Uncertainty Relation in Quantum Mechanics with Quantum Group(10)

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

rangesinD′.Theuncertaintyrelationimpliedminimalnonzerouncertaintiesinthepositionsandmomenta.

Nowiftherewasavλ∈D′thatiseigenvectore.g.ofx:

x.vλ=λvλ

onewouldthenofcoursehave

( x)2= vλ|(x vλ,x.vλ )2|vλ =0(53)(52)

whichwouldbeacontradiction.Wethusconcludethatthereisnodomainonwhichxandparesymmetricandhaveeigenvectors.Letusnowstudythefunctionalanalysisofxinmoredetail,theanalysisforpiscompletelyanalogous.

4.1Theoperatorsx,x andx

WestartbechoosingforxthedomainDx:=D(the nitelinearcombinationsofthevectors(a )r|0 withr=0,1,2,...),onwhichxandpareobviouslysymmetricandhavetheirimageinDx.WecanthusalreadyconcludefromabovethatxhasnoeigenvectorsinDx.Indeed,theeigenvalueproblem

x.vλ=λvλwithvλ=∞ fr(λ)(a )r

[r]q!|0 (54)

r=0

canbesolvedforallcomplexλ,butfromtherecursionformulathatweobtainforthecoe cientsfr(λ)ofvλitisclearthatin nitelymanyofthemarenonzero,thusvλ∈Dx:Explicitely,intheorthonormalisedbasis

er:=

thematrixelementsofxare

xrs=L( (a )r[r]q!|0 (55)

[s]qδr+1,s)(56)

andtherecursionformulathatweobtainforthecoe cientsfr(λ)ofthevectorvλthusreads: fr L