Exact solutions of a damped harmonic oscillator in a time dependent noncommutative space
aa r X i v : . [ qu a n t - ph ] J un Exact solutions of a damped harmonic oscillatorin a time dependent noncommutative space
Manjari Dutta a ∗ , Shreemoyee Ganguly b † , Sunandan Gangopadhyay c ‡ a,c Department of Theoretical Sciences , S.N. Bose National Centre for Basic Sciences , JD Block, Sector III, Salt Lake, Kolkata 700106, India b Department of Basic Science and Humanities,University of Engineering and Management (UEM),B/5, Plot No.III, Action Area-III, Newtown, Kolkata 700156
Abstract
In this paper we have obtained the exact eigenstates of a two dimensional damped harmonicoscillator in time dependent noncommutative space. It has been observed that for somespecific choices of the damping factor and the time dependent frequency of the oscillator,there exists interesting solutions of the time dependent noncommutative parameters follow-ing from the solutions of the Ermakov-Pinney equation. Further, these solutions enable us toget exact analytic forms for the phase which relates the eigenstates of the Hamiltonian withthe eigenstates of the Lewis invariant. We then obtain expressions for the matrix elementsof the coordinate operators raised to a finite arbitrary power. From these general resultswe then compute the expectation value of the Hamiltonian. The expectation values of theenergy are found to vary with time for different solutions of the Ermakov-Pinney equationcorresponding to different choices of the damping factor and the time dependent frequencyof the oscillator. ∗ [email protected] † [email protected] ‡ [email protected], [email protected] The study of time dependent classical as well as quantum harmonic oscillators has appealed totheoretical physicists since time immemorial. In the literature the work by Lewis et al. [1] has leadto an upsurge of analysis of the Hamiltonian for the time dependent quantum harmonic oscillatorusing a class of exact invariants designed for such systems [2, 3]. The problem becomes even morefascinating when one has a system of two such oscillators in two-dimensional space. Now, in orderto address practical situations one needs to include damping in the system. Although there areseveral studies on the one-dimensional damped quantum harmonic oscillator in the past [4]-[8],it’s two-dimensional equivalent is a less explored system [9]. The work by Lawson et.al. [9] is oneof the very few which analyses a two-dimensional damped quantum harmonic oscillator system.The solutions obtained by them for the mentioned system provides a platform to explore theconstruction of various coherent states with intriguing properties.In the present work we extend the study by Lawson et.al. [9] and consider the two-dimensionaldamped quantum harmonic oscillator in noncommutative (NC) space. It has been argued thatstudy of quantum mechanical systems in NC space is essential to ensure the attainment of grav-itational stability [10] in the present theories of quantum gravity, namely, string theory [11, 12]and loop quantum gravity [13]. The simplest quantum mechanical setting in two dimensional NCspace consists of replacing the standard set of commutation relations between the canonical coor-dinates by NC commutation relations [
X, Y ] = iθ , where θ is a positive real constant. Quantummechanical systems in such spaces have been studied extensively in the literature [14]-[24]. Thestudy of a two-dimensional quantum harmonic oscillator in NC space with time dependent NC pa-rameters was done in [25]. However, their system was an undamped oscillator. The parametrizedform of solutions obtained there offered an interesting possibility for study of generalized versionof Heisenberg’s uncertainty relations. Quantum damped harmonic oscillator on noncommutingtwo-dimensional space was studied in [26] where the exact propagator of the system was obtainedand the thermodynamic properties of the system was investigated using the standard canonicaldensity matrix.In this work, a two-dimensional damped quantum harmonic oscillator in NC space is consid-ered once again. However, our focus of study is different than the work carried out in [26]. Wefirst construct the Hamiltonian and then express it in terms of standard commutative variables.This is done in Section 2. Then we solve the Hamiltonian using the method of invariants [1]and obtain the corresponding eigenfunction in Section 3. In doing so, although we start withthe Hamiltonian and corresponding invariant in Cartesian coordinates, eventually we transformour operators to polar coordinates (following closely the procedure suggested in [25]) for ease ofsolution. The form of the Lewis invariant in Cartesian coordinates with a Zeeman term in theHamiltonian is an interesting result in itself and it also makes it easier to make a transition toit’s polar form. It is to be noted that the eigenfunction of the Hamiltonian is a product of theeigenfunction of the invariant and a phase factor. Both the eigenfunction and phase factor areexpressed in terms of time dependent parameters which obey the non-linear differential equationknown as Ermakov-Pinney (EP) equation [27, 28]. Next, in Section 4 we judiciously choose theparameters of the damped system such that they satisfy all the equations representing the systemas well as provide us with an exact closed form solution of the Hamiltonian. The solutions ofthe NC parameters obtained in our analysis turns out to be such that the phase factor in anintegral form given in [25] is exactly integrable for various kinds of dissipation. Then in Section 5we device a procedure to calculate the matrix element of a finite arbitrary power of the positionoperator with respect to the exact solutions for Hamiltonian eigenstates. Using these expressionswe proceed to calculate the expectation value of energy and study the evolution of the energy ex-pectation value of the system with time for various types of damping. In Section 6 we summarizeour results. The system we consider is a combination of two non-interacting damped harmonic oscillators intwo dimensional NC space. The oscillators have equal time dependent frequencies, time dependentcoefficients of friction and equal mass in NC space. Such a model of damped harmonic oscillatorwas considered in an earlier communication [9] in commutative space. In this work, we extendthe model by considering the system in NC space .The Hamiltonian of the system has the following form, H ( t ) = f ( t )2 M ( P + P ) + M ω ( t )2 f ( t ) ( X + X ) (1)where the damping factor f ( t ) is given by, f ( t ) = e − R t η ( s ) ds (2)with η ( s ) being the coefficient of friction. Here ω ( t ) is the time dependent angular frequency ofthe oscillators and M is their mass. It should be noted that in commutative space, the modelwith f ( t ) = e − Γ t and ω ( t ) = ω , with Γ and ω being positive constants, is said to be the two-dimensional Caldirola and Kanai Hamiltonian [29, 30]. The position and momentum coordinates( X i , P i ) are noncommuting variables in NC space, that is, their commutators are [ X , X ] = 0and [ P , P ] = 0. The corresponding canonical variables ( x i , p i ) in commutative space are suchthat the commutator [ x i , p j ] = i ~ δ i,j , [ x i , x j ] = 0 = [ p i , p j ]; ( i, j = 1 , We shall be considering NC phase space in our work. However, we shall generically refer this as NC space.
