Harmonic oscillator chain in noncommutative phase space with rotational symmetry
aa r X i v : . [ qu a n t - ph ] N ov Harmonic oscillator chain in noncommutative phase space withrotational symmetry
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
We consider a quantum space with rotationally invariant noncommutative algebra ofcoordinates and momenta. The algebra contains tensors of noncommutativity constructedinvolving additional coordinates and momenta. In the rotationally invariant noncommu-tative phase space harmonic oscillator chain is studied. We obtain that noncommutativityaffects on the frequencies of the system. In the case of a chain of particles with harmonicoscillator interaction we conclude that because of momentum noncommutativity the spec-trum of the center-of-mass of the system is discrete and corresponds to the spectrum ofharmonic oscillator.Key words: harmonic oscillator, composite system, tensors of noncommutativity
Owing to development of String Theory and Quantum Gravity [1, 2] studies of idea that spacecoordinates may be noncommutative has attracted much attention. Noncommutativity of coor-dinates leads to existence of minimal length, minimal area [3, 4], it leads to space quantization.Canonical version of noncommutative phase space is characterized by the following algebra[ X i , X j ] = i ¯ hθ ij , (1)[ P i , P j ] = i ¯ hη ij , (2)[ X i , P j ] = i ¯ h ( δ ij + γ ij ) . (3)where θ ij , η ij , γ ij are elements of constant matrixes. Parameters γ ij are considered to be definedas γ ij = P k θ ik η jk / θ ij , η ij , γ ij being constants is not rotationally invariant[6, 7]. Different generalizations of commutation relations (1)-(3) were considered to solve theproblem of rotational symmetry breaking in noncommutative space [8, 9, 10, 11]. Many papersare devoted to studies of position-dependent noncommutativity [12, 13, 14, 15, 16, 17, 18], spinnoncommutativity [19, 20]. The algebras of these types of noncommutativity are rotationallyinvariant by they are not equivalent to noncommutative algebra of canonical type. E-Mail address: [email protected]
1n paper [21] a rotationally invariant noncommutative algebra of canonical type was con-structed on the basis of idea of generalization of parameters of noncommutativity to a tensors.Introducing additional coordinates and additional momenta ˜ a i , ˜ b i ˜ p ai , ˜ p bi , we proposed to definethese tensors in the following form θ ij = c θ l P ¯ h X k ε ijk ˜ a k , (4) η ij = c η ¯ hl P X k ε ijk ˜ p bk . (5)Constants c θ , c η are dimensionless, l P is the Planck length. To preserve the rotational symmetrythe coordinates and momenta ˜ a i , ˜ b i ˜ p ai , ˜ p bi are supposed to be governed by a rotationally invariantsystems. The systems are considered to be harmonic oscillators H aosc = ¯ hω osc (cid:18) (˜ p a ) a (cid:19) , (6) H bosc = ¯ hω osc (˜ p b ) b ! , (7)with √ ¯ h/ √ m osc ω osc = l P and large frequency ω osc (the distance between energy levels is verylarge and oscillators are considered to be in the ground states). The algebra for additionalcoordinates and additional momenta is the following[˜ a i , ˜ a j ] = [˜ b i , ˜ b j ] = [˜ a i , ˜ b j ] = 0 , (8)[˜ p ai , ˜ p aj ] = [˜ p bi , ˜ p bj ] = [˜ p ai , ˜ p bj ] = 0 , (9)[˜ a i , ˜ p bj ] = [˜ b i , ˜ p aj ] = 0 , (10)[˜ a i , X j ] = [˜ a i , P j ] = [˜ p bi , X j ] = [˜ p bi , P j ] = 0 , (11)[˜ a i , ˜ p aj ] = [˜ b i , ˜ p bj ] = iδ ij . (12)Therefore, we have [ 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 as in thecase of canonical noncommutativity (1)-(3) with θ ij , η ij , γ ij being constants.In the present paper we study influence of noncommutativity of coordinates and noncom-mutativity of momenta on the spectrum of a harmonic oscillator chain. Studies of a systemof harmonic oscillators are important in various fields of physics among them molecular spec-troscopy and quantum chemistry [22, 23, 24, 25], quantum optics [26, 27, 28], nuclei physics[29, 30, 31], quantum information processing [32, 28, 33].Harmonic oscillator was intensively studied in the frame of noncommutative algebra [34,35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Recently experiments with micro- andnano-oscillators were implemented for probing minimal length [49]. In noncommutative space ofcanonical type two coupled harmonic oscillators