Particle in uniform field in noncommutative space with preserved time reversal and rotational symmetries
aa r X i v : . [ h e p - t h ] F e b Particle in uniform field in noncommutative space with preservedtime reversal and rotational symmetries
Kh. P. Gnatenko , Kh. I. Stakhur, A. V. Kryzhova Department for Theoretical Physics, Ivan Franko National University of Lviv12 Drahomanov Str., Lviv, 79005, Ukraine
Abstract
Quantized space described by time reversal invariant and rotationally invariant noncommutati-ve algebra of canonical type is studied. A particle in uniform field is considered. We findexactly the energy of a particle in uniform field in the quantized space and its wavefunctions.It is shown that the motion of the particle in the field direction in time reversal invariant androtationally invariant noncommutative space is the same as in the ordinary space (space withthe ordinary commutation relations for operators of coordinates and operators of momenta).Noncommutativity of coordinates has influence only on the motion of the particle in the di-rections perpendicular to the field direction. Namely, space quantization has effect on the massof the particle.
Key words: quantized space, noncommutative coordinates, time reversal symmetry, rotati-onal symmetry, particle in uniform fieldPACS number(s): 02.40.Gh, 03.65.-w
According to the String theory and Quantum gravity a minimal length of the order of thePlanck length exists [1, 2]. A space with minimal length can be described on the basis of ideaof deformation of the commutation relations for operators of coordinates and operators ofmomenta.Many different deformed algebras leading to the minimal length were proposed. One candistinguish three types of the algebras: noncommutative algebras of canonical type (see, forinstance, [3–6]), noncommutative algebras of Lie type (see, for instance, [7–9]), nonlineardeformed algebras (see, for instance, [10–12]). Noncommutative algebras of canonical type arethe most simple algebras which describe space quantization on the Planck scale. Algebra withnoncommutativity of coordinates of canonical type is characterized by the following relations [ X i , X j ] = i ¯ hθ ij , (1) [ X i , P j ] = i ¯ hδ ij , (2) [ P i , P j ] = 0 . (3) E-Mail address: [email protected] θ ij are called parameters of noncommutativity which are elements of constant matrixes.Properties of physical systems in the frame of noncommutative algebra of canonical type havebeen widely studied (see, for instance, [3–6,13–19], and references therein). Among the problemsnoncommutative gravitational quantum well has been examined [14,18,19]. It is worth stressingthat in noncommutative space characterized by commutation relations (1)-(3) the rotationaland time reversal symmetries are not preserved [5, 20–22]. In [23] noncommutative algebrawhich is rotationally invariant and equivalent to nocommutative algebra of canonical type wasproposed. In [24] effect of noncommutativity on the mass of a particle in uniform field wasfound in the frame of the rotationally invariant noncommutative algebra of canonical type [23].In the present paper we study a particle in uniform filed in the frame of rotationally invariantand time reversal invariant noncommutative algebra of canonical type proposed in [22]. Thetotal Hamiltonian is constructed and analyzed. We find exactly energy and wave functions of aparticle in uniform field in noncommutative space with preserved rotational and time reversalsymmetries.The paper is organized as follows. In Section 2 time reversal and rotationally invariantalgebra with noncommutativity of coordinates is presented. Section 3 is devoted to studies of aparticle in uniform field in rotationally invariant and time reversal invariant noncommutativespace. Conclusions are presented in Section 4. In [22] for preserving rotational and time reversal symmetries in noncommutative space theauthors considered the idea to generalize parameters of noncommutativity, defining tensor ofcoordinate noncommutativity as θ ij = c θ ¯ h X k ε ijk p ak , (4)here p ai are additional momenta, governed by a rotationally invariant system, c θ is a constant.For the reason of simplicity in [22] the additional momenta were assumed to be governed byharmonic oscillator H aosc = ( p a ) m osc + m osc ω osc a . (5)Here a i are additional coordinates conjugated to momenta p ai . The following relations are sati-sfied [ a i , p aj ] = i ¯ hδ ij , (6) [ a i , a j ] = [ p ai , p aj ] = 0 . (7)2he oscillator length is considered to be equal to the Planck length √ ¯ h/ √ m osc ω osc = l P andthe frequency of the oscillator is assumed to be very large [22]. So, the harmonic oscillator (5)remains in the ground state.The time reversal and rotationally invariant algebra with canonical noncommutativity ofcoordinates reads [ X i , X j ] = ic θ X k ε ijk p ak , (8) [ X i , P j ] = i ¯ hδ ij , [ P i , P j ] = 0 , (9) [ p ai , X j ] = [ p ai , P j ] = 0 . (10)It is convenient to represent coordinates and momenta which satisfy (8), (9) by coordinatesand momenta x i , p i satisfying the ordinary commutation relations [ x i , x j ] = [ p i , p j ] = 0 , (11) [ x i , p j ] = i ¯ hδ ij . (12)The representation is X i = x i + 12 [ θ × p ] i , P i = p i , (13)here θ = ( θ , θ , θ ) , θ i = c θ p ai ¯ h . (14)From relation (10) follows that algebra (8), (9) is equivalent to algebra with canonicalnoncommutativity of coordinates. Relations (8), (9) are invariant under the time reversaltransformation which include complex conjugation. Also, after this transformation the coordi-nates and the momenta change as X i → X i , P i → − P i , p ai → − p ai . Algebra (8), (9) istime reversal invariant [22]. After rotation the coordinates and momenta change as X ′ i = U ( ϕ ) X i U + ( ϕ ) , P ′ i = U ( ϕ ) P i U + ( ϕ ) p a ′ i = U ( ϕ ) p ai U + ( ϕ ) , U ( ϕ ) = exp( iϕ ( n · L t ) / ¯ h ) with L t = [ x × p ] + [ a × p a ] . Commutation relations (8), (9) are invariant under rotation, algebra isrotationally invariant [22].In the next section we study a particle in uniform field in the frame of rotationally invariantand time reversal invariant noncommutative algebra (8), (9). Let us study a particle with mass m in uniform field with the following Hamiltonian H p = P m − αX . (15)3n (15) coordinate and momenta satisfy relations (8), (9). Without loss of generality, for conveni-ence we consider the field pointed in the X direction (in (15) α characterize the force actingon the particle). Because algebra (8), (9) is rotationally invariant the results of this section canbe generalized to the case of arbitrary direction of the field.To construct time reversal invariant and rotationally invariant noncommutative algebra (8),(9) additional momenta p ai were involved, therefore to study a particle in uniform filed in thespace (8), (9) one should write the following Hamiltonian H = P m − αX + ( p a ) m osc + m osc ω osc a , (16)the last two terms in which correspond to harmonic oscillator (5). Then to find influence of spacequantization on the energy of a particle in uniform field it is convenient to use representation(13) and rewrite Hamiltonian (16) as follows H = p m − αx −
12 [ θ × p ] + ( p a ) m osc + m osc ω osc a p m − αx − αc θ h ( p a p − p a p ) + ( p a ) m osc + m osc ω osc a (17)Here we take into account (14).To find exact expression for the energy of a particle in uniform field in space described bycommutation relations (8), (9), we rewrite Hamiltonian (17) as H = (cid:18) − α c θ mm osc h (cid:19) p m + (cid:18) − α c θ mm osc h (cid:19) p m + p m − αx ++ 12 m osc (cid:16) p a − αc θ m osc h p (cid:17) + 12 m osc (cid:16) p a + αc θ m osc h p (cid:17) ++ ( p a ) m osc + + m osc ω osc a m osc ω osc a m osc ω osc a . (18)Note that operators ˜ H p = (cid:18) − α c θ m hω osc l P (cid:19) p m + (cid:18) − α c θ m hω osc l P (cid:19) p m + p m − αx , (19)and ˜ H osc = 12 m osc (cid:18) p a − αc θ ω osc l P p (cid:19) + 12 m osc (cid:18) p a + αc θ ω osc l P p (cid:19) ++ ( p a ) m osc + m osc ω osc a m osc ω osc a m osc ω osc a , (20)in (18) commute [ ˜ H p , ˜ H osc ] = 0 . (21)4riting (19), (20) we take into account that r ¯ hm osc ω osc = l P , (22)as was assumed in the paper [22], where the noncommutative algebra invariant upon timereversal and rotationally invariant was constructed.Operator ˜ H p can be rewritten as ˜ H p = ˜ H + ˜ H + ˜ H , (23)with ˜ H = p m eff , (24) ˜ H = p m eff , (25) ˜ H = p m − αx , (26) [ ˜ H , ˜ H ] = [ ˜ H , ˜ H ] = [ ˜ H , ˜ H ] = 0 . (27)The effective mass reads m eff = m (cid:18) − α c θ mm osc h (cid:19) − = m (cid:18) − α c θ m hω osc l P (cid:19) − . (28)Note, that operator of coordinate x and operator of momentum p in ˜ H satisfy the ordinarycommutation relations (11), (12). So, ˜ H corresponds to the Hamiltonian of a particle in uniformfiled in the ordinary space (in a space in which the operators of coordinates and operators ofmomenta satisfy