In order to express the NC Hamiltonian in terms of the standard commutative variablesexplicitly, we apply the standard Bopp-shift relations [31] ( ~ = 1): X = x − θ ( t )2 p ; X = x + θ ( t )2 p (3) P = p + Ω( t )2 x ; P = p − Ω( t )2 x . (4)Here θ ( t ) and Ω( t ) are the NC parameters for space and momentum respectively, such that[ X , X ] = iθ ( t ), [ P , P ] = i Ω( t ) and [ X , P ] = i [1 + θ ( t )Ω( t )4 ] = [ X , P ]; ( X ≡ X , X ≡ Y , P ≡ P x , P ≡ P y ).The Hamiltonian in terms of ( x i , p i ) coordinates is therefore given by the following relation, H = a ( t )2 ( p + p ) + b ( t )2 ( x + x ) + c ( t )( p x − p x ) . (5)The time dependent coefficients in the above Hamiltonian are given as, a ( t ) = f ( t ) M + M ω ( t ) θ ( t )4 f ( t ) (6) b ( t ) = f ( t )Ω ( t )4 M + M ω ( t ) f ( t ) (7) c ( t ) = 12 (cid:20) f ( t )Ω( t ) M + M ω ( t ) θ ( t ) f ( t ) (cid:21) . (8)Here it must be noted that although our Hamiltonian given by Eqn.(5) has the same form as thatin [25] to study a system of a two dimensional harmonic oscillator in NC space, the time dependentHamiltonian coefficients (given by Eqn(s).(8)) have very different form. This is because our systemis that of a damped harmonic oscillator in two-dimensional NC space. Thus, the damping factor f ( t ) modulates and alters the Hamiltonian coefficients from the form considered in earlier study[25]. In order to find the solutions of the model Hamiltonian H ( t ) (Eqn.(5)) representing the two-dimensional damped harmonic oscillator in NC space, we follow the route suggested by Lewis et.al. [1] in their work. First we construct the time-dependent Hermitian invariant operator I ( t )corresponding to our Hamiltonian operator H ( t ) (given by Eqn.(5)). This is because if one cansolve for the eigenfunctions of I ( t ), φ ( x , x ), such that, I ( t ) φ ( x , x ) = ǫφ ( x , x ) (9)where ǫ is an eigenvalue of I ( t ) corresponding to eigenstate φ ( x , x ), one can obtain the eigen-states of H ( t ), ψ ( x , x , t ), using the relation given by Lewis et. al. [1] which is as follows, ψ ( x , x , t ) = e i Θ( t ) φ ( x , x ) (10)where the real function Θ( t ) which acts as the phase factor will be discussed in details later. Next, following the approach taken by Lewis et.al. [1], we need to construct the operator I ( t )which is an invariant with respect to time, corresponding to the Hamiltonian H ( t ), as mentionedearlier, such that I ( t ) satisfies the condition, dIdt = ∂ t I + 1 i [ I, H ] = 0 . (11)The procedure is to choose the Hermitian invariant I ( t ) to be of the same homogeneous quadraticform defined by Lewis et. al. [1] for time-dependent harmonic oscillators. However, since we aredealing with a two-dimensional system in the present study, I ( t ) takes on the following form, I ( t ) = α ( t )( p + p ) + β ( t )( x + x ) + γ ( t )( x p + p x ) . (12)Here we will consider ~ = 1 since we choose to work in natural units. Now, using the form of I ( t ) defined by Eqn.(12) in Eqn.(11) and equating the coefficients of the canonical variables, weget the following relations, ˙ α ( t ) = − a ( t ) γ ( t ) (13)˙ β ( t ) = b ( t ) γ ( t ) (14)˙ γ ( t ) = 2 [ b ( t ) α ( t ) − β ( t ) a ( t ) ] (15)where dot denotes derivative with respect to time t .To express the above three time dependent parameters α , β and γ in terms of a single timedependent parameter, we parametrize α ( t ) = ρ ( t ). Substituting this in Eqn(s).(13, 15), we getthe other two parameters in terms of ρ ( t ) as, γ ( t ) = − ρ ˙ ρa ( t ) (16) β ( t ) = 1 a ( t ) (cid:20) ˙ ρ a ( t ) + ρ b + ρ ¨ ρa ( t ) − ρ ˙ ρ ˙ aa (cid:21) . (17)Now, substituting the value of β in Eqn.(14), we get a non-linear equation in ρ ( t ) which has theform of the non-linear Ermakov-Pinney (EP) equation with a dissipative term [25, 27, 28]. Theform of the non-linear equation is as follows,¨ ρ − ˙ aa ˙ ρ + abρ = ξ a ρ . (18)where ξ is a constant of integration. This equation has similar form to the EP equation obtainedin [25], which is expected since our H ( t ) has the same form as theirs. However, once again weshould recall the fact that the explicit form of the time-dependent coefficients are different dueto the presence of damping.Now, using the EP equation we get a simpler form of β as, β ( t ) = 1 a ( t ) (cid:20) ˙ ρ a ( t ) + ξ a ( t ) ρ (cid:21) . (19)Next, substituting the expressions of α , β and γ in Eqn.(12), we get the following expression for I ( t ), I ( t ) = ρ ( p + p ) + (cid:18) ˙ ρ a + ξ ρ (cid:19) ( x + x ) − ρ ˙ ρa ( x p + p x ) . (20)The form of the Lewis invariant in Cartesian coordinates will be used later to go over to it’s polarcoordinate form. The solution of the EP equation under various physically significant conditionsshall be discussed later. Now that we have the required Hermitian invariant I ( t ), we proceed to calculate it’s eigenstatesusing the operator approach. For this purpose we need to first construct some ladder operators.To do this, we first need to transform the form of I ( t ) (given by Eqn.(20)) to a more manageableform. For this we invoke a unitary transformation using a suitable unitary operator ˆ U havingthe following form, ˆ U = exp (cid:20) − i ˙ ρ a ( t ) ρ ( x + x ) (cid:21) , ˆ U † ˆ U = ˆ U ˆ U † = I . (21)Defining, φ ′ ( x , x ) = ˆ U φ ( x , x ) , I ′ ( t ) = ˆ U I ˆ U † (22)where φ ( x , x ) is an eigenfunction of I ( t ) as introduced in Eqn.(9), then, using Eqn(s).(9,22),we get, I ′ φ ′ = ˆ U I ˆ U † ˆ U φ = ˆ
