System of interacting harmonic oscillators in rotationally invariant noncommutative phase space
aa r X i v : . [ qu a n t - ph ] A ug System of interacting harmonic oscillators in rotationally invariantnoncommutative phase space
Kh. P. Gnatenko Ivan Franko National University of Lviv, Department for Theoretical Physics,12 Drahomanov St., Lviv, 79005, UkraineLaboratory for Statistical Physics of Complex SystemsInstitute for Condensed Matter Physics, NAS of Ukraine Lviv, 79011, Ukraine
Abstract
Rotationally invariant space with noncommutativity of coordinates and noncommuta-tivity of momenta of canonical type is considered. A system of N interacting harmonicoscillators in uniform filed and a system of N particles with harmonic oscillator interac-tion are studied. We analyze effect of noncommutativity on the energy levels of thesesystems. It is found that influence of coordinates noncommutativity on the energy levelsof the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally-invariant noncommutative phase space isalso analyzed. It is shown that the spectrum corresponds to the spectrum of a systemof N harmonic oscillators with frequency determined by the parameter of momentumnoncommutativity.Key words: noncommutative phase space; many-particle system, harmonic oscillator Recently much attention has been devoted to studies of a quantum space realized on thebasis of idea that the spatial coordinates might be noncommutative. The noncommutativespace of canonical type has been studied intensively. In the space the coordinates satisfythe following commutation relations[ X i , X j ] = i ¯ hθ ij , (1)where θ ij are elements of constant antisymmetric matrix, parameters of coordinate non-commutativity. In noncommutative phase space the momenta are supposed to be non-commutative too. The commutation relations read[ P i , P j ] = i ¯ hη ij . (2)Commutation relations for coordinates and momenta are generalized as[ X i , P j ] = i ¯ h ( δ ij + γ ij ) . (3)where η ij , γ ij are elements of constant matrixes. E-Mail address: [email protected] oordinates X i and momenta P i which satisfy (1), (2) can be represented as X i = x i − X j θ ij p j , (4) P i = p i + 12 X j η ij x j , (5)here x i , p i are coordinates and momenta which satisfy[ x i , x j ] = 0 , (6)[ x i , p j ] = i ¯ hδ ij , (7)[ p i , p j ] = 0 . (8)On the basis of (4), (5), the commutation relations for coordinates and momenta read[ X i , P j ] = i ¯ hδ ij + i ¯ h P k θ ik η jk / . So, parameters γ ij are considered in the following form γ ij = P k θ ik η jk / N interacting harmonic oscillators were examinedin [40].Studies of many-particle systems in the frame of noncommutative algebra of coordi-nates and momenta give a possibility to find new effects in the properties of wide class ofphysical systems caused by space quantization. Considered in the present paper systemof interacting harmonic oscillators has various applications. Studies of a system of N interacting harmonic oscillators are important in nuclei physics [41, 42, 43], in quantum hemistry and molecular spectroscopy [44, 45, 46, 47]. Recently networks of coupled har-monic oscillators have attracted much attention because of their importance for quantuminformation