Setup of deformed oscillators.

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Setup of deformed oscillators.
Properties of deformed oscillators.
Energy level degeneracies in the spectrum of
deformed oscillators.
Fibonacci property and deformed oscillators.
Application of deformed oscillators to Bose-gas
model.
Anastasiia Rebesh
ISSP 2009
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If instead of treating particles as
point-like structureless objects, one
attempts to take into account either nonzero
proper volume or composite nature of particles,
then it is natural to modify or deform the
standard commutation relations. For this and
other reasons so-called deformed oscillators
play important role in modern physics, and
various quantum or q-deformed algebras show their
efficiency in diverse problems of quantum
physics.
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The generating elements of the algebra of
deformed oscillators are
The defining commutation relations are
where the notion of the structure function
is used
For usual harmonic oscillator,
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ISSP 2009
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Vacuum and n-particle excited states are
The action of annihilation and creation operators
in q-analog of Fock space is
Hamiltonian is
Our subject is three-parameter family of
deformed oscillators given by
(here
are the deformation parameters).
The energy takes the form
Anastasiia Rebesh

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Usual quantum harmonic oscillator
Two-parametric p,q-deformed oscillator
plenty of one-parameter models
Tamm- Dancoff model
Arik-Coon q-oscillator
Biedenharn- Macfarlane model
- oscillator
A.M. Gavrilik, A.P. Rebesh, Mod. Phys. Lett. A
23, 921 (2008)
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Energy level degeneracies of p,q-oscillator
We assume that
and (0,0) is excluded.
Proposition. There exist infinite set of values
(p,q) (i.e., the continuum of points of curve
for which the degeneracy
takes place
The curve is given by continual, monotonously
decreasing function
To prove the statement one has to show that
A.M. Gavrilik, A.P.Rebesh, Ukr. J. Phys., V.53,
no. 6 (2008)

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At m1
E(4)E(5)
p
At m4
E(1)E(2)
q
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In the energy spectrum of (µp,q)-oscillator
can be the degeneracy, say of two neighbour
levels,
(there is no degeneracy if pq1 and
The particular case of two neighbouring energy
level degeneracy,
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ISSP 2009
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Fibonacci property of oscillators
Fibonacci numbers are such that
and the relation
does hold.
For the energy spectra of oscillators, we have
generalized relation
are constants.
Usual quantum oscillator
Fibonacci
oscillators
p,q-oscillator
-oscillator
?
n
n
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Using deformed oscillators, corresponding models
of Bose-gas are constructed. The two-particle
correlation functions are defined as
in terms of probabilities of one- and
two-particle generation.
we have
for
Introducing
and
intercept
Since
Intercept of two-particle correlation function in
terms of
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ISSP 2009
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A.M. Gavrilik, SIGMA, V.2, Paper 074 (2006)
Intercept calculated on the base of
Biedenharn-Macfarlane oscillator reads
Fig.3. Intercept
versus pions
transverse momentum
at T180 MeV
Fig.3. Two central surfaces (for central values
of exper. data) as implicit functions
p
is shown
q
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In the case of µ-oscillator
Using the relation for and a function
it is easy to get
Assuming that µ is small we expand the latter
expression as
To get right asymptotics, we choose suitable
approximation on
Then
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Thermal averages of
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µ 0.1
A
A, C T120 MeV
B, D T150 MeV
B
Curves are shown at different values of
deformation parameter µ and T.
µ 0.12
C
D
Asymptotics
Experimental data
Pion Interferometry in AuAu and CuCu Collisions
at
and 200 GeV, arXiv 0903.1296v2 nucl-exp.
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We have considered 3-parameter family of
(µp,q)-deformed oscillators. It contains, as
different limits, the known one-parameter
oscillators.
Some properties of deformed oscillators
were examined, in particular, energy level
degeneracies and Fibonacci property.
An approach based on some set of
q-oscillators along with their related model of
ideal gas of q-bosons is quite succesful if one
attempts to effectively describe the observed in
experiments on relativistic heavy-ion collisions
non-Bose properties of the intercept of
two-particle correlation function.
The graphics of 2-particle correlation
intercept obtained within µ-Bose gas model
show the trends (features) in good correspondence
with peculiarities of exper.data gathered
at RHIC on the intercept of two-pion
correlations.
Thank you for attention!