U Iφ = ˆ
U ǫφ = ǫφ ′ . (23)The transformed expression of the invariant, I ′ ( t ), using Eqn.(22), has the following form, I ′ ( t ) = ρ ( p + p ) + ξ ρ ( x + x ) . (24)This transformed form of the invariant, I ′ ( t ), has exactly the same form as that of the Hamiltonianfor a time dependent two-dimensional simple harmonic oscillator. So, we can introduce thecorresponding ladder operators for ˆ I ′ ( t ) to be given by,ˆ a ′ j = 1 √ ξ (cid:18) ξρ ˆ x j + iρ ˆ p j (cid:19) , ˆ a ′ † j = 1 √ ξ (cid:18) ξρ ˆ x j − iρ ˆ p j (cid:19) (25)where j = 1 , a ′ i , ˆ a ′ † j ] = δ ij .Now we make the reverse transformation to get the expression of the unprimed ladder operators:ˆ a j ( t ) = ˆ U † ˆ a ′ j ˆ U = 1 √ ξ (cid:20) ξρ x j + iρp j − i ˙ ρa ( t ) x j (cid:21) (26)ˆ a j † ( t ) = ˆ U † ˆ a ′ † j ˆ U = 1 √ ξ (cid:20) ξρ x j − iρp j + i ˙ ρa ( t ) x j (cid:21) . (27)It can be easily checked using the algebra of the primed ladder operators that [ˆ a i , ˆ a † j ] = δ ij .We now set ξ = 1 and consider two linear combinations of the above two operators such that,ˆ a ( t ) = − i √ a − i ˆ a ) = 12 (cid:20) ρ ( ˆ p − i ˆ p ) − (cid:18) iρ + ˙ ρa ( t ) (cid:19) ( ˆ x − i ˆ x ) (cid:21) (28)and ˆ a † ( t ) = i √ a † + i ˆ a † ) = 12 (cid:20) ρ ( ˆ p + i ˆ p ) + (cid:18) iρ − ˙ ρa ( t ) (cid:19) ( ˆ x + i ˆ x ) (cid:21) . (29)These also satisfy the commutation relation [ˆ a, ˆ a † ] = 1. With the above results in place, we now transform the invariant I ( t ) and the corresponding ladderoperators to polar coordinates for calculational convenience. For this we invoke the transformationof coordinates of the form, x = rcosθ ; y = rsinθ . (30)The canonical coordinates in polar representation takes the following form, p r = 12 (cid:16) x r p + p x r + x r p + p x r (cid:17) = x p + x p r − i r = − i (cid:18) ∂ r + 12 r (cid:19) (31) p θ = ( x p − x p ) = − i∂ θ . (32)The commutation relations between ( r , p r ) and ( θ , p θ ) have the form[ r, p r ] = [ θ, p θ ] = [ x , p ] = [ x , p ] = i. (33)The corresponding anticommutation relation can be found to be,[ r, p r ] + = [ x , p ] + + [ x , p ] + = 2( x p + p x ) (34)where [ A, B ] + = AB + BA represents anticommutator between operators A , B .In order to transform the invariant I ( t ) in polar coordinates, we need to have few other relationswhich are, ( p + p ) = (cid:18) p r + p θ r − r (cid:19) (35)( p + ip ) = e iθ (cid:20) p r + ir p θ + i r (cid:21) (36)( p − ip ) = e − iθ (cid:20) p r − ir p θ + i r (cid:21) . (37)Hence the invariant in polar coordinate system is given by, I ( t ) = ξ ρ r + (cid:18) ρp r − ˙ ρa r (cid:19) + (cid:16) ρp θ r (cid:17) − (cid:18) ρ ~ r (cid:19) (38)and the ladder operators in polar coordinate system have the following form,ˆ a ( t ) = 12 (cid:20)(cid:18) ρp r − ˙ ρa ( t ) r (cid:19) − i (cid:18) rρ + ρp θ r + ρ r (cid:19)(cid:21) e − iθ ˆ a † ( t ) = 12 e iθ (cid:20)(cid:18) ρp r − ˙ ρa ( t ) r (cid:19) + i (cid:18) rρ + ρp θ r + ρ r (cid:19)(cid:21) . (39)Now we note from Eqn(s).(38, 39) that both the invariant I ( t ) and the ladder operators havethe same form as those used in [25] to study the undamped harmonic oscillator in NC space.The time-dependent coefficients involved in the present study however differ due to the dampingpresent in our system. Thus, we can just borrow the expression of eigenfunction and the phasefactors from [25] for our present system. We depict the set of eigenstates of the invariant operator I ( t ) as | n, l i , following the conventionin [25]. Here, n and l are integers such that n + l >
