GENERAL I ARTICLE

Supersymmetry
Akshay Kulkarni and P Ramadevi

During the last few decades, theoretical physi-

cists have introduced symmetries (which mayor
may not have any geometrical interpretation) wi-
th the aim of solving difficult problems. In this
article, we shall first present the salient features
of one such symn1.etry called supersymmetry. Then,
Akshay Kulkarni is
we shall show the power of supersymmetry in
currently in his third year of
the BTech programme in
tackling quantum mechanical systems described
Engineering Physics at lIT by non-trivial potentials.
Bombay. This article is
based on a seminar he gave Introduction
on the topic 'Super-
symmetry.' He is primarily
Symmetries playa significant role in reducing the num-
interested in astrophysics, ber of degrees of freedom of any complex system. Hence,
and in physics and math- some features of a complex system can be determined
ematics in general. by exploiting the symmetry properties of the system. In
fact, the theory describing unifications of the fundamen-
tal particles and the strong, electro-weak forces of na-
ture, called standard model, is based all such symrnetry
principles [1,2]. Throughout the discovery of all those
symmetries, physicists always maintained that fermions
and bosons were two different classes of particles. Such
P Ramadevi is presently
a distinction between bosons and fermions poses cer-
assistant professor at the
Department of Physics, lIT tain problems in theoretical models. For instance, even
Bombay, Mumbai. Her though the standard model could explain most of the ex-
research interests are in the perimental observations about the elementary particles,
areas of string theory and there are some shortcomings of the model. For example,
topological field theory.
the Higgs scalar in the standard model is still undiscov-
ered. Further, there are problems with the Higgs scalar,
viz., we obtain correction terms to its mass which are
infinite (usually called divergence). This unphysical di-
vergence can be corrected by fine tuning, but it is an
Keywords unpleasant feature of the standard model. Therefore,
Symmetry, elementary particles, we require a completely non-traditional approach to re-
quantum mechanics .
move such divergences.

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RESONANCE I February 2003
 GENERAL I ARTICLE

This led theoretical physicists to come up with an in-

novative idea of introducing a novel symmetry called
supersymmetry which resulted in a finite mass for the
Higgs scalar. Unlike traditional symmetries, supersym-
metry does not tr~at bosons and fermions as two differ-
ent classes of particles. The supersymmetry operation
converts bosons into fermions and vice versa. Thus, su-
persymmetry introduces a fermionic (bosonic) partner
for every boson (fermion) differing by half a unit of spin
quantum number. The partner is called the superpart-
ner (sparticle) of the original particle. The particle and
its superpartner must be identical in all quantum num-
bers except the spin quantum number.
The theoretical proposition of supersymmetry needs to
be experimentally tested, i.e., sparticles of the elemen-
tary particles must be observed in high-energy collider
experiments. Present day accelerators work at energies
much below the energy needed for testing supersymme-
try. Even though it is still a debatable question whether
nature possesses supersymmetry or not, there are inter-
esting applications of supersymmetry. In fact, we will
show that quantum mechanical systems involving non-
trivial potentials can be easily solved by exploiting the
properties of supersymmetry.
The plan of this article is as follows: First we briefly
describe the meaning of symmetry. Then we discuss
bosonic and fermionic quantum harmonic oscillators, whi-
ch are combined with the next section to show the nat-
ural emergence of a supersymmetric oscillator. Later we
present the formalism of obtaining any supersymmetric
system from two partner systems. Using such a super-
symmetry formalism, we relate the famous example of
a particle in an infinite potential well, which is an ex- Supersymmetry
actly solvable system, to a partner system described by introduces a
a non-trivial potential. fermionic (bosonic)
partner for every
boson (fermion).