processing [48, 49, 50]. The studies are also important for searching signa-tures towards the Planck scale physics which are observable on macroscopic scales. Thesystems under consideration can be realized on the classical level as a system of oscillatorscoupled by springs.Our paper is organized as follows. In Section 2 noncommutative algebra which isrotationally invariant and equivalent to noncommutative algebra of canonical type is pre-sented. Section 3 is devoted to studies of the total hamiltonian in rotationally invariantnoncommutative phase space. Effect of noncommutativity on the spectrum of a systemof N interacting harmonic oscillators in uniform field is examined in the Section 4. Alsoin this section a system of free particles in uniform filed and a system of particles withharmonic oscillator interaction are studied. Spectrum of a system of two interacting har-monic oscillators and spectrum of three interacting harmonic oscillators are analyzed inSection 5 and Section 6, respectively. Section 7 is devoted to conclusions. In our paper [18] we considered the tensors of noncommutativity to be defined as θ ij = c θ l P ¯ h X k ε ijk ˜ a k , (9) η ij = c η ¯ hl P X k ε ijk ˜ p bk . (10)where c θ , c η are dimensionless constants, l P is the Planck length. We use notations ˜ a i , ˜ b i ˜ p ai , ˜ p bi for additional dimensionless coordinates and momenta conjugate to them which aregoverned by a spherically symmetric systems. For simplicity these systems are consideredto be harmonic oscillators H aosc = ¯ hω osc (cid:18) (˜ p a ) a (cid:19) , (11) H bosc = ¯ hω osc (˜ p b ) b ! . (12)The values of parameters of noncommutativity are supposed to be of the order of thePlanck scale. So, we put √ ¯ h/ √ m osc ω osc = l P . The frequency of the oscillators ω osc isconsidered to be very large which leads to the statement that the oscillators put into theground states remain in them [18]. So, in [18] we proposed the following noncommutative lgebra [ X i , X j ] = ic θ l P X k ε ijk ˜ a k , (13)[ X i , P j ] = i ¯ h (cid:16) δ ij + c θ c η ˜a · ˜p b ) δ ij − c θ c η a j ˜ p bi (cid:17) , (14)[ P i , P j ] = c η ¯ h l P X k ε ijk ˜ p bk . (15)Commutation relations for ˜ a i , ˜ b i ˜ p ai , ˜ p bi were considered to be as follows[˜ a i , ˜ a j ] = [˜ b i , ˜ b j ] = [˜ a i , ˜ b j ] = [˜ p ai , ˜ p aj ] == [˜ p bi , ˜ p bj ] = [˜ p ai , ˜ p bj ] = 0 , (16)[˜ a i , ˜ p aj ] = [˜ b i , ˜ p bj ] = iδ ij , (17)[˜ a i , ˜ p bj ] = [˜ b i , ˜ p aj ] = 0 (18)[˜ a i , X j ] = [˜ a i , P j ] = [˜ p bi , X j ] = [˜ p bi , P j ] = 0 . (19)So, like in the case of canonical version of noncommutativity with θ ij , η ij , γ ij beingconstants, in the case of θ ij , η ij being defined as (9), (10) we can write[ θ ij , X k ] = [ θ ij , P k ] = [ η ij , X k ] = [ η ij , P k ] = [ γ ij , X k ] = [ γ ij , P k ] = 0 (20)From this one can state that the proposed algebra[ X i , X j ] = ic θ l P X k ε ijk ˜ a k , (21)[ X i , P j ] = i ¯ h (cid:16) δ ij + c θ c η a · ˜ p b ) δ ij − c θ c η a j ˜ p bi (cid:17) , (22)[ P i , P j ] = c η ¯ h l P X k ε ijk ˜ p bk , (23)is equivalent to noncommutative algebra of canonical type at the same time it is rotation-ally invariant.The noncommutative coordinates and noncommutative momenta which satisfy (21)-(23) can be represented as X i = x i + c θ l P h [˜ a × p ] i = x i + 12 [ θ × p ] i , (24) P i = p i − c η ¯ h l P [ x × ˜ p b ] i = p i −