0. So we have the condition l > − n .Thus, if l = − n + m , then m is a positive integer; and the corresponding eigenfunction in polarcoordinate system has the following form (restoring ~ ), φ n,m − n ( r, θ ) = h r, θ | n, m − n i (40)= λ n ( i √ ~ ρ ) m √ m ! r n − m e iθ ( m − n ) − a ( t ) − iρ ˙ ρ a ( t ) ~ ρ r U (cid:18) − m, − m + n, r ~ ρ (cid:19) (41)where λ n is given by λ n = 1 πn !( ~ ρ ) n . (42)Here, U (cid:18) − m, − m + n, r ~ ρ (cid:19) is Tricomi’s confluent hypergeometric function [32, 33] and theeigenfunction φ n,m − n ( r, θ ) satisfies the following orthonormality relation, Z π dθ Z ∞ rdrφ ∗ n,m − n ( r, θ ) φ n ′ ,m ′ − n ′ ( r, θ ) = δ nn ′ δ mm ′ . (43)Again following [25], the expression of the phase factor Θ( t ) is given by,Θ n , l ( t ) = ( n + l ) Z t (cid:18) c ( T ) − a ( T ) ρ ( T ) (cid:19) dT . (44)For a given value of l = − n + m , it would be given by [25],Θ n , m − n ( t ) = m Z t (cid:18) c ( T ) − a ( T ) ρ ( T ) (cid:19) dT . (45)We shall use this expression to compute the phase explicitly as a function of time for variousphysical cases in the subsequent discussion.The eigenfunction of the Hamiltonian therefore reads (using Eqn(s).(10, 41, 45)) ψ n,m − n ( r, θ, t ) = e i Θ n,m − n ( t ) φ n,m − n ( r, θ )= λ n ( i √ ~ ρ ) m √ m ! exp (cid:20) im Z t (cid:18) c ( T ) − a ( T ) ρ ( T ) (cid:19) dT (cid:21) × r n − m e iθ ( m − n ) − a ( t ) − iρ ˙ ρ a ( t ) ~ ρ r U (cid:18) − m, − m + n, r ~ ρ (cid:19) . (46) In this paper we are primarily interested in damped oscillators in NC space. For this purpose wewant to find the eigenfunctions of the corresponding Hamiltonian under various types of damping.The various kinds of damping are represented by various forms of the time dependent coefficientsof the Hamiltonian, namely, a ( t ), b ( t ) and c ( t ). However, the various forms must be constructedin such a way that they satisfy the non-linear EP equation given by Eqn.(18). The procedureof this construction of exact analytical solutions is based on the Chiellini integrability condition[34] and this formalism was followed in [25]. We shall do the same in this paper. So, for variousforms of a ( t ) and b ( t ), we get the corresponding form of ρ ( t ) using the EP equation togetherwith the Chiellini integrability condition. In other words, the set of values of a ( t ), b ( t ) and ρ ( t )that we use must be a solution set of the EP equation consistent with the Chiellini integrabilitycondition. In the subsequent discussion we shall proceed to obtain solutions of the EP equationfor the damped NC oscillator.0 The simplest kind of solution set of EP equation under damping is the exponentially decaying setused in [25]. The solution set is given by the following relations, a ( t ) = σe − ϑt , b ( t ) = ∆ e ϑt , ρ ( t ) = µe − ϑt/ (47)where σ, ∆ and µ are constants. Here, ϑ is any positive real number. Substituting the expressionof a ( t ) , b ( t ) and ρ ( t ) in the EP equation, we can easily verify the relation between these constantsto be as follows, µ = ξ σ σ ∆ − ϑ . (48) We now write down the eigenfunctions of the Hamiltonian for the choosen set of time-dependentcoefficients. For this endeavour we need to choose explicit forms of the damping factor f ( t ) andangular frequency of the oscillator ω ( t ). The eigenfunction of the invariant I ( t ) (which is givenby Eqn.(41)) takes on the following form for the solution set-I: φ n,m − n ( r, θ ) = λ n ( iµe − ϑt/ ) m √ m ! r n − m e iθ ( m − n ) − σ + iµ ϑ σµ e − ϑt r U (cid:18) − m, − m + n, r e ϑt µ (cid:19) (49)where λ n is given by λ n = 1 π n ! [ µ exp ( − ϑt )] n . (50)In order to obtain explicit expressions of the phase factors for various cases of the damping factor,we choose both the functions ω ( t ) and η ( t ) as follows. h A i Solution Set-Ia
Firstly, we choose the damping factor f ( t ) = 1. Thus, in this case the damping in the system isdue to the exponentially decaying frequency ω ( t ). For this purpose we set, η ( t ) = 0 ⇒ f ( t ) = 1 (51) ω ( t ) = ω exp ( − Γ t/ . (52)1Substituting the expressions for a ( t ), b ( t ), ω ( t ) and f ( t ) in the Eqn(s).(6, 7), we get the timedependent NC parameters as, θ ( t ) = 2 M ω exp [Γ t/ p M σ exp ( − ϑt ) − t ) = 2 q M [∆ exp ( ϑt ) − M ω exp ( − Γ t )] . (54)It can be checked that in the limit Γ →
0, that is, for constant frequency, the expressions for θ ( t )and Ω( t ) reduce to those in [25]. When ϑ = Γ, then the solutions take the form, θ ( t ) = 2 M ω p M σ − e Γ t (55)Ω( t ) = 2 q M [∆ exp (Γ t ) − M ω exp ( − Γ t )] . (56)Substituting these relations in the expression for c ( t ) in Eqn.(8), we get an expression for thephase in a closed form as, c ( t ) = r ∆ exp (Γ t ) − M ω exp ( − Γ t ) M + ω exp ( − Γ t/ p M σ exp ( − Γ t ) − . (57)Substituting the expressions of a ( t ), ρ ( t ) and c ( t ) in Eqn.(45), we get,Θ n , l ( t ) = ( n + l ) ω √ M σ Γ " log e e Γ t − M σ − p M σ ( M σ − e Γ t )1 − M σ − p M σ ( M σ − − Γ t − p M σ ( M σe − t − e − Γ t ) + 2 p M σ ( M σ − i + 2( n + l )Γ "r ∆ M e Γ t − ω e − Γ t − r ∆ M − ω +2 iω (cid:26) e − Γ t/ F (cid:18) − , , , ∆ e t M ω (cid:19) − F (cid:18) − , , , ∆ M ω (cid:19)(cid:27)(cid:21) − σµ ( n + l ) t (58)where F ( a, b, c ; t ) is said to be the Gauss hypergeometric function. It is interesting to note thatthe solutions of the time dependent NC parameters enable us to get an exact analytic expressionfor the phase factor. It is further interesting to observe that the phase has a complex part whichindicates that the wave function decays with time. h B i Solution Set-Ib