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 GENERAL I ARTICLE

What is a Symmetry?
A symmetry of an object (system) is any transformation
under which that object (system) remains invariant, i.e.,
object remains the same before and after the transfor-
mation. For example, consider a square. It is invariant
under reflection about a diagonal. So, reflection about
a diagonal is a symmetry of a square. In physical sit-
uations, there can be other symmetries which may not
have such a geometrical interpretation. Hence, to de-
scribe the symmetries (both geometrical as well as non-
geometrical) possessed by physical systems, it will be
useful to know a quantitative definition.
A formal definition of symmetry is. as follows:
For a system of particles, a symmetry is any transforma-
'An operator which defermines
energy states of the system. tion under which the Hamiltonian! describing the sys-
tem remains invariant. Such a symmetry transformation
is generated by an operator usually referred to as gen-
erator. A system is said to possess a symmetry if the
Hamiltonian commutes with the generator of the sym-
metry transformation. The mathematical meaning of
the words 'generator, commutes' will be obvious from
the following example.
Consider translation as the symmetry transformation. If
1/;( f) is the wave function of the system before transla-
tion, and 'l/J(f + ii) is the wave function after translation
by a constant vector ii, then
1/;(f + ii) = eiii.p/ h 1/;(f},
where p = -in'\l is the momentum operator. The above
equation is nothing but a Taylor series expansion of the
wave function. The momentum operator p is called the
generator of translations bec·ause it generates the wave
function at f + ii (coordinate after transformation) from
the wave function at r (coordinate before transforma-
tion). We know that if the potential V is constant in
space, then the Hamiltonian H = - 1i~;:2 + V is invari-
ant under translation. Also, if V is constant, then the

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RESONANCE I February 2003
 GENERAL I ARTICLE

commutator bracket [H,pJ = Hp - fill is zero, i.e., H

commutes with p. We will now see the consequence of
the zero commutator bracket on the wave function.
Consider the wave function 'ljJ( f) satisfying the following
eigenvalue equation:
H'ljJ(T) = E'ljJ(T) ,
where E is the energy eigenvalue. Then, the zero com-
mutator bracket operating on 'ljJ(f) gives
H [foj;(f)] = pH 'ljJ(f) = E[foP(f)]
Thus, we see that for a given energy E, there are two
eigenfunctions 'ljJ(f) and foI;(r). These wave functions
are usually referred to as degenerate eigenfunctions.
Our aim is to find a system which is invariant under
the new symmetry transformation, called supersymme-
try transformation, and also determine the explicit form
for the supersymmetric generator. The first step in this
direction will be to study two simple systems - a bosonic
and a fer monic harmonic oscillator, which are the build-
ing blocks for constructing a supersymmetric oscillator.
Then, we can study the properties of supersymmetry
which convert bosons into fermions.
Harmonic Oscillator
We shall now study some aspects of bosonic and ferm~­
onic oscillators.
The Hamiltonian H for a simple harmonic oscillator is
1i 2 d? 1 2
H = - 2m dx 2 + 2"kx ,

where m is the mass and k is the spring constant. We

can introduce a dimensionless variable y = V7x, where
W = If is the angular frequency. Then, H can be fac-
torised as
(1)

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RESONANCE I February 2003 31
 GENERAL I ARTICLE

where at = - ;y a ;y
+ Y and = + y. The energy eigen-
states which are solutions of the Schrodinger equation
are the Hermite polynomials

where In)'s are usually referred to as number states, with

n denoting the excited level of the harmonic oscillator
or the number of particles. The operators a and at are
called annihilation and creation operators because their
action on the number states is as follows:

yn+1In+1), (2)
v'nln - 1), aiD) = D. (3)

In other words, a (at) lowers (raises) the state In) to

In - 1) (In + 1)).
Bosons and fermions are distinguished by Pauli's exclu-
sion principle. That is, no two fermions can exist in
the same quantum state, whereas any number of bosons
can exist in the same quantum state. This allows the
number of particles n in a quantum state In) to take
any value for a bosonic oscillator, whereas for a fermi-
onie oscillator, n must be either zero or one. We shall
see in the following subsections how such a distinction
is implemented.
Bosanic H armanic Oscillator
We shall denote the operators and the number states
of the bosonic harmonic oscillator with a subscript b
to remember their bosonic nature. The operators ab, a!
satisfy

Using the above commutation relation in (1), we can

write the Hamiltonian for the bosonic oscillator Hb as

Hb = nwb ( atab + ~) , (5)

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 GENERAL I ARTICLE

whose energy eigenvalues Enb are obtained from

Hblnb) = Enblnb), where Enb = (nb + 1/2)nwb, (6)
where nb = 0,1,2, 00. It is appropriate to point out
that the commutator bracket does not impose any re-
striction on the number of particles nb, and hence plays
a crucial role in maintaining the bosonic nature of the
oscillator.
Fermionic Harmonic Oscillator
As mentioned earlier, Pauli's exclusion principle demands
that the fermionic states can be either 10) or 11). Equiv-
alently, the state 12) must be zero. Defining a} and a,
as the fermionic creation and annihilation operators, the
number state 12) from (2) will be
12) ex (a})210),
which is zero if and only if (a})2 = ~{a},a,t} = 0,
where the bracket in parenthesis is called anticommu-
tator bracket defined as {A, B} = AB + BA for any
two operators A, B. Hence, the property of fermions is
achieved by the anticommutation relations.
Therefore, the restriction imposed by Pauli's exclusion
principle results in the following relations for the oper-
ators a" a} and the fermionic Hamiltonian H,:

{ a" a, } {a}, a}} = 0 and {a" a}} = 1, (7)

Hf nwf (a}af - ~) , (8)

H,ln,) En! In,), where
En! = (n, - 1/2)nwf, (9)
where n, = 0,1. In fact, the anticommutation relation
in (7) has been used to derive the Hamiltonian H, from
(1).
With this background on bosonic and fermionic har-
monic oscillators, we shall study the supersymmetric os-
cillator obtained from them in the next section.

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 GENERAL I ARTICLE

Supersymmetry and Supersymmetric Oscillator

Consider a system which is a combination of one bosonic
and one fermionic oscillator. Assume that there is no
interaction between the two oscillators, i.e., fermionic
operators commute with bosonic operators. The state
of the combined system can be represented as Inb' nf).
With the aim of understanding supersymmetry, we de-
fine an operator Q = abaj and its conjugate Qt = afa!.
The action of these operators on any state of the system
IS

abajlnb, nf = 0) = Vnblnb - 1,
= 1),
nf
afablnb, nf = 1) = Vnb + 11nb + 1,
nf = 0).

Note that aj, af act only on the fermionic state, leav-

ing the bosonic state untouched, because the two sys-
tems are non-interacting. Similarly al, ab act only on
the bosonic state.
The operator Q changes nf = 0 to nf = 1, and Qt does
the reverse. Given any state containing even number
of fermions, it is equivalent to a state containing only
bosons. In other words, Inb' nf - even number) is con-
sidered to behave like a bosonic state. Similarly, a state
with n f odd and any number of bosons is equivalent to
a fermionic state. Therefore, we can say that the oper-
ator Q transforms a bosonic state into a fermionic state
and the operator Qt transforms a fermionic state into
a bosonic state. Such a transformation performed by
the operator Q or Qt is called supersymmetric trans-
formation and Q , Qt are called the generators of the
supersymmetric transformation.
We have basically combined a bosonic system and a
fermionic system. Therefore, the Hamiltonian for such
a system, with no interactions between the fermionic

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 GENERAL I ARTICLE

and bosonic oscillators, will be the sum of the individ-

ual Hamiltonians, i.e., H = Hb + H f . From here on, we
choose the same frequency w for both the bosonic and
fermionic oscillators. Thus

Our next step is to determine the symmetry possessed

by the combined system. Recalling the formal definition
of any symmetry, we shall evaluate the commutator of H
and the generators Q, Qt. This will determine whether
the combined system is invariant under supersymmetry'
transformation. Expanding one such commutator, we
get
[H, Q] = w[alab + a}af ,ab a }] =

w (alababa} + a}afaba} - aba}alab - aba}a}af )

= w (akababa} + ab(l - afa} )a} - (abab + l)aba}) = O.

