© 2001 by Chapman & Hall/CRC

CHAPTER 2

Basic Principles of

SUSYQM

2.1 SUSY and the Oscillator Problem

By now it is well established that SUSYQM provides an elegant

description of the mathematical structure and symmetry properties

of the Schroedinger equation. To appreciate the relevance of SUSY in

simple nonrelativistic quantum mechanical syltems and to see how it

works in these systems let us begin our discussion with the standard

harmonic oscillator example. Its Hamiltonian HB is given by

’2

d2

h 1

+ mωB x2

2

HB = ’ (2.1)

2

2m dx 2

where ωB denotes the natural frequency of the oscillator and ’ =h

h

2π , h the Planck™s constant. Unless there is any scope of confusion

we shall adopt the units ’ = m = 1.

h

Associated with HB is a set of operators b and b+ called, re-

spectively, the lowering (or annihilation) and raising (or creation)

d

operators [1-6] which can be de¬ned by p = ’i dx

i

√

b= (p ’ iωB x)

2ωB

i

b+ = ’√ (p + iωB x) (2.2)

2ωB

© 2001 by Chapman & Hall/CRC

Under (2.2) the Hamiltonian HB assumes the form

1

HB = ω B b+ , b (2.3)

2

where {b+ , b} is the anti-commutator of b and b+ .

As usual the action of b and b+ upon an eigenstate |n > of

harmonic oscillator is given by

√

b|n > = n|n ’ 1 >

√

b+ |n > = n + 1|n + 1 > (2.4)

The associated bosonic number operator NB = b+ b obeys

NB |n >= n|n > (2.5)

with n = nB .

+n

The number states are |n > (b n! |0 > (n = 0, 1, 2, . . .) and the

√)

lowest state, the vacuum |0 >, is subjected to b|0 >= 0.

The canonical quantum condition [q, p] = i can be translated in

terms of b and b+ in the form

[b, b+ ] = 1 (2.6)

Along with (2.6) the following conditons also hold

[b, b] = 0,

b+ , b+ =0 (2.7)

[b, HB ] = ωB b,

b+ , HB = ’ωB b+ (2.8)

We may utilize (2.6) to express HB as

1 1

HB = ωB (b+ b + ) = ωB NB + (2.9)

2 2

which¬‚eads to the energy spectrum

1

EB = ω B nB + (2.10)

2

© 2001 by Chapman & Hall/CRC

The form (2.3) implies that the Hamiltonian HB is symmetric under

the interchange of b and b+ , indicating that the associated particles

obey Bose statistics.

Consider now the replacement of the operators b and b+ by the

corresponding ones of the fermionic oscillator. This will yield the

fermionic Hamiltonian

ωF +

HF = a ,a (2.11)

2

where a and a+ , identi¬ed with the lowering (or annihilation) and

raising (or creation) operators of a fermionic oscillator, satisfy the

conditions

{a, a+ } = 1, (2.12)

{a, a} = 0, {a+ , a+ } = 0 (2.13)

We may also de¬ne in analogy with NB a fermionic number operator

NF = a+ a. However, the nilpotency conditions (2.13) restrict NF to

the eigenvalues 0 and 1 only

2

= (a+ a)(a+ a)

NF

= (a+ a)

= NF

NF (NF ’ 1) = 0 (2.14)

The result (2.14) is in conformity with Pauli™s exclusion principle.

The antisymmetric nature of HF under the interchange of a and a+

is suggestive that we are dealing with objects satisfying Fermi-Dirac

statistics. Such objects are called fermions. As with b and b+ in (2.2),

the operators a and a+ also admit of a plausible representation. In

terms of Pauli matrices we can set

1 1

a = σ’ , a+ = σ+ (2.15)

2 2

where σ± = σ1 ± iσ2 and [σ+ , σ’ ] = 4σ3 . Note that

01 0 ’i 10

σ1 = , σ2 = , σ3 = (2.16)

10 i 0 0 ’1

© 2001 by Chapman & Hall/CRC

We now use the condition (2.12) to express HF as

1

HF = ω F N F ’ (2.17)

2

which has the spectrum

1

EF = ω F nF ’ (2.18)

2

where nF = 0, 1.