Here the oscillator is damped due to the damping factor f ( t ) and the frequency ω ( t ) is a constant.This situation can be depicted by the following relations, f ( t ) = exp ( − Γ t ) ; ω ( t ) = ω . (59)2Substituting these relations in Eqn(s).(6, 7), we get the time dependent NC parameters as, θ ( t ) = 2 M ω p M σ exp ( − ϑt ) − exp ( − Γ t ) e − Γ t/ (60)Ω( t ) = 2 e Γ t p M [∆ exp ( ϑ − Γ) t − M ω ] . (61)It can be checked that in the limit Γ →
0, that is, for constant frequency, the expressions for θ ( t )and Ω( t ) reduce to those in [25]. When ϑ = Γ, then the solutions take the form, θ ( t ) = 2 M ω √ M σ − e − Γ t (62)Ω( t ) = 2 e Γ t p M [∆ − M ω ] . (63)Substituting these relations in the expression for c ( t ) in Eqn.(8), we get, c ( t ) = r ∆ − M ω M + ω √ M σ − constant . (64)Substituting the expressions of a ( t ) , ρ ( t ) and c ( t ) in Eqn.(45) , we get an expression for the phasein a closed form as,Θ n , l ( t ) =( n + l ) " − σµ + r ∆ − M ω M + ω √ M σ − t . (65)Once again we are able to obtain an exact expression for the phase, in this case varying linearlywith time. It is important to note that the reality of the phase in this case depends crucially on theparameters ∆, M , σ , ω . The phase Θ n,l is real if ∆ − M ω ≥ M σ ≥
1, else it is complex. h C i Solution Set-Ic
Here the oscillator is damped due to the damping factor f ( t ) and the time-dependent frequency ω ( t ); both of which are exponentially decaying. Thus, we set, f ( t ) = exp ( − Γ t ) ; ω ( t ) = ω exp ( − Γ t/ . (66)Substituting these relations in Eqn.(s)(6, 7), we get the time dependent NC parameters to be, θ ( t ) = 2 M ω q ( M σe − ( ϑ − Γ) t − e − Γ t/ (67)Ω( t ) = 2 p M [∆ exp ( ϑt ) − M ω ] e Γ t/ . (68)It can be checked that in the limit Γ →
0, that is, for constant frequency, the expressions for θ ( t )and Ω( t ) reduce to those in [25]. When ϑ = Γ, then the solutions take the form, θ ( t ) = 2 M ω p ( M σ − e − Γ t/ (69)Ω( t ) = 2 p M [∆ exp (Γ t ) − M ω ] e Γ t/ . (70)3Substituting these relations in the expression for c ( t ) in Eqn.(8), we get, c ( t ) = r ∆ − M ω exp [ − Γ t ] M + ω e − Γ t/ √ M σ − . (71)Substituting the expressions of a ( t ), ρ ( t ) and c ( t ) in Eqn.(45), we obtain an expression for thephase in a closed form as,Θ n , l ( t ) = ( n + l )Γ √ M (cid:20) √ ∆ Γ t + 2 q ∆ − M ω − q ∆ − M ω exp ( − Γ t )+ 2 √ ∆ log ∆ + p ∆[∆ − M ω exp ( − Γ t )]∆ + p ∆[∆ − M ω ] ! − ( n + l ) (cid:20) σ tµ + 2Γ ω (cid:0) e − Γ t/ − (cid:1) √ M σ − (cid:21) . (72) We now consider rationally decaying solutions of the EP equation similar to that used in [25]which is of the form, a ( t ) = σ (cid:18) k (cid:19) ( k +2) /k (Γ t + χ ) ( k +2) /k b ( t ) = ∆ (cid:18) kk + 2 (cid:19) (2 − k ) /k (Γ t + χ ) ( k − /k ⇒ ∆ (cid:18) k (cid:19) ( k − /k (Γ t + χ ) ( k − /k ρ ( t ) = µ (cid:18) k (cid:19) /k (Γ t + χ ) /k (73)where σ , ∆, µ , Γ and χ are constants such that (Γ t + χ ) = 0, and k is an integer. Substitutingthe expressions of a ( t ), b ( t ), and ρ ( t ) in the EP equation, we can easily verify the relation betweenthese constants to be as follows, Γ µ = ( k + 2) ( σ ∆ µ − ξ σ µ ) . (74)4 The eigenfunction of the invariant operator I ( t ) (given by Eqn.(41)) for this solution Set-II isgiven by, φ n , m − n ( r, θ ) = λ n ( iµ ) m √ m ! (cid:20) k + 2 k (Γ t + χ ) (cid:21) m/k r n − m e iθ ( m − n ) − [ σ ( k + 2) + iµ Γ] (Γ t + χ ) /k k /k σ ( k + 2) ( k +2) /k µ r × U − m, − m + n, r [ k (Γ t + χ )] /k µ ( k + 2) /k ! (75)where λ n is given by λ n = 1 π n ! µ n +2 (cid:20) k (Γ t + χ ) k + 2 (cid:21) n ) /k . (76)In order to get the eigenfunction of the Hamiltonian H ( t ), we need to calculate the associatedphase factor. Once again for this we need to fix up the forms of the damping factor f ( t ) andangular frequency ω ( t ) of the oscillator. In order to explore the solution of H ( t ) for rationallydecaying coefficients, we choose a rationally decaying form for ω ( t ) and set f ( t ) = 1. Thus, wehave the following relations, η ( t ) = 0 ⇒ f ( t ) = 1 (77) ω ( t ) = ω (Γ t + χ ) . (78)Substituting these relations in Eqns.(6, 7), we get the time dependent NC parameters as, θ ( t ) = 2 (Γ t + χ ) M ω s M σ (cid:20) ( k + 2) k (Γ t + χ ) (cid:21) ( k +2) /k − t ) = 2 s M ∆ (cid:20) k + 2 k (Γ t + χ ) (cid:21) ( k − /k − M ω (Γ t + χ ) . (80)We now consider k = 2. This enables us to integrate the expression for the phase factor (givenby Eqn.(45)). The simplified forms of a ( t ), b ( t ) and ρ ( t ) for k = 2 read, a ( t ) = 4 σ (Γ t + χ ) , b ( t ) = ∆ , ρ ( t ) = (cid:20) µ Γ t + χ (cid:21) / . (81)Substituting these relations in the expression for c ( t ) in Eqn.(8) gives, c ( t ) = ω (Γ t + χ ) s σ M (Γ t + χ ) − s ∆ M − ω (Γ t + χ ) . (82)5Substituting these expressions for a ( t ), ρ ( t ) and c ( t ) for k = 2 in Eqn.