Here we have used the commutation and anticommuta-
tion relations of the fermionic and bosonic operators,
and the non-interacting property of the bosonic and
fermionic oscillators, to simplify the bracket, and we see
that the Hamiltonian of the combined system H com-
mutes with the generator Q. This crucially depended
on choosing identical frequency w for the bosonic and
fermionic oscillators. Similarly, we can also show that
[H, Qt] = O. The zero commutator bracket implies that
the combined system described by the Hamiltonian H
is invariant under the supersymmetric transformation.
Hence, it is appropriate to call such a combined system
as supersymmetric oscillator.
The action of zero commutator bracket [H, Q] = 0 on a
state Inb' nf = 0) will prove that the states \nb, nf = 0)
and Q\nb, nf = 0) =
\nb - 1, nf = 1) are degenerate
eigenfunctions of the Hamiltonian H. It is not difficult
to verify that the generators Q, Qt obey the following

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 GENERAL I ARTICLE

relations:

Though we have illustrated supersymmetry and its gen-

erators using supersymmetric oscillator as an example,
we can take the above relations as the defining equations
for any supersymmetric system. In the next section, we
shall present the formalism of obtaining any supersym-
metric system from two systems called partner systems.
Supersymmetric Systems
Let us consider any general Hamiltonian HI, describing
a system in a potential Vi (x):

1i 2 d2
HI = - 2m dx 2 + V1(x),
whose energy eigenvalues and eigenstates are E~l) and
'l/J~l)(X), respectively. Using Schrodinger equation, we
can rewrite the form of the potential VI (x) in terms of
Ea
ground state energy 1) and ground state eigenfunction
'l/Jo(x) as
2 (I)"
Vi(x) = ~ 'l/J o (x) + EU).
2m 'If;a1) (x) 0

In order to obtain a supersymmetric system, we need

to combine this system with another system which will
satisfy all the defining relations (10) of supersymmetry.
For this purpose, it will be useful to factorise HI in the
following way:

Ii d t Ii d
Al = v2mdx + W(x),A 1 = - V2mdx + W(x),
where the function W (x) is called the superpotential.
The factorised form of the Hamiltonian implies VI (x) =
W2( x) - k
W' (x) +
1
Ed
) Extending the role of Al and

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 GENERAL I ARTICLE

A1 as the creation and annihilation operator for the sys-

tem described by HI, we can impose the following condi-
1
tion on the ground state wave function: AI1/J6 )(x) = O.
Hence, the solution for W(x) in terms of the ground
state wave function is

Now, using At and AI, we can construct a new Hamil-

tonian, H 2 , as follows:
2 2
_ t (1) _ 1i d
H2 - A1A1 + Eo - - 2mdx 2 + V2 (x),
whose energy eigenvalues and eigenfunctions are E~2)
and 'ljJ~) (x), respectively. The potential V2 (x) in terms
of superpotential is

The two Hamiltonians HI and H2 and their correspond-

ing eigenfunctions, 'ljJ~I)(X) and 'ljJ~2)(X), can be rewritten
in a compact form as follows: Suppose we construct a
matrix Hamiltonian

H = [HI -OEal) 0 ]
H2 - E6 1
) ,

whose eigenfunctions are the column vectors Wn(x) =

(~~~;~:D It is straightforward to check that the ma-
trix Hamiltonian H commutes with the following matrix
operators:

Q= [11 ~] and Qt = [~ ~t 1
Further, these matrices obey the defining relations (10)
of supersymmetry. Hence, we can say that the com-
bined system described by matrix Hamiltonian H is a
supersymmetric system.

