Uncertainty Relation in Quantum Mechanics with Quantum Group(5)

We study the commutation relations, uncertainty relations and spectra of position and momentum operators within the framework of quantum group % symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an effect of the underlying noncom

[xs,xr]=i

Fors<ronegets:

[xs,pr]=iKrq+1Ks

q+1

Ls{xs,pr}{ps,pr}(16)(17){xs,pr}(18)q+1

Toseethis,solveEqs.7forthea’sanda ’s,expressEqs.1-3intermsofthex’sandp’sand nde.g.Eq.15fromEq.1+(Eq.1)++Eq.3+(Eq.3)+.Ifq2=1theconstantsKandLdropoutofthecommutationrelations,re ectingthatinordinaryquantummechanicsalengthoramomentumscalecanonlybesetbytheHamiltoniani.e.bychoosingaparticularsystem.Here,forq2=1theKandLappearinthecommutationrelations,thusthesescalesbecomeapropertyofthequantummechanicalformalismitself.[ps,pr]= i

2.2Amaximalsetofcommutingobservables

Theoperatorsxi(aswellastheoperatorspi)nolongercommuteamongthemselves.Thusweconcludethatthepositionoperatorscannotbesimultaneouslydiagonalisedandthesameforthemomentumoperators.

Beforestudyingthestructureofthenoncommutative’con gurationspace’andthenoncommutativemomentumspaceinmoredetail,letusmentionthatforexamplethefollowingsymmetricoperatorshicanstillserveasasetofcommutingobservables:

(r=1,...,n)4Ki2

Thesymmetryisobvious.Toprovecommutativityweusethat

[a iai,ajaj]=0hi:=x2i(19) i,j(20)

whichfollowsimmediatelyfromEqs.1-3.

Withthede nitionsEqs.7follows

a rar=x2r

24Kr 1

4iLrKr ih¯+ih¯(q2 1)s≤r q2+1