The q‐deformed three‐dimensional harmonic oscillator is defined in terms of the q‐bosons corresponding to the spherical components of a nondeformed three‐dimensional oscillator. It is shown that the dynamical algebra is spq(6,R). Two important subalgebra chains are identified: spq(6,R)⊇suq(3)⊇soq(3) and spq(6,R) ⊇ spq2(2,R) ⊕ soq(3). The basis states of the q‐deformed oscillator are classified according to these subalgebras. Finally, the Hamiltonian eigenvalues are discussed.

1.
P.
Kulish
and
N.
Reshetikhin
,
J. Sov. Math.
23
,
2435
(
1983
).
2.
V.
Drinfeld
,
Sov. Math. Dokl.
32
,
254
(
1985
).
3.
M.
Jimbo
,
Lett. Math. Phys.
10
,
63
(
1985
);
M.
Jimbo
,
11
,
247
(
1986
).,
Lett. Math. Phys.
4.
C. Zachos, “Paradigms of quantum algebras,” Argonne National Laboratory preprint ANL-HEP-PR-90-61 (1992), and references therein.
5.
L. C.
Biedenharn
,
J. Phys. A
22
,
L873
(
1989
).
6.
A. J.
Macfarlane
,
J. Phys. A
22
,
4581
(
1989
).
7.
D.
Bonatsos
,
P. P.
Raychev
,
R. P.
Roussev
, and
Yu. F.
Smirnov
,
Chem. Phys. Lett.
175
,
300
(
1990
).
8.
D.
Bonatsos
,
E. N.
Argyres
, and
P. P.
Raychev
,
J. Phys. A
24
,
L403
(
1991
).
9.
D.
Bonatsos
,
C.
Daskaloyannis
, and
K.
Kokkotas
,
J. Phys. A
24
,
L795
(
1991
).
10.
D.
Bonatsos
,
C.
Daskaloyannis
, and
K.
Kokkotas
,
J. Math. Phys.
33
,
2958
(
1992
).
11.
M. G. Mayer and J. H. D. Jensen, Elementary Theory of Nuclear Shell Structure (Wiley, New York, 1955).
12.
F.
Iachello
and
R. D.
Levine
,
J. Chem. Phys.
77
,
3046
(
1982
).
13.
C.
Quesne
and
M.
Moshinsky
,
J. Math. Phys.
12
,
1780
(
1971
).
14.
B. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974).
15.
M.
Moshinsky
and
C.
Quesne
,
J. Math. Phys.
12
,
1772
(
1971
).
16.
G.
Couvreur
,
J.
Deenen
, and
C.
Quesne
,
J. Math. Phys.
24
,
779
(
1983
).
17.
C.
Quesne
,
J. Phys. A
18
,
2675
(
1985
).
18.
J.
Van der Jeugt
,
J. Phys. A
25
,
L213
(
1992
).
19.
J. Van der Jeugt, Subalgebras of Quantum Enveloping Algebras and Applications, Prod. XIX Int. Coll. Group Theoretical Methods in Physics 1992, edited by J. Mateos, M. A. del Olmo, and M. Santander (in press).
20.
A. Sciarrino, Deformed U(Gl(3)) fromSOq(3), Proc. of Symmetries in Science VII: Spectrum Generating Algebras and Dynamics in Physics 1992, edited by B. Gruber (in press).
21.
P. P.
Kulish
and
E. V.
Damaskinsky
,
J. Phys. A
23
,
L415
(
1990
).
This content is only available via PDF.
You do not currently have access to this content.