Uncertainty Relation in Quantum Mechanics with Quantum Group(7)

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

3.1Minimaluncertaintiesinpositionandmomentum

Inordertoanalysethecontentofthisuncertaintyrelationweexpressitin’polarcoordinates’

x:=2Lrcosαand p:=2Krsinα(31)

whereittakestheform:

r≥21+(q 1)2 x 2

4K2

q2+1(33)

FromEq.33followsthattheminimalαislargerthan0andthemaximalαissmallerthanπ/2.Thusthehyperbolaoftheordinaryuncertaintyrelation,havingthe xandthe paxesasasymptoteshasturnedintoagraphwithasymptotesthatarenolongerparalleltotheaxes.FromEq.32followsthatrisalwayslargerthan0.Wethusconcludethatthereareminimalnonzerouncertaintiesinthepositionsandthemomenta.

Inordertocalculatethemwede ne

f( x, p):= x p h¯

4L2+( p)2+ p 2

p

whichhasthesolution:f( x, p)=0andf( x, p)=0(35)

( xmin)=L22q2 1

4L2+ p 2

q2 1

q2 1