the ordinary commutation relations).Introducing ˜ p a = p a − αc θ ω osc l P p , (29) ˜ p a = p a + αc θ ω osc l P p , (30) ˜ p a = p a , (31)one can write (20) as ˜ H osc = (˜ p a ) m osc + m osc ω osc a . (32)Operators a i and ˜ p ai satisfy the ordinary commutation relations [ a i , a j ] = [˜ p ai , ˜ p aj ] = 0 , (33) [ a i , ˜ p aj ] = i ¯ hδ ij , (34)5herefore operator (32) corresponds to Hamiltonian of three-dimensional harmonic oscillatorwith mass m osc and frequency ω osc in the ordinary space. Spectrum of the oscillator in theordinary space is well known. Let us recall that the frequency ω osc is very large [22], andoscillator putted in the ground state remains in it. So, the oscillator energy is hω osc / .Operators ˜ H , ˜ H , ˜ H , ˜ H osc commute with each other (see (21), (27)). So, the spectrum ofa particle in uniform filed in rotationally invariant and time reversal invariant noncommutativespace reads E = ¯ h k m (cid:18) − α c θ m hω osc l P (cid:19) + ¯ h k m (cid:18) − α c θ m hω osc l P (cid:19) + E + 32 ¯ hω osc . (35)Note that the motion of a particle in the directions perpendicular to the field direction is free. In(35) k , k denote components of the wave vector which correspond to this motion, E denotescontinious eigenvalues of ˜ H . The last term in (35) corresponds to the ground state energy ofthe harmonic oscillator (32).Let us also write the eigenfunctions of the total Hamiltonian (18). Because relations (21),(27) are satisfied, we can write ψ ( x , a ) = ˜ ψ ( x ) ˜ ψ ( x ) ˜ ψ ( x ) ˜ ψ ( a ) (36)where ˜ ψ i ( x i ) are eigenfunctions of ˜ H i given by (24)-(26). Note that ψ (3) ( x ) is eigenfunction ofa particle in uniform field in the ordinary space, which is well known (see, for instance, [26]).It reads ψ (3) ( x ) = (cid:18) m π α ¯ h (cid:19) Φ (cid:18) mα ¯ h (cid:19) (cid:18) − x − E α (cid:19)! , (37)here Φ is the Airy function Φ( x ) = 1 √ π Z ∞ cos (cid:18) t tx (cid:19) dt. (38)Functions ˜ ψ ( a ) in (36) are eigenfunctions of H ′ osc = 12 m osc (cid:18) p a − αc θ ¯ hk ω osc l P (cid:19) + 12 m osc (cid:18) p a + αc θ ¯ hk ω osc l P (cid:19) ++ ( p a ) m osc + m osc ω osc a m osc ω osc a m osc ω osc a . (39)Hamiltonian (39) obtained replacing p and p by ¯ hk , ¯ hk , respectively, in (20). The eigenfuncti-on of (39) corresponding to the ground state reads ˜ ψ ( a ) = 1 π l P e − a l P − iβ ( k a − k a ) . (40)Here for convenience we use the following notation β = αc θ ω osc l P . (41)6o, we can write the eigenfunctions of the total Hamiltonian (18). They read ψ ( x , a ) = Ce ik x e ik x Φ (cid:18) mα ¯ h (cid:19) (cid:18) − x − E α (cid:19)! e − a l P − iβ ( k a − k a ) , (42)where C is the normalization constant.Let us analyze the obtained results. It is important to note that features of space structureon the Planck scale have effect only on the motion of a particle in the directions perpendicularto the direction of the field. The first two terms in (35) can be rewritten by effective mass (28).So, space quantization has effect on the mass of the particle in uniform field in rotationallyinvariant and time reversal invariant space with noncommutativity of coordinates. Conclusion
In the paper we have considered algebra with noncommutativity of coordinates which is rotati-onally and time reversal invariant (8), (9). This algebra describes space quantization at thePlanck scale. Influence of space quantization on the motion of a particle in uniform field hasbeen studied. Taking into account that the rotationally invariant and time reversal invariantnoncommutative algebra contains additional momenta, we have constructed and examined totalHamiltonian of a particle in uniform field in time reversal invariant and rotationally invariantnoncommutative space (16). Energy and wave functions of the particle have been found exactly(35), (42). We obtain that the motion of a particle in the field direction in rotationally and time-reversal invariant noncommutative space is the same as in the space with ordinary commutationrelations for operators of coordinates and operators of momenta. Features of space structuredescribed by noncommutative algebra (8), (9) have effect on the motion of a particle in thedirections perpendicular to the field direction. Similarly as in the ordinary space the motionof a particle in these directions is free. The noncommutativity has only effect on the particlemass.
Acknowledgement
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