12 [ x × η ] i , (25)where coordinates and momenta x i , p i satisfy (6)-(8) and for convenience the followingvectors θ = ( θ , θ , θ ) , η = ( η , η , η ) , (26) θ i = 12 X jk ε ijk θ jk , (27) η i = 12 X jk ε ijk η jk , (28) re introduced.After rotation one has X ′ i = U ( ϕ ) X i U + ( ϕ ), P ′ i = U ( ϕ ) P i U + ( ϕ ) a ′ i = U ( ϕ ) a i U + ( ϕ ), p b ′ i = U ( ϕ ) p bi U + ( ϕ ). The rotation operator U ( ϕ ) = exp( iϕ ( n · L t ) / ¯ h ), contains the totalangular momentum which reads L t = [ x × p ]+¯ h [˜ a × ˜ p a ]+¯ h [˜ b × ˜ p b ] [18]. The commutationrelations for coordinates and momenta remain the same[ X ′ i , X ′ j ] = ic θ l P X k ε ijk ˜ a ′ k , (29)[ X ′ i , P ′ j ] = i ¯ h (cid:16) δ ij + c θ c η a ′ · ˜ p b ′ ) δ ij − c θ c η a ′ j ˜ p b ′ i (cid:17) , (30)[ P ′ i , P ′ j ] = c η ¯ h l P X k ε ijk ˜ p b ′ k . (31) Let us study a system of N interacting harmonic oscillators of masses m and frequencies ω in uniform field in noncommutative phase space with rotational symmetry (21)-(23).The hamiltonian of the system reads H s = X n ( P ( n ) ) m + X n mω ( X ( n ) ) k X m,nm = n ( X ( n ) − X ( m ) ) ++ κ X n X ( n )1 . (32)Here κ and k are constants. The direction of the field for convenience is chosen to coincidewith the X axis direction. For κ = 0, Hamiltonian (32) corresponds to nondissipativesymmetric network of coupled harmonic oscillators [49].Coordinates and momenta satisfy the following commutation relations[ X ( n ) i , X ( m ) j ] = i ¯ hδ mn θ ( n ) ij , (33)[ X ( n ) i , P ( m ) j ] = i ¯ hδ mn δ ij + X k θ ( n ) ik η ( m ) jk , (34)[ P ( n ) i , P ( m ) j ] = i ¯ hδ mn η ( n ) ij , (35)with θ ( n ) ij = c ( n ) θ l P ¯ h X k ε ijk ˜ a k , (36) η ( n ) ij = c ( n ) η ¯ hl P X k ε ijk ˜ p bk , (37) ere indexes m, n = (1 ...N ) label the particles. Note that we consider the general casewhen different particles satisfy noncommutative algebra with different tensors of non-commutativity. The problem of description of composite system in rotationally invariantnoncommutative phase space was discussed in our previous paper [54]. In the paper weproposed condition on the parameters c ( n ) θ , c ( n ) η in tensors of noncommutativity on whichthe list of important results can be obtained (among them the noncommutative coordi-nates are independent on mass and noncommutative momenta are proportional to massas it has to be, coordinates and momenta of the center-of-mass commute with the coordi-nates and momenta of the relative motion [54], the weak equivalence principle is recovered[55]). The conditions read c ( n ) θ m n = ˜ γ = const, c ( n ) η m n = ˜ α = const. (38)Constants ˜ γ , ˜ α are the same for particles with different masses. We would like also tonote that the idea to relate parameters of algebra for coordinates and momenta with massis also important in deformed space with minimal length [56, 57, 58], two-dimensionalnoncommutative space of canonical type [59], four-dimensional noncommutative phasespace of canonical type [60, 61].In the case of system of harmonic