(45), we get the followingexpression for the phase factor in a closed form as,Θ n,l ( t ) = ( n + l )Γ ω tan − ω q ∆ M (Γ t + χ ) − ω + r ∆ (Γ t + χ ) M − ω − σµ log e ( χ + Γ t ) χ − r ∆ M χ − ω − ω tan − ω q ∆ M χ − ω + ω ( n + l )Γ " p σ M − χ χ − p σ M − ( χ + Γ t ) ( χ + Γ t )+ ilog e ( χ + Γ t ) + p ( χ + Γ t ) − σ Mχ + p χ − σ M . (83)We can now get the eigenfunction of this rationally decaying damped system using Eqn.(10). We now propose a simple method of obtaining a solution of the EP equation. The method is asfollows. Choosing ρ ( t ) to be any arbitrary time dependent function and taking it’s time derivativeas proportional to a ( t ), that is, a ( t ) = constant × ˙ ρ and setting b ( t ) = constant × aρ , we observethat these would always satisfy the EP equation along with a certain constraint relation amongthe constants.Here we consider a simple solution which is a special case of the above solution for the EPequation. We call this the elementary solution which reads, a ( t ) = σ , b ( t ) = ∆(Γ t + χ ) , ρ ( t ) = µ (Γ t + χ ) (84)where Γ, χ , µ , σ and ∆ are constants. The above solution set satisfy the EP equation with thefollowing constraint relation, ∆ µ = ξ σ . (85)6 The eigenfunctions of the invariant operator I ( t ) for this solution set is given by, φ n,m − n ( r, θ ) = λ n [ iµ (Γ t + χ )] m √ m ! r n − m e iθ ( m − n ) − σ − iµ Γ(Γ t + χ )2 σµ (Γ t + χ ) r × U (cid:18) − m, − m + n, r µ (Γ t + χ ) (cid:19) (86)where λ n is given by λ n = 1 πn ![ µ (Γ t + χ )] n . (87)In order to get an eigenfunction of the Hamiltonian, we calculate the phase factor for a particularcase of the damped harmonic oscillator where the angular frequency ω ( t ) is rationally decayingand the damping factor f ( t )=1. Thus, we set, η ( t ) = 0 ⇒ f ( t ) = 1 (88) ω ( t ) = ω (Γ t + χ ) (89)where Γ and χ are real constants. Substituting these relations in Eqns.(6, 7), we get the timedependent NC parameters as, θ ( t ) = 2 (Γ t + χ ) ω M √ M σ − t ) = 2 s M ∆(Γ t + χ ) − M ω (Γ t + χ ) . (91)Substituting these relations in the expression for c ( t ) in Eqn.(8), we get, c ( t ) = s ∆ M (Γ t + χ ) − ω (Γ t + χ ) + ω (Γ t + χ ) √ M σ − . (92)Substituting these expressions of a ( t ), ρ ( t ) and c ( t ) in Eqn.(45), we obtain an expression for thephase factor in a closed form as,Θ n , l ( t ) = ( n + l ) (cid:20) ω √ M σ −
1Γ log (Γ t + χ ) χ − σtµ χ (Γ t + χ ) (cid:21) + ( n + l )Γ "s ∆ M χ − ω − s ∆ M (Γ t + χ ) − ω + ω tan − ω χ r ∆ M − χ ω − tan − ω (Γ t + χ ) r ∆ M − ω (Γ t + χ ) . (93)We can now get the eigenfunction of this system by using Eqn.(10).7 In this section, we intend to calculate the expectation value of energy. For this we need tocalculate the expectation value of the Hamiltonian H ( t ) in it’s own eigenstates. The expectationvalue h H i is given by (using Eqn.(5)), h H i = a ( t )2 ( h p i + h p i ) + b ( t )2 ( h x i + h x i ) + c ( t )( h p x i − h p x i ) . (94)To calculate this we need to get the expectation value of the individual canonical operators. Toset up our notation we denote the eigenstates of the Hamiltonian H ( t ) by | n, l i H . We start by calculating the matrix element of an arbitrary power of x , H h n, l | x k | n, l i H , which isgiven by H h n, m − n | x k | n, m ′ − n i H = Z rdrdθ H h n, m − n | r, θ ih r, θ | r k cos k θ | n, m ′ − n i H = 12 k e i (Θ n,m ′− n − Θ n,m − n ) Z r k +1 drdθ ( e iθ + e − iθ ) k × φ ∗ n,m − n ( r, θ ) φ n,m ′ − n ( r, θ ) (95)where we have used the relations, | n, l i H = e i Θ n,l | n, l i where | n, l i H and | n, l i are eigenstatesof the Hamiltonian H ( t ) and Lewis invariant I ( t ) respectively. We have also used the relation h r, θ | n, m ′ − n i = φ n,m ′ − n ( r, θ ), with φ being the eigenfunction of I ( t ). Now, Eqn.(95) can berewritten as, H h n, m − n | x k | n, m ′ − n i H = π k − k X r =0 k C r δ m ′ ,m +2 r − k A ( n, m, m + 2 r − k ) × Z ∞ r dr r n − m − r + k ) e − r ~ ρ × U (cid:18) − m, − m + n, r ~ ρ (cid:19) × U (cid:18) − m − r + k, − m − r + k + n, r ~ ρ (cid:19) (96)where A ( n, m, m + 2 r − k ) = e i (Θ n,m − n +2 r − k − Θ n,m − n ) λ n ( − i ~ / ρ ) m ( i ~ / ρ ) m +2 r − k p m !( m + 2 r − k )! .8Now defining w = − r ~ ρ , we have, H h n, m − n | x k | n, m ′ − n i H = k X r =0 π k e i (Θ n,m − n +2 r − k − Θ n,m − n ) ( − k + r i − k k C r δ m ′ ,m +2 r − k × λ n ( ~ / ρ ) n + k +2 p m !