were studied in [50, 51, 52]. In [53] a spectrumof a system of N oscillators interacting with each other (symmetric network of coupled harmonicoscillators) has been examined in rotationally invariant noncommutative phase space. In [54]2lassical N interacting harmonic oscillators were examined in noncommutative space-time. In[55, 56] influence of noncommutativity of coordinates and noncommutativity of momenta onthe properties of a system of free particles was examined.The paper is organized as follows. In Section 2 we study energy levels of a harmonicoscillator chain in rotationally invariant noncommutative phase space. Particular case of achain of particles with harmonic oscillator interaction is examined. Conclusions are presentedin Section 3. Let us consider a chain of N interacting harmonic oscillators with masses m and frequencies ω in a space with (1)-(3) and (4), (5) in the case of the closed configuration of the system. So,let us study the following Hamiltonian H s = N X n =1 ( P ( n ) ) m + N X n =1 mω ( X ( n ) ) k N X n =1 ( X ( n +1) − X ( n ) ) (13)with periodic boundary conditions X ( N +1) = X (1) , k is a constant.In general case coordinates and momenta which correspond to different particles satisfynoncommutative algebra with different tensors of noncommutativity. We have[ X ( n ) i , X ( m ) j ] = i ¯ hδ mn θ ( n ) ij , (14)[ X ( n ) i , P ( m ) j ] = i ¯ hδ mn δ ij + X k θ ( n ) ik η ( m ) jk ! , (15)[ P ( n ) i , P ( m ) j ] = i ¯ hδ mn η ( n ) ij , (16) θ ( n ) ij = c ( n ) θ l P ¯ h X k ε ijk ˜ a k , (17) η ( n ) ij = c ( n ) η ¯ hl P X k ε ijk ˜ p bk , (18)where indexes m, n = (1 ...N ) label the particles [57].Because of presence of additional coordinates and momenta in (17), (18) we have to studyHamiltonian which include Hamiltonians of harmonic oscillators H = H s + H aosc + H bosc (19)3he noncommutative coordinates and noncommutative momenta can be represented as X ( n ) i = x ( n ) i + 12 [ θ ( n ) × p ( n ) ] i , (20) P ( n ) i = p ( n ) i −
12 [ x ( n ) × η ( n ) ] i , (21)where coordinates and momenta x ( n ) i , p ( n ) i satisfy the ordinary commutation relations[ x ( n ) i , x ( m ) j ] = [ p ( n ) i , p ( m ) j ] = 0 , (22)[ x ( n ) i , p ( m ) j ] = i ¯ hδ mn . (23)and vectors θ ( n ) , η ( n ) have the components θ ( n ) i = P jk ε ijk θ njk / , η ( n ) i = P jk ε ijk η ( n ) jk / . In ourpaper [57] we proposed the constants c ( n ) θ , c ( n ) η in tensors of noncommutativity to be determinedby mass as c ( n ) θ m n = ˜ γ = const , c ( n ) η /m n = ˜ α = const with ˜ γ , ˜ α being the same for differentparticles. Therefore one has θ ( n ) ij = ˜ γl P m n ¯ h X k ε ijk ˜ a k , (24) η ( n ) ij = ˜ α ¯ hm n l P X k ε ijk ˜ p bk . (25)Determination of tensors of noncommutativity in forms (24), (25) gives a possibility to considernoncommutative coordinates as kinematic variables [57], to recover the weak equivalence prin-ciple [58]. Taking into account (24), (25), in the case when the system consists of oscillatorswith the same masses one has θ ( n ) ij = θ ij , η ( n ) ij = η ij . Using (20)-(21) the Hamiltonian H s reads H s = N X n =1 (cid:18) ( p ( n ) ) m + mω ( x ( n ) ) k ( x ( n +1) − x ( n ) ) − ( η · [ x ( n ) × p ( n ) ])2 m −− mω ( θ · [ x ( n ) × p ( n ) ])2 −− k ( θ · [( x ( n +1) − x ( n ) ) × ( p ( n +1) − p ( n ) )])++ [ η × x ( n ) ] m + mω θ × p ( n ) ] ++ k θ × ( p ( n +1) − p ( n ) )] (cid:19) . (26)In [57] we showed that up to the second order in ∆ H defined as∆ H = H s − h H s i ab , (27)Hamiltonian H = h H s i ab + H aosc + H bosc , (28)4an be studied because up to the second order in the perturbation theory the correctionsto spectrum of H caused by terms ∆ H = H − H = H s − h H s i ab vanish. Here notation h ... i ab is used for averaging over the eigenstates of H aosc H bosc which are well known h ... i ab = h ψ a , , ψ b , , | ... | ψ a , , ψ b , , i . For the harmonic oscillator chain we have∆ H = N X n =1 (cid:18) [ η × x ( n ) ] m + mω θ × p ( n ) ] −− mω ( θ · [ x ( n ) × p ( n ) ])2 − ( η · [ x ( n ) × p ( n ) ])2 m −− k θ · [( x ( n +1) − x ( n ) ) × ( p ( n +1) − p ( n +1) )]++ k θ × ( p ( n +1) − p ( n ) )] − h η i ( x ( n ) ) m −− h θ i mω ( p ( n ) ) − k h θ i ( p ( n +1) − p ( n ) ) (cid:19) . (29)here we take into account that h ψ a , , | ˜ a i | ψ a , , i = h ψ b , , | ˜ p i | ψ b , , i = 0 and use the followingnotations 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 , (30) 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 . (31)So, analyzing the form of ∆ H (29), we have that up to the second order in the parametersof noncommutativity one can study Hamiltonian H . This Hamiltonian for convenience can berewritten as H = N X n =1 ( p ( n ) ) m eff + m eff ω eff ( x ( n ) ) k ( x ( n +1) − x ( n ) ) ++ k h θ i ( p ( n +1) − p ( n ) ) + H aosc + H bosc (cid:19) , (32)with m eff = m (cid:18) m ω h θ i (cid:19) − , (33) ω eff = (cid:18) ω + h η i m (cid:19) (cid:18) m ω h θ i (cid:19) . (34)5he terms H aosc + H bosc commute with H . Coordinates and momenta x ( n ) , p ( n ) satisfy (22),(23). Let us rewrite H as H =¯ hω eff X n (cid:18) km eff h θ i πnN (cid:19) ˜ p ( n ) (˜ p ( n ) ) † ++ ¯ hω eff X n km eff ω eff sin πnN ! ˜ x ( n ) (˜ x ( n ) ) † , (35)using x ( n ) = s ¯ hN m eff ω eff N X l =1 exp (cid:18) πinlN (cid:19) ˜ x ( l ) , (36) p ( n ) = r ¯ hm eff ω eff N N X l =1 exp (cid:18) − πinlN (cid:19) ˜ p ( l ) (37)(see, for example, [28]). Introducing operators a ( n ) j defined as a ( n ) j = 1 √ w n (cid:16) w n ˜ x ( n ) j + i ˜ p ( n ) j (cid:17) , (38) w n = km eff ω eff sin πnN ! ×× (cid:18) km eff h θ i πnN (cid:19) − (39)we have H = ¯ hω eff N X n =1 3 X j =1 (cid:18) km eff h θ i πnN (cid:19) ×× km eff ω eff sin πnN ! (cid:18) ( a ( n ) j ) † a ( n ) j + 12 (cid:19) . (40)The spectrum of H reads E { n } , { n } , { n } = ¯ h N X a =1 (cid:18) ω eff + 8 km eff sin πaN (cid:19) ×× (cid:18) km eff h θ i πaN (cid:19) (cid:16) n ( a )1 + n ( a )2 ++ n ( a )3 + 32 (cid:19) = N X a =1 ¯ hω a (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) , (41)6here n ( a ) i are quantum numbers ( n ( a ) i = 0 , , ... ). Taking into account (33), (34) the frequen-cies reads ω a = (cid:18) ω + h η i m (cid:19) (cid:18) m ω h θ i k m h θ i πaN (cid:19) + 8 km sin πaN ++ 32 k h θ i πaN . (42)For a chain of particles with harmonic oscillator interaction, describing by Hamiltonian (13)with ω = 0, up to the second order in the parameters of noncommutativity one has E { n } , { n } , { n } == N X a =1 ¯ hω a (cid:18) n ( a )1 + n ( a )2 + n ( a )3 + 32 (cid:19) , (43)with ω a = 8 km sin πaN + h η i m + 32 k h θ i πaN . (44)It is worth noting that in the case of a space with noncommutative coordinates and commutativemomenta (1)-(3) with (4) and η ij = 0 the spectrum of a chain of particles with harmonicoscillator reads (43) with ω a = 8 km sin πaN + 32 k h θ i πaN . (45)Note that ω N = 0 and corresponds to the spectrum of the center-of-mass of the system. Non-commutativity of momenta leads to discrete spectrum of the center-of-mass of a chain of in-teracting particles. From (43), (44) we have that the spectrum of the center-of-mass of thesystem corresponds to the spectrum of three dimensional harmonic oscillator with frequencydetermined as ω N = h η i m . (46)In the limit h θ i → h η i → ω a = ω + km sin πaN . Rotationally invariant algebra with noncommutativity of coordinates and noncommutativityof momenta has been considered. The algebra is constructed involving additional coordinates7nd additional momenta (1)-(3) with (4), (5). We have studied influence of noncommutativityon the spectrum of harmonic oscillator chain with periodic boundary conditions. For thispurpose the total Hamiltonian has been examined (19) and energy levels of harmonic oscillatorchain have been obtained up to the second order in the parameters of noncommutativity.We have found that noncommutativity does not change the form of the chain’s spectrum (41).Noncommutativity of coordinates and noncommutativity of momenta affects on the frequenciesof the system (42).The case of a chain of particles with harmonic oscillator interaction describing by Hamil-tonian (13) with ω = 0 has been studied. We have obtained that the spectrum of the center-of-mass of the system is discrete because of noncommutativity of momenta. This spectrumcorresponds to the spectrum of harmonic oscillator with frequency (46). Acknowledgement.
The author thanks Prof. V. M. Tkachuk for his advices and supportduring research studies. The publication contains the results of studies conducted by President’sof Ukraine grant for competitive projects (F-75).
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