78
K. K. Leelamma
of more complicated systems expressed in terms of their normal modes. Thus its
applications are not limited to molecular spectroscopy, but extend to a variety of
branches of modern physics such as condensed matter physics, nuclear structure,
quantum field theory, quantum optics, quantum statistical mechanics and so forth.
The studies carried out by the author is essentially based on the q-deformed harmonic
oscillator introduced in 1989 by Biedenharn [33] and independently by Macfarlane
[34].
The creation, annihilation and number operators a
† , a and N (= a
† a) satisfy the
commutation relations
[a, a
†
] = 1
( 3 4 )
[N , a
†
] = a
†
(35)
[N , a] = −a
(36)
The generators of su(2) can be realised in the form
J + = −
1
2
a
† 2 ; J − =
1
2
a
2
; J z =
1
4
(aa
†
+ a
† a)
(37)
To define a q-analogue to the harmonic oscillator, the q-deformed operators a q and
a
†
q , and the q-boson vacuum ket |0 q are considered. Biedenharn and Macfarlane
independently found two-commutation relations for a q and a
†
q , which are equivalent
and which can be written in an alternate form [35] as
[a q , a
†
q ] q = a q a
†
q − qa
†
q a q = q
−N q
(38)
where the operator N q satisfies the commutation relations
[N q , a
†
q ] = a
†
q ; [N q , a q ] = −a q
(39)
N q is Hermitian, but
N q = a
†
q a q
(40)
[a q , a
†
q ] q is the q-commutator. The algebra defined by (38) and (39) is referred to as
q-oscillator algebra. In the limit q → 1, it tends to the standard oscillator algebra.
The operators a
†
q , a q and N q are referred to as q-boson creation operator, q-boson
annihilation operator and q-boson number operator, respectively.
To realise the Lie algebra of the generators of SU q (2), a pair of mutually commuting q-harmonic oscillator systems with operators a iq and a
†
iq with i = 1, 2 is
considered. Then the q-analogue of the Jordan-Schwinger map is defined:
J + = a
†
1q a 2q ; J − = a
†
2q a 1q = J
†
+ ; J z =
1
2
(N 1q − N 2q )
(41)
K. K. Leelamma
of more complicated systems expressed in terms of their normal modes. Thus its
applications are not limited to molecular spectroscopy, but extend to a variety of
branches of modern physics such as condensed matter physics, nuclear structure,
quantum field theory, quantum optics, quantum statistical mechanics and so forth.
The studies carried out by the author is essentially based on the q-deformed harmonic
oscillator introduced in 1989 by Biedenharn [33] and independently by Macfarlane
[34].
The creation, annihilation and number operators a
† , a and N (= a
† a) satisfy the
commutation relations
[a, a
†
] = 1
( 3 4 )
[N , a
†
] = a
†
(35)
[N , a] = −a
(36)
The generators of su(2) can be realised in the form
J + = −
1
2
a
† 2 ; J − =
1
2
a
2
; J z =
1
4
(aa
†
+ a
† a)
(37)
To define a q-analogue to the harmonic oscillator, the q-deformed operators a q and
a
†
q , and the q-boson vacuum ket |0 q are considered. Biedenharn and Macfarlane
independently found two-commutation relations for a q and a
†
q , which are equivalent
and which can be written in an alternate form [35] as
[a q , a
†
q ] q = a q a
†
q − qa
†
q a q = q
−N q
(38)
where the operator N q satisfies the commutation relations
[N q , a
†
q ] = a
†
q ; [N q , a q ] = −a q
(39)
N q is Hermitian, but
N q = a
†
q a q
(40)
[a q , a
†
q ] q is the q-commutator. The algebra defined by (38) and (39) is referred to as
q-oscillator algebra. In the limit q → 1, it tends to the standard oscillator algebra.
The operators a
†
q , a q and N q are referred to as q-boson creation operator, q-boson
annihilation operator and q-boson number operator, respectively.
To realise the Lie algebra of the generators of SU q (2), a pair of mutually commuting q-harmonic oscillator systems with operators a iq and a
†
iq with i = 1, 2 is
considered. Then the q-analogue of the Jordan-Schwinger map is defined:
J + = a
†
1q a 2q ; J − = a
†
2q a 1q = J
†
+ ; J z =
1
2
(N 1q − N 2q )
(41)