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 GENERAL I ARTICLE

Applying the zero commutator bracket [H, Q] = 0 and

[H, Qt] = 0 on wn(x) gives the following equations:

H2 (Al1j;~I) (x)) AIHl1j;~I)(X) = E~I) (Al1j;~I)(X)),

HI (At 1j;~2) (x)) AtH21j;~2)(x) = E~2) (At1j;~2)(X))

Thus, when 1j;~I)(X)'S are eigenfunctions of HI, Al~~I)(X)

are eigenfunctions of H 2 . Similarly, when 1j;~2)(X)'S are
eigenfunctions of H 2, then A l1j;~2) (x) are eigenfunctions
of HI. Using a little algebra, we can show the following
relations:
E(2)
n
= E(I) . ,,1.(2)(x) = (E(I) _
n+l' If'n n+l
E(I»)-1/2 A"I.(I)
0
(x)·
If'n+l'

Thus, we have exploited supersymmetry to relate the

energy eigenfunctions of one Hamiltonian HI in terms
of the energy eigenfunctions of the partner Hamiltonian
H 2 • Therefore, HI and H2 are usually called supersym-
metric partner Hamiltonians, and their corresponding
potentials VI and 112 are called supersymmetric partner
potentials.
An Example
The concept of supersymmetric partner potentials im-
mediately suggests the following use: If we have a po-
tential which is difficult to solve analytically, but its
partner potential is relatively easy to solve, then by in-
voking the properties of supersymmetry we immediately
have all the energy eigenvalues and eigenfunctions of the
unsolvable potential. Let us illustrate the power of su-
persymmetry through a simpl.e example.
Consider a system described by a potential V1 (x) (infi-
ni te potential well):

o for 0 S; x S; L,
00 otherwise,

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 GENERAL I ARTICLE

whose wave functions and energy eigenvalues are known:

2
,,1.(1)( ) = {2. ((n + 1)1[) d E(l) = 1[2(n + 1)21i
If' n X YL sIn L x an n 2m£2'

where n = 0, 1,2, .. For algebraic simplicity, let us 2 The analysis of the problem is
take 1i = 2m = 1 2. identical even when we include
h and 2m.
The superpotential W (x) can be obtained using the above
ground state wave function in (11):
W - _ 1
1
d1/J6 ) (x) _ . II:!!.
YLL cos(:!!.x)
L _
- 1/J~l)(x) dx - \If sin( fx) -

= -L cot (Lx)
Hence, the supersymmetric partner potential V; (x) from
(12) for this example will be
1[2 1[
V;(x) = W'(x) + W 2 (x) + Ea 1) = £2cosec2(£x) +
1[2 1[ 1[2
2
£2 cot (£ x) + £2
21[2 1[
2
£2 cosec (L x).

Suppose we started with the above potential V2 (x), we

would have no clue about finding the energy eigenval-
ues and energy eigenfunctions. However, we used super-
symmetry formalism to derive the potential as a partner
potential. Therefore, using(13), the energy eigenvalues
and eigenfunctions are related to the energies and wave
functions of the particle in a potential well:

