Uncertainty Relation in Quantum Mechanics with Quantum Group(11)

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

Toseethiswerewritetherecursionformulainmatrixform:

fr+1

fr

Theiterationmatrixsimpli esforlargerto

=1[r]q[r+1]q Lλ2L [r 1]q[r+1]q fr 1fr 2 (59)(60)

[r 1]q/[r]q

1/|q|0

0 1/|q| andeventuallygoeslike(61)

Sinceq2>1wecanthusapplythequotientcriterium(behaviourlikeageometricalseries)toconcludethat

r=0∞ (λ)fr(λ)<∞fr(62)

i.e.thatallvλarenormalisable3.TheyareobviouslycontainedinDx andarethuseigenvectorsofx .Sincetherearenonrealeigenvaluesweconcludethatx isnotsymmetric.Thisallowsitseigenvectorstobelinearilydependend.TheyareactuallyingenerallinearilydependendoneachothersincetheHilbertspaceHisseperableandthereisanuncountablein nitenumberofeigenvectorsvλ.Ananalyticexpressionforthescalarproductoftwonormalisedeigenvectors v λ,v λ′ hasnotyetbeenworkedout.However,thenumericalapproximationconvergesasquicklyasageometricalseries.

Theoperatorx ismuchbetterbehavedthanx ,sinceitisclosedandsymmetric,aseverybi-adjointofadenslyde nedsymmetricoperator.

Itsdomain

Dx ={v∈H| w∈H a∈Dx : v,x .a = w,a }(63)

isinbetweenthoseofxandx :Dx Dx Dx anditdoesnotcontainanyeigenvectorsvλ.

4.2Self-adjointextensions

Wenowapplythestandardprocedure,seee.g.[18,19]4,forcheckingforself-adjointextensionsofclosedsymmetricoperators: