q-Oscillators and a q-deformed Debye Model for Lattice Heat Capacity
83
i.e., in the limit q → 1, the energy spectrum of q-deformed harmonic oscillator
coincides with that of the standard harmonic oscillator. Besides the energy spectrum, other properties of q-oscillators are also well studied. For example, coherent states and squeezed states of q-harmonic oscillators have been investigated by
Vinod et al. [37].
1.4.3 Studies of Physical Systems using q-Oscillator Algebra
Numerous applications of quantised algebra to real physical systems have been
worked out by various authors. A few of them are cited here. When used to describe
the vibrational spectra of diatomic molecules [38] such as H 2 , it is seen that when
q is chosen as a pure phase, the results show fair agreement with the experimental data, for η 0.06. A q-rotator model with su q (2) symmetry has been set up to
describe the rotational spectra of diatomic molecules [39]. For deformation parameter η 0.01, the spectra of the q-rotator model coincide with the observed spectra
to satisfactory accuracy. A complete quantum group theoretic treatment of vibrating
and rotating diatomic molecules has also been given [40] by assuming the deformation parameter q to depend on the rotational quantum number J . The coincidence
between the predictions of the model and conventional phenomenological formulae
is remarkable. The su q (2) algebra has been used for the description of energy spectra
of the deformed even-even nuclei [41], and it is shown that there is good agreement
with experimental results when q is chosen as a phase with η 0.04. The manybody problem of q-oscillators has been investigated by several authors [42–44]. The
spectra of the system are found to be rich, exhibiting interactions between the levels of the individual oscillators. The deformed algebra has also been employed to
the many-body problem of composite particles. The q-oscillator models in two and
higher dimensions are applied to the spectra of triatomic molecules such as H 2 O
and superdeformed nuclei. The nature of an electromagnetic field of high intensity
modelled by q-oscillators also has been studied [45].
2 q-Oscillator Debye Model for Lattice Heat Capacity
The Debye model for lattice heat capacity is modified retaining all the basic assumptions except that each mode is here treated as a q-deformed harmonic oscillator [17].
The two basic experimental facts about the heat capacity of solids which any theory
must explain are (i) at room temperature, the heat capacity of most solids is close to
3k B per atom so that for molecules consisting of n atoms, the molar heat capacity is
close to 3n R where R is the universal gas constant. Accurate measurements indicate
temperature dependence of heat capacity in this region. (ii) At low temperatures, the
heat capacities decrease and vanish at T = 0 K. The decrease goes as T
3 .
The Debye model has been successful in describing the experimental observations
at low temperatures in many pure crystalline solids. In the low temperature regime,
the Debye’s theory predicts
83
i.e., in the limit q → 1, the energy spectrum of q-deformed harmonic oscillator
coincides with that of the standard harmonic oscillator. Besides the energy spectrum, other properties of q-oscillators are also well studied. For example, coherent states and squeezed states of q-harmonic oscillators have been investigated by
Vinod et al. [37].
1.4.3 Studies of Physical Systems using q-Oscillator Algebra
Numerous applications of quantised algebra to real physical systems have been
worked out by various authors. A few of them are cited here. When used to describe
the vibrational spectra of diatomic molecules [38] such as H 2 , it is seen that when
q is chosen as a pure phase, the results show fair agreement with the experimental data, for η 0.06. A q-rotator model with su q (2) symmetry has been set up to
describe the rotational spectra of diatomic molecules [39]. For deformation parameter η 0.01, the spectra of the q-rotator model coincide with the observed spectra
to satisfactory accuracy. A complete quantum group theoretic treatment of vibrating
and rotating diatomic molecules has also been given [40] by assuming the deformation parameter q to depend on the rotational quantum number J . The coincidence
between the predictions of the model and conventional phenomenological formulae
is remarkable. The su q (2) algebra has been used for the description of energy spectra
of the deformed even-even nuclei [41], and it is shown that there is good agreement
with experimental results when q is chosen as a phase with η 0.04. The manybody problem of q-oscillators has been investigated by several authors [42–44]. The
spectra of the system are found to be rich, exhibiting interactions between the levels of the individual oscillators. The deformed algebra has also been employed to
the many-body problem of composite particles. The q-oscillator models in two and
higher dimensions are applied to the spectra of triatomic molecules such as H 2 O
and superdeformed nuclei. The nature of an electromagnetic field of high intensity
modelled by q-oscillators also has been studied [45].
2 q-Oscillator Debye Model for Lattice Heat Capacity
The Debye model for lattice heat capacity is modified retaining all the basic assumptions except that each mode is here treated as a q-deformed harmonic oscillator [17].
The two basic experimental facts about the heat capacity of solids which any theory
must explain are (i) at room temperature, the heat capacity of most solids is close to
3k B per atom so that for molecules consisting of n atoms, the molar heat capacity is
close to 3n R where R is the universal gas constant. Accurate measurements indicate
temperature dependence of heat capacity in this region. (ii) At low temperatures, the
heat capacities decrease and vanish at T = 0 K. The decrease goes as T
3 .
The Debye model has been successful in describing the experimental observations
at low temperatures in many pure crystalline solids. In the low temperature regime,
the Debye’s theory predicts