oscillators with masses m taking into account (36),(37), (38) one has θ ( n ) ij = θ ij = c θ l P ¯ h X k ε ijk ˜ a k , (39) η ( n ) ij = η ij = c η ¯ hl P X k ε ijk ˜ p bk , (40)with c θ = ˜ γ/m , c η = ˜ αm .Using representation (24)-(25) and (39), (40) the hamiltonian of a system can bewritten in the following form H s = X n ( p ( n ) ) m + mω ( x ( n ) ) κx ( n )1 ! + k X m,nm = n ( x ( n ) − x ( m ) ) ++ X n − ( η · L ( n ) )2 m − mω ( θ · L ( n ) )2 + κ θ × p ( n ) ] + mω θ × p ( n ) ] ++ [ η × x ( n ) ] m ! − k X m,nm = n θ · [( x ( n ) − x ( m ) ) × ( p ( n ) − p ( m ) )] ++ X m,nm = n k θ × ( p ( n ) − p ( m ) )] , (41)where L ( n ) = [ x ( n ) × p ( n ) ]. Because of involving of additional coordinates and additionalmomenta ˜ a i , ˜ b i ˜ p ai , ˜ p bi we have to consider the total hamiltonian which is the sum of H s and Hamiltonians of harmonic oscillators H aosc , H bosc H = H s + H aosc + H bosc = H + ∆ H. (42) ere H = h H s i ab + H aosc + H bosc , (43)∆ H = H − H = H s − h H s i ab , (44) h ... i ab denotes averaging over degrees of freedom of harmonic oscillators H aosc H bosc in theground states h ... i ab = h ψ a , , ψ b , , | ... | ψ a , , ψ b , , i (45) ψ a , , , ψ b , , are eigenstates of tree-dimensional harmonic oscillators H aosc , H bosc in theground states in the ordinary space (space with commutative coordinates and commuta-tive momenta).In our previous paper [54] we concluded that up to the second order in ∆ H one canconsider Hamiltonian H .For a system of interacting harmonic oscillators using h ψ a , , | θ i | ψ a , , i = h ψ b , , | η i | ψ b , , i = 0 , (46) h θ i θ j i = c θ l P ¯ h h ψ a , , | ˜ a i ˜ a j | ψ a , , i = c θ l P h δ ij = h θ i δ ij , (47) h η i η j i = ¯ h c η l P h ψ b , , | ˜ p bi ˜ p bj | ψ b , , i = ¯ h c η l P δ ij = h η i δ ij , (48)and calculating h [ η × x ( n ) ] i ab = 23 h η i ( x ( n ) ) , h [ θ × p ( n ) ] i ab = 23 h θ i ( p ( n ) ) , (49) h [ θ × ( p ( n ) − p ( m ) )] i ab = 23 h θ i ( p ( n ) − p ( m ) ) (50)the expression for ∆ H can be written as∆ H = X n − ( η · L ( n ) )2 m − mω ( θ · L ( n ) )2 + κ θ × p ( n ) ] + mω θ × p ( n ) ] ++ [ η × x ( n ) ] m ! − k X m,nm = n θ · [( x ( n ) − x ( m ) ) × ( p ( n ) − p ( m ) )] ++ X m,nm = n k θ × ( p ( n ) − p ( m ) )] − X n h η i ( x ( n ) ) m + h θ i mω ( p ( n ) ) ! −− k X m,nm = n h θ i ( p ( n ) − p ( m ) ) . (51)So, on the basis of conclusion presented in [54] up to the second order in ∆ H (or takinginto account (51), up to the second order in the parameters of noncommutativity) for a ystem of interacting harmonic oscillators in uniform field we can consider the Hamiltonianas H = X n ( p ( n ) ) m + mω ( x ( n ) ) κx ( n )1 ! + k X m,nm = n ( x ( n ) − x ( m ) ) ++ X n h η i ( x ( n ) ) m + h θ i mω ( p ( n ) ) ! ++ k X m,nm = n h θ i ( p ( n ) − p ( m ) ) + H aosc + H bosc . (52) N interacting oscillators Let us find energy levels of a system of N interacting harmonic oscillators. It is convenientto introduce effective mass and effective frequency m eff = m (cid:18) m ω h θ i (cid:19) − , (53) ω eff = (cid:18) ω + h η i m (cid:19) (cid:18) m ω h θ i (cid:19) (54)and rewrite (52) as H = X n ( p ( n ) ) m eff + m eff ω eff (˜ x ( n ) ) ! − N κ m eff ω