For the development of SUSY it is interesting to consider [7] the

composite system emerging out of the superposition of the bosonic

and fermionic oscillators. The energy E of such a system, being the

sum of EB and EF , is given by

1 1

E = ω B nB + + ω F nF ’ (2.19)

2 2

We immediately observe from the above expression that E remains

unchanged under a simultaneous destruction of one bosonic quantum

(nB ’ nB ’1) and creation of one fermionic quantum (nF ’ nF +1)

or vice-versa provided the natural frequencies ωB and ωF are set

equal. Such a symmetry is called “supersymmetry” (SUSY) and the

corresponding energy spectrum reads

E = ω(nB + nF ) (2.20)

where ω = ωB = ωF . Obviously the ground state has a vanishing

energy value (nB = nF = 0) and is nondegenerate (SUSY unbroken).

This zero value arises due to the cancellation between the boson and

fermion contributions to the supersymmetric ground-state energy.

Note that individually the ground-state energy values for the bosonic

and fermionic oscillators are ω2 and ’ ω2 , respectively, which can be

B F

seen to be nonzero quantities. However, except for the ground-state,

the spectrum (2.20) is doubly degenerate.

It also follows in a rather trivial way that since the SUSY degen-

eracy arises because of the simultaneous destruction (or creation) of

one bosonic quantum and creation (or destruction) of one fermionic

quantum, the corresponding generators should behave like ba+ (or

b+ a). Indeed if we de¬ne quantities Q and Q+ as

√

ωb — a+ ,

Q=

√+

Q+ = ωb — a (2.21)

© 2001 by Chapman & Hall/CRC

it is straightforward to check that the underlying supersymmetric

Hamiltonian Hs can be expressed as

Hs = ω b+ b + a+ a

Q, Q+

= (2.22)

and it commutes with both Q and Q+

[Q, Hs ] = 0

Q+ , Hs =0 (2.23)

Further,

{Q, Q} = 0

Q+ , Q+ =0 (2.24)

Corresponding to Hs a basis in the Hilbert space composed of HB —

HF is given by {|n > —| 0 >F , |n > — a+ | 0 >F } where n = 0, 1, 2 . . .

and 0 >F is the fermionic vacuum.

In view of (2.23), Q and Q+ are called supercharge operators or

simply supercharges. From (2.22) - (2.24) we also see that Q, Q+ , and

Hs obey among themselves an algebra involving both commutators

as well as anti-commutators. As already mentioned in Chapter 1

such an algebra is referred to as a graded algebra.

It is now clear that the role of Q and Q+ is to convert a bosonic

(fermionic) state to a fermionic (bosonic) state when operated upon.

This may be summarised as follows

√

Q |nB , nF > = ωnB | nB ’ 1, nF + 1 >, nB =,0 F

n =1

Q+ |nB , nF > = ω(nB + 1) nB + 1, nF ’ 1 >, nF = 0 (2.25)

However, Q+ |nB , nF >= 0 and Q|nB , nF >= 0 for the cases (nB =

0, nF = 1) and nF = 0, respectively.

To seek a physical interpretation of the SUSY Hamiltonian Hs

let us use the representations (2.2) and (2.15) for the bosonic and

fermionic operators. We ¬nd from (2.22)

12 1

p + ω 2 x2 •

Hs = + ωσ3 (2.26)

2 2

© 2001 by Chapman & Hall/CRC

where • is the (2 — 2) unit matrix. We see that Hs corresponds to a

bosonic oscillator with an electron in the external magnetic ¬eld.

The two components of Hs in (2.26) can be projected out in a

manner

1 d2 1 22

ω x ’ ω ≡ ωb+ b

H+ = ’ +

2

2 dx 2

1 d2 1 22

ω x + ω ≡ ωbb+

H’ = ’ + (2.27a, b)

2

2 dx 2

Equivalently one can express Hs as

Hs ≡ diag (H’ , H+ )

1• ω

= ω b+ b + + σ3 (2.28)

2 2

by making use of (2.6).

From (2.27) it is seen that H+ and H’ are nothing but two real-

izations of the same harmonic oscillator Hamiltonian with constant

shifts ±ω in the energy spectrum. We also notice that H± are the

outcomes of the products of the operators b and b+ in direct and

reverse orders, respectively, the explicit forms being induced by the

representations (2.2) and (2.15). Indeed this is the essence of the

factorization scheme in quantum mechanics to which we shall return

in Chapter 5 to handle more complicated systems.

2.2 Superpotential and Setting Up a Super-

symmetric Hamiltonian

H+ and H’ being the partner Hamiltonians in Hs , we can easily

isolate the corresponding partner potentials V± from (2.27). Actually

these potentials may be expressed as

1

W 2 (x) “ W (x)

V± (x) = (2.29)

2

with W (x) = ωx. We shall refer to the function W (x) as the super-

potential. The representations (2.29) were introduced by Witten [8]

to explore the conditions under which SUSY may be spontaneously