-
66
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sa}eulprooc arlJ
(d .g : ll4
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,(qlo - tq?o),oz : f[y,ty]
gpasodord a.re suor?€lal uor+ts?ntuuroc 3ulno11o; aq1 'og
aqr sr d1 areqn ,ot : T [ 1nd am
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,{lrriffinuiruocuou Jo ralau"r"d e.IJ
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-
lot, PA : lb6, Xil : lb6, Pil: 0. So, X* Pt and 9ii : #@eb1 -
aib) satisfy the same commutation
relations as in the case of the canonical version of
noncommutativity (1). Moreover, algebra (2)
is rotationally invariant. 5
Energy levels of the hydrogen atom in noncommutative space
Ip order to find corrections to thelenergy levels of the
hydrogen atom in rotationally invariant
noncommutative space (2) we consider the following
Hamiltonian
H : Hn* Hor",
where Ho,": %.-.W"+ry{e+ryE and, H7: #-*t"the Hamiltonian of the
hydrogen
atom, here R : \f DrX7.
It is convenient to use the followin$ representation Xi:
ri+tl?xPl,, k : pi)whete B : ff[axbJ,coordinates r; and momenta p,;
satisfy the ordinary commutation relations. Thereforepe can
rewrite Hamiltonian (3) as follows H : Ho* 11, here H0 : * -
* * ror",, : 1fDurf;, and
7 is the perturbation caused by the noncommutativity of
coordinates I/ : -e2lR+ e2 f r.
Note that in the case of &r -+ oo harmonic oscillator put
into the ground state remains in it.Therefore, using the
perturbation theory, up to the second order in 0 we obtain
L'Er1 :a5-n5 1)(2t + 1) }t(t + 1)(2t + t)(2t + 3)(21 - 1)
(4)
(3)
tr2 e2 (02) (- - K r l
&hn" \
1
X6'(r +6n'
@+ 15n
2 - 2 t ( t + t )
+ bnz -Jl( l + 1) + 1
2(r + 2)(2t + 1)(21 + 3)( l - r ) (2t - r ) 6 t ( t + t ) ( t +
Dot + 1 ) (21+ 3) ( ' - 1 ) (2 t - 1 ) ) '
where (0') : kb3,o,orlr8,o,ol0'1,,h3,0,0,,h8,0,0) : ffi,n"r"
,lr3,o,o, t\,o,n ur"the well known eigenfunc-
tions of the harmonic oscillators, and as is the Bohr
radius.5
It is worth mentioning that in the case of I : 0 or I : 1 the
corrections (4) are divergent. Theproblem of divergence of the
corrections to the ns energy levels was studied in Ref. 5.
Conclusion
We have considered the generalization of the constant
antisymmetric matrix 06i to a tensor thatgives the possibility to
construct rotationally invariant noncommutative algebra. The
hydrogeuatom have been studied in noncommutative space (2). We have
found the corrections to the
energy levels of the hydrogen atom.
Acknowledgments
The author is grateful to Prof. V. M. Tkachuk for his advices
and great support during research
studies. The author thanks Dr. A. A. Rovenchak for a careful
reading of these proceedings. The
author also thanks the organizers of the Tfans-European School
of High Energy Physics for thepossibility to attend the School.
100
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