( m + 2 r − k )! × Z ∞ dww n − m − r + k e − w L ( n − m ) m ( w ) L ( n − m − r + k ) m +2 r − k ( w ) (97)where we have used the following result on special functions [32, 33], L ( ζ ) n ( w ) = ( − n n ! U ( − n, ζ + 1 , w ) (98)where L ( ζ ) n ( w ) are associated Laguerre polynomials.Now, we get using the relation for phase given in [25],Θ n,l = ( n + l ) Z t (cid:20) c ( τ ) − a ( τ ) ρ ( τ ) (cid:21) dτ (99)the following relation, e i (Θ n,m − n +2 r − k − Θ n,m − n ) = e i [ { ( n + m − n +2 r − k ) − ( n + m − n ) } R t ( c ( τ ) − a ( τ ) ρ τ ) ) dτ ] = e i (0+2 r − k ) R t ( c ( τ ) − a ( τ ) ρ τ ) ) dτ = e i Θ , r − k . (100)So, we finally get the following relation for the matrix element of x k , H h n, m − n | x k | n, m ′ − n i H = k X r =0 π k e i Θ , r − k ( − k + r i − k k C r δ m ′ ,m +2 r − k × λ n ( ~ / ρ ) n + k +2 p m !( m + 2 r − k )! × Z ∞ dw w n − m − r + k e − w L ( n − m ) m ( w ) L ( n − m − r + k ) m +2 r − k ( w ) . (101)This is a new result in this paper and can be used to obtain the matrix element or expectationvalue of any power of x . For the sake of completeness, we also write down the matrix element of x k in the eigenstates of the Lewis invariant I ( t ), which reads h n, m − n | x k | n, m ′ − n i = k X r =0 π k ( − k + r i − k k C r δ m ′ ,m +2 r − k × λ n ( ~ / ρ ) n + k +2 p m !( m + 2 r − k )! × Z ∞ dw w n − m − r + k e − w L ( n − m ) m ( w ) L ( n − m − r + k ) m +2 r − k ( w ) . (102)Note that the phase factor does not appear in the above result.9Now, we proceed to evaluate the matrix element H h n, m − n | x | n, m ′ − n i H using the expressionobtained in Eqn.(101). This reads H h n, m − n | x | n, m ′ − n i H = H h n, m − n | x k | k =1; r =0 | n, m ′ − n i H + H h n, m − n | x k | k =1; r =1 | n, m ′ − n i H . (103)Evaluating the above matrix elements give, H h n, m − n | x k | k =1; r =0 | n, m ′ − n i H = − i ρ ~ / ) √ me − i Θ , δ m,m ′ +1 (104) H h n, m − n | x k | k =1; r =1 | n, m ′ − n i H = i ρ ~ / ) √ m ′ e i Θ , δ m ′ ,m +1 . (105)In order to obtain Eqn(s).(104, 105), we used the following relations involving the associatedLaguerre polynomials, L ( ζ ) n ( w ) = L ( ζ +1) n ( w ) − L ( ζ +1) n − ( w ) Z ∞ dw w ζ e − w L ( ζ ) n ( w ) L ζm ( w ) = ( n + ζ )! n ! δ n,m . (106)Combining Eqn(s).(104, 105), we get the following expression, H h n, m − n | x | n, m ′ − n i H = i ρ ~ / )[ √ m ′ e i Θ , δ m ′ ,m +1 − √ me − i Θ , δ m,m ′ +1 ] . (107)Next, we evaluate, H h n, m − n | x | n, m ′ − n i H = H h n, m − n | x k | k =2; r =0 | n, m ′ − n i H + H h n, m − n | x k | k =2; r =1 | n, m ′ − n i H + H h n, m − n | x k | k =2; r =2 | n, m ′ − n i H . (108)Evaluation of the above matrix elements yield, H h n, m − n | x k | k =2; r =0 | n, m ′ − n i H = −
14 ( ~ ρ ) e − i Θ , δ m ′ ,m − p m ( m − H h n, m − n | x k | k =2; r =1 | n, m ′ − n i H = 12 ( ~ ρ ) e − i Θ , δ m,m ′ ( m + n + 1) H h n, m − n | x k | k =2; r =2 | n, m ′ − n i H = −
14 ( ~ ρ ) e i Θ , δ m ′ ,m +2 p ( m + 2)( m + 1) . (109)In order to calculate the above expressions, apart from the relations between special functionsgiven by Eqn.(106), we need the following relation, Z ∞ dw w k + p e − w L kn ( w ) L kn ( w ) = ( n + k )! n ! × (2 n + k + 1) p . (110)0So we have, H h n, m − n | x | n, m ′ − n i H = ( ~ ρ )2 δ m,m ′ ( m + n + 1) − ( ~ ρ )4 h e − i Θ , δ m ′ ,m − p m ( m − e i Θ , δ m ′ ,m +2 p ( m + 2)( m + 1) i . (111)It is to be noted that the matrix elements for x and x in the eigenstates of the Hamiltonian[given by Eqn(s).(107, 111) respectively], matches exactly with the corresponding expression givenin [25], although the result quoted in [25] is in the eigenstate of the invariant I ( t ).The matrix element of y k in the eigenstates of the Hamiltonian can be obtained similarly, andreads, H h n, m − n | y k | n, m ′ − n i H = k X r =0 π k e i Θ , r − k k C r δ m ′ ,m +2 r − k × λ n ( ~ / ρ ) n + k +2 p m !( m + 2 r − k )! × Z ∞ dw w n − m − r + k e − w L ( n − m ) m ( w ) L ( n − m − r + k ) m +2 r − k ( w ) . (112)Once again we write down the matrix element of y k in the eigenstates of the Lewis invariant I ( t ).This reads h n, m − n | y k | n, m ′ − n i = k X r =0 π k k C r δ m ′ ,m +2 r − k × λ n ( ~ / ρ ) n + k +2 p m !( m + 2 r − k )! × Z ∞ dw w n − m − r + k e − w L ( n − m ) m ( w ) L ( n − m − r + k ) m +2 r − k ( w ) . (113)Using Eqn.(112), we may evaluate the matrix element of y and y in the eigenstate of theHamiltonian. We find, H h n, m − n | y | n, m ′ − n i H = H h n, m − n | y k | k =1; r =0 | n, m ′ − n i H + H h n, m − n | y k | k =1; r =1 | n, m ′ − n i H = −
12 ( ρ ~ / )[ √ me − i Θ , δ m ′ ,m − + √ m + 1 e i Θ , δ m ′ ,m +1 ] . (114) H h n, m − n | y | n, m ′ − n i H = H h n, m − n | y k | k =2; r =0 | n, m ′ − n i H + H h n, m − n | y k | k =2; r =1 | n, m ′ − n i H + H h n, m − n | y k | k =2; r =2 | n, m ′ − n i H = ~ ρ δ m ′ ,m − p m ( m − e − i Θ , + 12 δ m,m ′ ( ~ ρ )( m + n + 1) + ~ ρ δ m ′ ,m +2 p ( m + 2)( m + 1) e i Θ , . (115)1From the above analysis, we find that even the expression for the matrix element of the operator y k in the eigenstate of H ( t ) matches with that found in [25] for k = 1 ,
2, though again they hadinappropriately quoted the results in the eigenstate of the Lewis invariant.
