q-Oscillators and a q-deformed Debye
Model for Lattice Heat Capacity
K. K. Leelamma
Abstract Quantum groups and quantum algebras are deformations of classical Lie
groups and their structure is much more complex than that of Lie groups. They
are symmetry groups of non-commutative spaces. The representation theory of the
quantum group SU q (2) has led to the development of q-deformed harmonic oscillator
algebra. It has found applications in several branches of physics such as vibrational
spectroscopy, nuclear physics, many-body theory and quantum optics. This article
introduces the q-oscillator and gives a brief description of the studies carried out by
the author using the concept of q-oscillator algebra, under the supervision of Prof.
K. Babu Joseph. It includes a q-oscillator Debye model for the lattice heat capacity
of crystals and a q-anharmonic oscillator with quartic interaction.
Keywords Quantum group · q-Oscillator · Debye model · q-Commutators ·
Anharmonic oscillator
1 Introduction
History of physics has several stories of deformations. In classical mechanics itself,
the Lorentz transformation between two inertial frames is a deformation of Galilean
transformation with β =
v
c
as the deformation parameter. In the limit β → 0, the
original non-relativistic mechanics is regained. Thus special relativity is a deformation of Galilean relativity. Similarly, quantum mechanics is a deformation of classical
mechanics with as the deformation parameter. In the limit → 0, the results of
quantum mechanics merge with the classical results. Quantum groups and quantum
algebras are deformations of classical Lie groups. Quantum group first appeared
in physics literature as a deformation of the universal enveloping algebra of a Lie
algebra in the study of integrable quantum systems. In the beginning of the 1980s,
there was much progress made in the field of quantum integrable field theories. One
K. K. Leelamma (B)
Department of Physics, Union Christian College, Aluva, Kerala, India
e-mail: leelamma_kk@yahoo.co.in
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_6
71
Model for Lattice Heat Capacity
K. K. Leelamma
Abstract Quantum groups and quantum algebras are deformations of classical Lie
groups and their structure is much more complex than that of Lie groups. They
are symmetry groups of non-commutative spaces. The representation theory of the
quantum group SU q (2) has led to the development of q-deformed harmonic oscillator
algebra. It has found applications in several branches of physics such as vibrational
spectroscopy, nuclear physics, many-body theory and quantum optics. This article
introduces the q-oscillator and gives a brief description of the studies carried out by
the author using the concept of q-oscillator algebra, under the supervision of Prof.
K. Babu Joseph. It includes a q-oscillator Debye model for the lattice heat capacity
of crystals and a q-anharmonic oscillator with quartic interaction.
Keywords Quantum group · q-Oscillator · Debye model · q-Commutators ·
Anharmonic oscillator
1 Introduction
History of physics has several stories of deformations. In classical mechanics itself,
the Lorentz transformation between two inertial frames is a deformation of Galilean
transformation with β =
v
c
as the deformation parameter. In the limit β → 0, the
original non-relativistic mechanics is regained. Thus special relativity is a deformation of Galilean relativity. Similarly, quantum mechanics is a deformation of classical
mechanics with as the deformation parameter. In the limit → 0, the results of
quantum mechanics merge with the classical results. Quantum groups and quantum
algebras are deformations of classical Lie groups. Quantum group first appeared
in physics literature as a deformation of the universal enveloping algebra of a Lie
algebra in the study of integrable quantum systems. In the beginning of the 1980s,
there was much progress made in the field of quantum integrable field theories. One
K. K. Leelamma (B)
Department of Physics, Union Christian College, Aluva, Kerala, India
e-mail: leelamma_kk@yahoo.co.in
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_6
71