eff ++ k X m,nm = n (˜ x ( n ) − ˜ x ( m ) ) + k X m,nm = n h θ i ( p ( n ) − p ( m ) ) + H aosc + H bosc . (55)where the vector ˜ x ( n ) = x ( n )1 + κm eff ω eff , x ( n )2 , x ( n )3 ! , (56)is introduced. Coordinates and momenta ˜ x ( n ) , p ( n ) satisfy the ordinary commutationrelations [˜ x ( n ) i , ˜ x ( m ) j ] = 0 , (57)[˜ x ( n ) i , p ( m ) j ] = i ¯ hδ nm δ ij , (58)[ p ( n ) i , p ( m ) j ] = 0 . (59)Note also that [ H , H aosc ] = [ H , H bosc ] = 0 . (60) herefore, the spectrum of H reads E { n } , { n } , { n } = N X a =1 ¯ hω a (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) − N κ m eff ω eff ++3¯ hω osc . (61)where ω = ω eff , (62) ω = ω = ... = ω N == ω eff + 2 kNm eff + kN h θ i m eff ω eff k h θ i N ! . (63) n ( a ) i are quantum numbers ( n ( a ) i = 0 , , ... ). In (61) we take into account that theoscillators H aosc , H bosc are in the ground states. The first term in (61) corresponds tothe spectrum of the center-of-mass of the system. The terms with a = 2 ..N correspondsto the spectrum of the relative motion. This can be shown introducing coordinates andmomenta of the center-of-mass x c = P n x ( n ) /N , p c = P n p ( n ) , and coordinates andmomenta of relative motion ∆ x ( n ) = x ( n ) − x c , ∆ p ( n ) = p ( n ) − p c /N . From (55) we canwrite H = H c + H rel + H aosc + H bosc , (64) H c = ( p c ) N m eff + N m eff ω eff (˜ x c ) − N κ m eff ω eff , (65) H rel = X n (∆ p ( n ) ) m eff + m eff ω eff (∆ x ( n ) ) ! ++ k X m,nm = n (∆ x ( n ) − ∆ x ( m ) ) + k X m,nm = n h θ i (∆ p ( n ) − ∆ p ( m ) ) , (66)[ H c , H rel ] = [ H c , H aosc + H bosc ] = [ H rel , H aosc + H bosc ] = 0 , (67)where ˜ x c = (cid:16) x c + κ/ ( m eff ω eff ) , x c , x c (cid:17) . So, from (61) we have that the noncommuta-tivity of coordinates and noncommutativity of momenta effects on the frequencies in thespectra of the center-of-mass and relative motion of the system. The presents of uniformfield shifts the spectrum on a constant.In the limit h θ i → h η i → E { n } , { n } , { n } reduces to knownspectrum for system of N interacting harmonic oscillators in uniform field in the ordinaryspace E { n } , { n } , { n } = ¯ hω (cid:18) n (1)1 + n (1)2 + n (1)3 + 32 (cid:19) ++ N X a =2 ¯ h (cid:18) ω + 2 N km (cid:19) (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) − N κ mω (68) rom (61) setting ω = 0, one can write the following expression for the spectrum of asystem of N particles of mass m with harmonic oscillator interaction. E { n } , { n } , { n } = ¯ h h η i m (cid:18) n (1)1 + n (1)2 + n (1)3 + 32 (cid:19) ++¯ h (cid:18) kNm + h η i m + 2 k h θ i N (cid:19) N X a =2 (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) −− N κ m h η i + 3¯ hω osc . (69)The first term in (69) corresponds to the spectrum of the center-of-mass of the system. Inthe contrast to the ordinary space (space with commutative coordinates and commutativemomenta) because of momentum noncommutativity the spectrum of the center-of-massof the system of particles with harmonic oscillator interaction is discreet. The spectrumcorresponds to the spectrum of harmonic oscillator with the frequency ¯ h h η i / m . Thefrequency in the spectrum of the relative motion is affected