,,1.(2)
If'n
= [E(l)
n+l
_ E(1)]-1/2 A"I.(l)
0 If'n+l
=

= [1I"2(n+2)2 _ 11"2] -1/2

L2 L2
(.fL
dx
+ w) !I. sin (n+2)1I" X
VI L

-V
-
2
L[(n+2)2-1]
((
n
+ 2) cos (n+2)1I"
L x - co t IX
11" . (n+2)1I" )
SIn L x

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 GENERAL I ARTICLE

Thus, using the concept of supersynlmetry, we have sol-

ved a potential which is very difficult to solve by tradi-
tional methods.
Hierarchy of Hamiltonians
So far, we have seen that the ground state wave func-
tion of HI was used to determine the explicit form of
the creation and annihilation operators At and AI. Us-
ing supersymmetry, we derived the partner Hamiltonian
H2 whose eigenfunctions and energies are given by (13).
This procedure can be continued to obtain a tower of
Hamiltonians tied by supersymmetry relations. For in-
2
stance, we can refactorise H2 = A~A2 + ), where Ea
the forms of the new creation and annihilation opera-
tors (At and A 2 ) can be deduced from the ground state
wave function of H 2 . Similar to the procedure of de-
riving Hamiltonian H2 from its supersymmetric partner
HI, we can construct another Hamiltonian H3 from H 2 ·
This process can be continued. It may be noticed from
(13) that each newly constructed Hamiltonian will have
one fewer energy eigenstate than the previous one. So
if HI has s energy eigenstates, then we can construct a
hierarchy of (s - 1) Hamiltonians, all having the same
energy spectra, except that the mth Hamiltonian has
s + 1 - m energy eigenstates. This procedure will relate
the energy eigenvalues and eigenfunctions of the m-th
Hamiltonian Hm in terms of eigenvalues and wave func-
tions of HI in the following way:
(m) - E(m-I) - (1)
En - n+I - = E n +m - I ,
"I,(m) = (E(I) _ E(I) )-1/2 (
E(I) _ E(l))-I/2
\fin n+m-I m-2 n+m-I 0

Am - I . AI7jJ~~m-I'
and the corresponding potentials are
2
d (1)
Vm(x) = Vl(X) - 2 dX2 In(7jJo
Thus, knowing all the eigenvalues and eigenfunctions
of HI, we immediately know all the energy values and

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RESONANCE I February 2003
 GENERAL I ARTICLE

eigenfunctions of the hierarchy of (8 - 1) Hamiltonians.

Thus, a large number of potentials which are unsolvable
by traditional methods can be solved by this method.
Conclusion
In this article, we have presented the salient features of
symmetry with emphasis on supersymmetry. We have
shown that systems with non-trivial potentials can be
exactly solved by exploiting the properties of supersym-
metry.
Acknowledgements
We would like to thank S H Patil, U A Yajnik and
S Umasankar for their valuable comments and sugges-
tions. P R would like to thank N Ananthkrishnan for
critically reading this manuscript and suggesting some
nlodification in the presentation.
Suggested Reading
Address for Correspondence
[1] R M Godbole,Resonance, Vo1.5, No.2, p.16, 2000. Akshay Kulkarni and
[2] Sourendu Gupta, Resonance, Vol. 6, No.2, p. 29,2001. P Ramadevi
[3] S K Soni and Simmi Singh, Supersymmerry, Narosa Publishing House, Physics Department
India, 2000. Indian Institute of Technology
[4] A Khare,Physics Reports, Vol. 251, p. 267, 1995. Mumbai 400 076, India.
[5] R Dut~ A Khare and U Sukhatme,Am. J. o/Physics,Vol. 56, p.163, 1988.

~
The French "Gazette des mathematiciens" for January 2002 represented I G Petrovski; to
its readers as a "great mathematician. known for his parabolic equation" in the paper by
J Troue!. An analogous definition for Hadamard would have been "known as the author of
.1 I'. Hadamard's lemma which allows the division ofa smoothfunction by its argument", and
Hilbert would have been defined as the "creator of Hilbert space ", and Riemann as the
"inventor of Riemannian metrics n.

Petrovskii was one ofthefounders ofthe Moscow Mathematical School, the rector ofMoscow
University during about 20 of its best years and was one of the deepest and most creative
mathematicians ofthe 20th centwy. The work of PetrovskU on Hilbert's J6th probl~m laid
the foundations for the new subject of real algebraic geometry which continues to develop
actively to this day.
V I Arnold in Russian Mathematical Surveys, Vo1.57, 2002

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