q-Oscillators and a q-deformed Debye Model for Lattice Heat Capacity
77
[x∂ x , x] = x
(28)
[x∂ x , D x ] = −D x
(29)
Here a and b are constants and f and g are functions of x.
1.3 su q (2) Algebra
One of the well-studied quantum groups is SU q (2) which is the q-deformation of
the classical group SU (2), the group of angular momentum. The Lie algebra su(2)
consists of three elements L + , L − and L z which satisfy the commutation relations
[L z , L ± ] = ±L ±
(30)
[L + , L − ] = 2L z
(31)
with
L
†
+ = L −
(32)
Kulish and Reshetikhin [4], while studying the solution of YBE, introduced the
algebra of three elements J + , J − and J z :
[J z , J ± ] = ±J ± ;
[J + , J − ] = [2J z ] q
(33)
In the limit q → 1, this algebra goes into su(2) algebra. Thus it is called qdeformation of su(2) algebra and is denoted by su q (2). Both su q (2) and sl q (2) are
quantum algebras with a single deformation parameter q. Going to higher dimensions with more than two non-commuting co-ordinates, one has to use more than one
deformation parameter. Several authors have worked on two-parameter deformations
[31, 32].
1.4 q-Harmonic Oscillator
The simple harmonic oscillator (SHO) problem has an indispensable role in physics.
It is customary to use the SHO to illustrate the basic concepts and new methods
in classical as well as quantum physics. The wave mechanical theory of oscillators
provides the basis for understanding the properties of a wide variety of systems which
are analysable in terms of harmonic oscillators. It is useful not only in the study of
vibrations of diatomic and polyatomic molecules, but also in the study of vibrations
77
[x∂ x , x] = x
(28)
[x∂ x , D x ] = −D x
(29)
Here a and b are constants and f and g are functions of x.
1.3 su q (2) Algebra
One of the well-studied quantum groups is SU q (2) which is the q-deformation of
the classical group SU (2), the group of angular momentum. The Lie algebra su(2)
consists of three elements L + , L − and L z which satisfy the commutation relations
[L z , L ± ] = ±L ±
(30)
[L + , L − ] = 2L z
(31)
with
L
†
+ = L −
(32)
Kulish and Reshetikhin [4], while studying the solution of YBE, introduced the
algebra of three elements J + , J − and J z :
[J z , J ± ] = ±J ± ;
[J + , J − ] = [2J z ] q
(33)
In the limit q → 1, this algebra goes into su(2) algebra. Thus it is called qdeformation of su(2) algebra and is denoted by su q (2). Both su q (2) and sl q (2) are
quantum algebras with a single deformation parameter q. Going to higher dimensions with more than two non-commuting co-ordinates, one has to use more than one
deformation parameter. Several authors have worked on two-parameter deformations
[31, 32].
1.4 q-Harmonic Oscillator
The simple harmonic oscillator (SHO) problem has an indispensable role in physics.
It is customary to use the SHO to illustrate the basic concepts and new methods
in classical as well as quantum physics. The wave mechanical theory of oscillators
provides the basis for understanding the properties of a wide variety of systems which
are analysable in terms of harmonic oscillators. It is useful not only in the study of
vibrations of diatomic and polyatomic molecules, but also in the study of vibrations