by the noncommutativity ofcoordinates and noncommutativity of momenta (see second term in (69)).It is worth mentioning that from (61) and (69) we have that the effect of coordinatesnoncommutativity on the spectrum of interacting harmonic oscillators (a system of par-ticles with harmonic oscillator interaction) increases with increasing of the number ofparticles in the system.For a system of N free particles in uniform field in rotationally invariant noncommu-tative phase space setting k = 0 in (69) energy levels are as follows E { n } , { n } , { n } = N X a =1 ¯ h h η i m (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) − N κ m h η i + 3¯ hω osc . (70)Note that the spectrum of a system of free particles is affected only by momentum noncom-mutativity. The spectrum corresponds to the spectrum of N oscillators with frequenciesdetermined by the parameter of momentum noncommutativity ¯ h h η i / m .We would like also to mention that the presence of uniform field κ shift the spectra(61), (69), (70) by constant. Let us study particular case when a system consists of two oscillators with masses m , m and frequencies ω , ω and is described by the following Hamiltonian H s = ( P (1) ) m + ( P (2) ) m + m ω ( X (1) ) m ω ( X (2) ) k ( X (1) − X (2) ) . (71)where X ( n ) , P ( n ) satisfy (33)-(35), ( n = 1 , olecular physics [44, 45], used for description of states of light in the framework oftwo-photon quantum optics [52, 53].Taking into account (43) and using representation (24)-(25), we have H = ( p (1) ) m (1) eff + ( p (2) ) m (2) eff + m (1) eff ( ω (1) eff ) ( x (1) ) m (2) eff ( ω (2) eff ) ( x (2) ) ++ k ( x (1) − x (2) ) + k (cid:16) h ( θ (1) ) i ( p (1) ) + h ( θ (2) ) i ( p (2) ) −− h θ (1) θ (2) i ( p (1) · p (2) ) (cid:17) + H aosc + H bosc . (72)with m ( n ) eff = m n m n ω n h ( θ ( n ) ) i ! − , (73) ω ( n ) eff = (cid:18) ω n + h ( η n ) i m n (cid:19) m n ω n h ( θ ( n ) ) i ! , (74) h θ ( n ) θ ( m ) i = c ( n ) θ c ( m ) θ l P ¯ h h ψ a , , | ˜ a | ψ a , , i = 3 c ( n ) θ c ( m ) θ l P h , (75) h ( η ( n ) ) i = ¯ h ( c ( n ) η ) l P h ψ b , , | (˜ p b ) | ψ b , , i = 3¯ h ( c ( n ) η ) l P , (76)Coordinates x ( n ) i and momenta p ( n ) i satisfy the ordinary commutation relations. So, thespectrum of H is as follows E { n } , { n } , { n } = ¯ hω + (cid:18) n (1)1 + n (1)2 + n (1)3 + 32 (cid:19) ++¯ hω − (cid:18) n (2)1 + n (2)2 + n (2)3 + 32 (cid:19) + 3¯ hω osc . (77)where ω ± = 12 X n ( ω ( n ) eff ) + 2 km ( n ) eff + km ( n ) eff ( ω ( n ) eff ) h ( θ ( n ) ) i k (cid:16) h ( θ ( n ) ) i + h θ (1) θ (2) i (cid:17)(cid:19) ± √ D, (78) = X n ( ω ( n ) eff ) + X n km ( n ) eff + X n km ( n ) eff ( ω ( n ) eff ) h ( θ ( n ) ) i X n k (cid:16) h ( θ ( n ) ) i + h θ (1) θ (2) i (cid:17)! − Y n ( ω ( n ) eff ) + 2 km ( n ) eff ++ km ( n ) eff ( ω ( n ) eff ) h ( θ ( n ) ) i k (cid:16) h ( θ ( n ) ) i + h θ (1) θ (2) i (cid:17) ++4 km (2) eff + km (1) eff ( ω (1) eff ) h θ (1) θ (2) i k (cid:16) h ( θ (2) ) i + h θ (1) θ (2) i (cid:17) × km (1) eff + km (2) eff ( ω (2) eff ) h θ (1) θ (2) i k (cid:16) h ( θ (1) ) i + h θ (1) θ (2) i (cid:17) . (79)In the case of m = m we have m ( n ) eff = m eff , ω ( n ) eff = ω eff and the expressions reduce to ω − = ω