Uncertainty Relation in Quantum Mechanics with Quantum Group(4)

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

achievedifthecommutationrelationscomeoutintheform[x,p]=ih¯+f(q,x,p)i.e.withthecentraltermbeingih¯withoutanyq-factors.Theuncertaintyrelationh¯ x p≥ f(q,x,p) shouldthenreducetotheusualrelationswhere f is2negligible.Theactualcommutationrelationscomeoutasfollows:

2.1Commutationrelations

WeexpressthecommutationrelationsEqs.4intermsofthex’sandp’s:

[x4iLrKrr,pr]=

x2s

q2+1s≤r4Ks2 1

t<r

2s

4L2+p

s

2

q2+1 ih¯LrKq 1

rLtKt

SolvingthisequationwegetrelationsbetweentheLrandKr:

Lh¯rKr:=2 r

Fromthisfollowsβrimmediately:

β 1 1

r=4Lq2

rKr2 r

Thusthecommutationrelationstake nalform:

[xr,pr]=ih¯+ih¯(q2 1) s

s q2+1≤r4L2+p2

s

q 1

Kr(11)(12)(13)