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© 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

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