eff , (80) ω + = ω eff + 4 km eff + 2 k h θ i m eff ω eff k h θ i ! . (81)which corresponds to (62), (63) with N = 2. Let us study a system of three interacting oscillators with masses m , m = m = m , andfrequencies ω , ω = ω = ω . The Hamiltonian reads H s = ( P (1) ) m + ( P (2) ) m + ( P (3) ) m + m ω ( X (1) ) mω ( X (2) ) mω ( X (3) ) k ( X (1) − X (2) ) + k ( X (2) − X (3) ) + k ( X (3) − X (3) ) . (82)In the case when ω n = 0 the Hamiltonian (82) is used as a model for description ofconfining forces between quarks [41, 42, 43]. Up to the second order in the parameters ofnoncommutativity one can consider H = X n ( p ( n ) ) m ( n ) eff + X n m ( n ) eff ( ω ( n ) eff ) ( x ( n ) ) k X m,nm = n ( x ( n ) − x ( m ) ) + k X m,nm = n (cid:16) h ( θ ( n ) ) i ( p ( n ) ) + h ( θ ( m ) ) i ( p ( m ) ) −− h θ ( n ) θ ( m ) i ( p ( n ) · p ( m ) ) (cid:17) + H aosc + H bosc . (83)where m ( n ) eff , ω ( n ) eff , h θ ( n ) θ ( m ) i are defined as (73)-(75). he spectrum of (83) reads E { n } , { n } , { n } = X a =1 ¯ h ˜ ω a (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) + 3¯ hω osc . (84)˜ ω = 1 √ ω eff + ( ω (1) eff ) + 2 km eff + 4 km (1) eff + A − √ D , (85)˜ ω = 1 √ ω eff + ( ω (1) eff ) + 2 km eff + 4 km (1) eff + A + √ D , (86)˜ ω = (cid:18) ω eff + 6 km eff (cid:19) (cid:0) km eff h θ i (cid:1) , (87)with D = ω eff − ( ω (1) eff ) + 4 km eff − km (1) eff + A + (cid:18) km + A (cid:19) (cid:16) ω (1) eff ) −− ω eff − km + 8 km (1) eff + 8 (cid:18) km + A (cid:19) (cid:18) km + A (cid:19) (cid:18) km + A (cid:19) − + A , (88) A = km eff ω eff k ! h θ i + km (1) eff ( ω (1) eff ) k h ( θ (1) ) i ++ 8 k h θθ (1) i , (89) A = km eff ω eff k ! h θ i − km (1) eff ( ω (1) eff ) k h ( θ (1) ) i −− k h θθ (1) i , (90) A = k km eff ω eff ! h θ i − k h θθ (1) i , (91) A = km (1) eff ( ω (1) eff ) k h θθ (1) i + 2 k h θ i , (92) A = km eff ( ω eff )3 + 2 k ! h θθ (1) i + 4 k h ( θ (1) ) i , (93) A = − (cid:0) km eff ω eff + 4 k (cid:1) h θ i + km (1) eff ( ω (1) eff ) k h ( θ (1) ) i ++ 2 k h θθ (1) i . (94)here for convenience we use notations m eff = m (2) eff = m (3) eff , ω eff = ω (2) eff = ω (3) eff , and θ = θ (2) = θ (3) . n the case when the masses and frequencies of the oscillators are equal, m = m , ω = ω , the result (84) reproduce (61) with N = 3. We have˜ ω = ω eff , (95)˜ ω = ˜ ω = (cid:18) ω eff + 6 km eff + k h θ i m eff ω eff + 6 k h θ i (cid:19) . (96)For Hamiltonian (82) with ω n = 0 which is considered for description of confiningforces between quarks the spectrum is given by (84) with (85), (86), (87) and m (1) eff = m , m eff = m , ω (1) eff = p h ( η ) i / p m , ω eff = p h ( η ) i / √ m . Note, that because ofnoncommutativity of coordinates and noncommutativity of momenta the spectrum of thecenter-of-mass of the system is discrete and corresponds to the spectrum of harmonicoscillator with frequency ˜ ω (85).In a rotationally-invariant space with noncommutativity of coordinates (space whichis characterized by (33), (34) and [ P ( n ) i , P ( m ) j ] = 0), the spectrum of a system describedby Hamiltonian (82) with ω n = 0 has the form (84) with frequencies ˜ ω = 0 , (97)˜ ω = 1 √ (cid:18) km + 4 km (1) + 2 k h θ i + 8 k h ( θ (1) ) i + 8 k h