Uncertainty Relation in Quantum Mechanics with Quantum Group(8)

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

DuetoEq.25thereweretwofreeparameters:ThelengthLandq.Insteadwecannowuse x0and p0asthefreeparametersandexpressL,Kandqintermsofthese:

L= x0 4( x0)2( p0)2+(¯h)2

2 x0 p0+h¯+

4 x0 p0

4( x0)2( p0)2+h¯2(40) q=/h¯(41)

ThecommutationrelationEq.24thentakestheform:

[x,p]=ih¯+ig( x0, p0)

where

g( x0, p0):=4 x0 p0 x2( p0)2 (42)4( x0 p0)2+h¯2 h¯

4( x0 p0)2+h¯+h¯2(43)

Letusnowidentifythephysicalregionwheretheordinaryquantummechanicalbe-haviourisrecovered:

Sincephysicallyweknowthat x0and p0canonlybeverysmall,say x0 p0 h¯/2,weexpandgtothe rstnonzeroorderinthisproductandarriveatthesimpli edcommutationrelation:

[x,p]=ih¯+4i

4( p0)2

( p0) p 22(45)h¯2