A, A+

2Hs = (2.35)

2 2

© 2001 by Chapman & Hall/CRC

Expressed in a matrix structure Hs is diagonal

Hs ≡ diag (H’ , H+ )

1

diag AA+ , A+ A

= (2.36)

2

Note that Hs as in (2.30) is just a manifestation of (2.34). In the

literature it is customery to refer to H+ and H’ as “bosonic” and

“fermionic” hands of Hs , respectively.

The components H± , however, are deceptively nonlinear since

any one of them, say H’ , can always be brought to a linear form by

the transformation W = u /u. Thus for a suitable u, W (x) may be

determined which in turn sheds light on the structure of the other

component.

It is worth noting that both H± may be handled together by

taking recourse to the change of variables W = gu /u where, g,

which may be positive or negative, is an arbitrary parameter. We

see that H± acquire the forms

2

d2 u u

2H± = ’ 2 + g 2 ± g “g (2.37)

dx u u

It is clear that the parameter g e¬ects an interchange between the

“bosonic” and “fermionic” sectors : g ’ ’g, H+ ” H’ . To show

how this procedure works in practice we take for illustration [64] the

superpotential conforming to supersymmetric Liouville system [24]

√

2g

described by the superpotential W (x) = a exp ax , g and a are

√2

parameters. Then u is given by u(x) = exp 2 2 exp ax /a2 . 2

2

d

The Hamiltonian H+ satis¬es ’ dx2 + W 2 ’ W ψ+ = 2E+ ψ+ .

√

42

g exp ax

Transforming y = , the Schroedinger equation for H+

a2 2

becomes

d2 1d 1 1 8E+

ψ+ + ψ+ ’ ψ+ + 2 2 ψ+ = 0 (2.38)

dy 2 y dy 2g 4 ay

The Schroedinger equation for H’ can be at once ascertained from

(2.38) by replacing g ’ ’g which means transforming y ’ ’y. The

relevant eigenfunctions turn out to be given by con¬‚uent hypogeo-

metric function.

© 2001 by Chapman & Hall/CRC

The construction of the SUSYQM scheme presented in (2.30) -

(2.33) remains incomplete until we have made a connection to the

Schroedinger Hamiltonian H. This is what we™ll do now.

Pursuing the analogy with the harmonic oscillator problem, specif-

ically (2.27a), we adopt for V the form V = 1 W 2 ’ W + » in-

2

Wwhich the constant » can be adjusted to coincide with the ground-

state energy E0 oh H+ . In other words we write

1

W2 ’ W

V (x) ’ E0 = (2.39)

2

indicating that V and V+ can di¬er only by the amount of the ground-

state energy value E0 of H.

If W0 (x) is a particular solution, the general solution of (2.39) is

given by

exp [2 x W0 („ )d„ ]

W (x) = W0 (x) + , β∈R (2.40)

β ’ x exp [2 y W0 („ )d„ ] dy

On the other hand, the Schroedinger equation

1 d2

’ + V (x) ’ E0 ψ0 = 0 (2.41)

2 dx2

subject to (2.39) has the solution

x x

ψ0 (x) = A exp ’ W („ )d„ + B exp ’ W („ )d„

x y

exp 2 W („ )d„ dy (2.42)

where A, B, ∈ R and assuming ψ(x) ∈ L2 (’∞, ∞). If (2.40) is sub-

stituted in (2.42), the wave function is the same [65] whether a par-

ticular W0 (x) or a general solution to (2.39) is used in (2.42).

In N = 2 SUSYQM, in place of the supercharges Q and Q+ de-

¬ned in (2.31), we can also reformulate the algebra (2.32) - (2.35) by

introducing a set of hermitean operators Q1 and Q2 being expressed

as

Q = (Q1 + iQ2 ) /2, Q+ = (Q1 ’ iQ2 ) /2 (2.43)

While (2.32) is converted to Hs = Q2 = Q2 that is

1 2

{Qi , Qj } = 2δij Hs (2.44)

© 2001 by Chapman & Hall/CRC

(2.33) becomes

[Qi , Hs ] = 0, i = 1, 2 (2.45)

In terms of the superpotential W (x), Q1 and Q2 read

1 p

√ σ1 W ’ σ2 √

Q1 =

m

22

1 p

√ σ1 √ + σ2 W

Q2 = (2.46)

m

22

™

On account of (2.45), Q1 and Q2 are constants of motion: Q1 = 0

™

and Q2 = 0.

From (2.44) we learn that the energy of an arbitrary state is

strictly nonnegative. This is because [66]

Eψ = < ψ|Hs |ψ >

= < ψ|Q+ Q1 |ψ >

1

= < φ|φ >≥ 0 (2.47)

where |φ >= Q1 |ψ >, and we have used in the second step the

representation (2.44) of Hs .

For an exact SUSY

Q1 |0 > = 0

Q2 |0 > = 0 (2.48)

So |φ > = 0 would mean existence of degenerate vacuum states0 >|

and |0 > related by a supercharge signalling a spontaneous symmetry

breaking.

It is to be stressed that the vanishing vacuum energy is a typ-

ical feature of unbroken SUSY models. For the harmonic oscillator

whose Hamiltonian is given by (2.3) we can say that HB remains

invariant under the interchange of the operators b and b+ . However,

the same does not hold for its vacuum which satis¬es b|0 >. In the

case of unbroken SUSY both the Hamiltonian Hs and the vacuum

are invariant with respect to the interchange Q ” Q+ .

2.3 Physical Interpretation of Hs

As for the supersymmetric Hamiltonian in the oscillator case here

also we may wish to seek [66, 36] a physical interpretation of (2.30).

© 2001 by Chapman & Hall/CRC

To this end let us restore the mass parameter m in Hs which then

reads

1 p2 1W

+ W 2 • + σ3 √

Hs = (2.49)

2m 2 m

Comparing with the Schroedinger Hamiltonian for hhe electron (mass

m and charge ’e) subjected to an external magnetic ¬eld namely

p2 e2 ’2

1 ie e’’ |e| ’ ’

’

div A ’ A. p + σ .B

H= +A+ (2.50)

2 mm 2m m 2m

’ ’ ’

where A = 1 B — r is the vector potential, we ¬nd that (2.50) goes

2 √

’ m

over to (2.49) for the speci¬c case when A = 0. 2|e| W, 0 . The point

to observe is the importance of the electron magnetic moment term

in (2.50) without which it is not reducible to (2.49). We thus see

that a simple problem of an electron in the external magnetic ¬eld

exhibies SUSY.

Let us dwell on the Hamiltonian H a little more. If we assume

’

the magnetic ¬eld B to be constant and parallel to the Z axis so that

’

B = B k, it follows that

1

’’

A. p = BLz

2

’2 ’’ 2

= r2 B 2 ’ r .B

4A

x2 + y 2 B 2

= (2.51)

As a result H becomes

1 1

p2 + (p2 + p2 ) + mω 2 x2 + y 2 ’ ω (Lz ’ σ3 ) (2.50a)

H=

2m z x y

2

Apart from a free motion in the z direction, H describes two harmonic

oscillators in the xy-plane and also involves a coupling to the orbital

eB

and spin moments. In (2.50a) ω is the Larmor frequency: ω = 2m

’ ’

and S = 1 σ .

2

In the standard approach of quantization of oscillators the cou-

pling terms look like

ω (Lz ’ σ3 ) = ’iω b+ by ’ b+ bx + ωσ3 (2.52)

x y

© 2001 by Chapman & Hall/CRC

However, setting

1

B+ = √ b+ + ib+

x y