θθ (1) i + √ D (cid:19) , (98)˜ ω = (cid:18) km + 6 k h θ i (cid:19) , (99)where D = (cid:18) km − km (1) + 10 k h θ i − k h ( θ (1) ) i − k h θθ (1) i (cid:19) + (cid:18) km ++ 8 k h θ i − k h θθ (1) i (cid:19) (cid:18) − km + 8 km (1) + 8 (cid:18) km + 4 k h θθ (1) i + 2 k h θ i (cid:19) ×× (cid:18) km + 2 k h θθ (1) i + 4 k h ( θ (1) ) i (cid:19) (cid:18) km + 8 k h θ i − k h θθ (1) i (cid:19) − −− k h θ i + 16 k h ( θ (1) ) i + 2 k h θθ (1) i (cid:19) . (100)The frequencies are obtained putting ω eff = ω (1) eff = 0, m (1) eff = m , m eff = m in (85),(86), (87). Note, that the spectrum of the center-of-mass of the system is not affectedby noncommutativity of coordinates (97). The noncommutativity has influence on thefrequencies in the spectrum of the relative motion (98), (99). We have considered noncommutative phase space of canonical type with rotational sym-metry (21)-(23). The corresponding noncommutative algebra is constructed with the help f generalization of parameters of noncommutativity to tensors determined by additionalcoordinates and additional momenta [18].In the frame of the rotationally invariant noncommutative algebra we have examineda system of N harmonic oscillators with harmonic oscillator interaction in uniform field.The total hamiltonian has been constructed and analyzed (42). We have found energylevels of the system up to the second order in the parameters of noncommutativity. Wehave obtained that noncommutativity affects on the frequencies of the system (61). Uni-form field causes shift of the spectrum on a constant (61). A system of two interactingoscillators and a system of three interacting oscillators have been studied in details andthe corresponding spectra have been obtained (77), (84).As particular cases, a system of particles with harmonic oscillator interaction anda system of free particles have been studied in uniform field in rotationally-invariantnoncommutative phase space. We have obtained that the spectrum of free particles inuniform field corresponds to the spectrum of a system of N oscillators with frequencies¯ h h η i / m and is not affected by the coordinates noncommutativity (70). For a systemof particles with harmonic oscillator interaction in uniform field we have found that thespectrum of the center-of-mass of the system is affected by noncommutativity of momentaand corresponds to the spectrum of harmonic oscillator (see first term in (69)). Thespectrum of the relative motion of the system corresponds to the spectrum of harmonicoscillators with frequencies determined by parameters of momentum noncommutativityand coordinate noncommutativity (see second term in (69)). We have concluded thateffect of coordinates noncommutativity on the spectra of systems with harmonic oscillatorinteraction (system of interacting harmonic oscillators, system of particles with harmonicoscillator interaction) increases with increasing of the number of particles in the systems(61), (69). Acknowledgments
The author thanks Prof. V. M. Tkachuk for his advices and support during researchstudies. This work was partly supported by the by the State Found for FundamentalResearch under the project F-76.
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