As we have already seen from Eqn.(5), in order to calculate the expectation value of energy oneneeds the expectation values h p i , h p i , h x i , h x i , h p x i and h p x i . As we have seenin the previous subsection, our calculated generalized expressions for matrix elements H h n, m − n | x k | k =1; r =0 | n, m ′ − n i H and H h n, m − n | y k | k =1; r =0 | n, m ′ − n i H matched exactly with thecalculations in [25] for k = 1 ,
2. Hence, we use the matrix elements quoted in the said work tocalculate the following expectation values, h x j i = ρ n + m + 1) ; h p j i = 12 (cid:18) ρ + ˙ ρ a (cid:19) ( n + m + 1) ; h x j p k i = 12 ǫ jk ( m − n ) ; (116)where j, k = 1 , ǫ jk = − ǫ kj with ǫ = 1. So, the expectation value of energy h E n,m − n ( t ) i with respect to energy eigenstate ψ n,m − n ( r, θ, t ) can be expressed as, h E n,m − n ( t ) i = 12 ( n + m + 1) (cid:20) b ( t ) ρ ( t ) + a ( t ) ρ ( t ) + ˙ ρ ( t ) a ( t ) (cid:21) + c ( t ) ( n − m ) . = 12 (cid:20) ( n + m + 1) (cid:18) b ( t ) ρ ( t ) + a ( t ) ρ ( t ) + ˙ ρ ( t ) a ( t ) (cid:19) + ( n − m ) (cid:18) f ( t )Ω( t ) M + M ω ( t ) θ ( t ) f ( t ) (cid:19)(cid:21) . (117)It is interesting to note that even when the frequency of oscillation ω →
0, the expectation valueof energy is non-zero. This is because all the three parameters of the Hamiltonian a ( t ), b ( t ) and c ( t ) are finite even as ω →
0, as is clear from the Eqn(s).(6,7,8). Now we will proceed to studythe time-dependent behaviour of h E n,m − n ( t ) i for various types of damping. For the exponentially decaying solution given by Eqn.(47), the energy expectation value takes thefollowing form, h E n,m − n ( t ) i = ( n + m + 1) µ ∆ + c ( t ) ( n − m ) (118)where we have set the constant ξ to unity and used the constraint relation given by Eqn.(48). h A i Solution Set-Ia
For this case we consider f ( t ) = 1 and ω ( t ) = ω e − Γ t/ . The expectation value of energy for theground state has the following expression, h E n, − n ( t ) i = ( n + 1) µ ∆ + n "r ∆ exp (Γ t ) − M ω exp ( − Γ t ) M + ω exp ( − Γ t/ p M σ exp ( − Γ t ) − i . (119)2 Γ t 〈 Ε 〉 / ω Set IASet IBSet IC
Figure 1:
A study of the variation of expectation value of energy, scaled by ω ( h E i ω ) in order tomake it dimensionless, as we vary Γ t (again a dimensionless quantity). Here we consider massM=1, µ =1, ∆ = , σ = , ω = and Γ =1 in natural units. The expectation value of energy h E i is calculated for exponentially decaying Hamiltonian parameters when h A i Set-IA f ( t ) = 1 and ω ( t ) = ω e − Γ t/ ; h B i Set-IB f ( t ) = e − Γ t and ω ( t ) = ω and h C i Set-IC f ( t ) = e − Γ t and ω ( t ) = ω e − Γ t/ . While for h A i the energy first decreases, then increases with time, for h B i theenergy remains constant as we vary time. For h C i the energy decays off with time. From Eqn.(119), we see that the expectation value of the energy becomes complex beyond acertain time limit. The condition for getting the expectation value of energy to be real is asfollows,
M σ e − Γ t > ⇒ t ≤ ln ( M σ )Γ . (120)We see from Fig.(1), that the energy initially decays but then increases with time. This is becausefor large time at which exp ( − Γ t/ ≈
0, the approximated expression of energy reads E n, − n ( t ) ≈ ( n + 1) µ ∆ + n r ∆ exp (Γ t ) M (121)which is still increasing with time. The reason for the increase of energy with time is the form ofthe coefficient b ( t ) in the Hamiltonian. Although the coefficient a ( t ) is exponentially decayingwith time, the coefficient b ( t ) exponentially increases with time in order to satisfy EP equation.However, since there is an upper limit of time within which the energy remains real, so the energyremains finite within the allowed time interval. h B i Solution Set-Ib
Here we set f ( t ) = e − Γ t and ω ( t ) = ω . With this the energy expression for the ground state3takes the form, h E n, − n ( t ) i = ( n + 1) µ ∆ + n "r ∆ − M ω M + ω √ M σ − . (122)We note from Fig.(1), that the expectation value of the energy remarkably remains constant aswe vary time, as is observed from Eqn.(122). This must be because the effect of the exponen-tially decaying Hamiltonian coefficient a ( t ) and damping term f ( t ) gets balanced out by theexponentially increasing Hamiltonian coefficient b ( t ). h C i Solution Set-Ic
Here we set f ( t ) = e − Γ t and ω ( t ) = ω e − Γ t/ . With this the expectation value of the energyexpression takes the form, h E n, − n ( t ) i = ( n + 1) µ ∆ + n "r ∆ − M ω exp [ − Γ t ] M + ω exp ( − Γ t/ √ M σ − . (123)The above expression gives a very nice decaying expression for the expectation value of energywith respect to time, and finally approaching a constant value in the limit t → ∞ . This behaviouris also exhibited in the nature of the plot of variation of the expectation value of energy with timeseen in Fig.(1). In this case the expectation value of energy for k = 2 reads E n, − n ( t ) = ( n + 1)2(Γ t + χ ) (cid:20) (cid:18) σµ + ∆ µ (cid:19) + µ Γ σ (cid:21) + n " ω Γ t + χ s σ M (Γ t + χ ) − s ∆ M − ω (Γ t + χ ) . (124)Note that although it has a nice decaying property like the damping case on commutative plane,there is an upper bound of time above which the energy ceases to be real. The upper bound ontime reads, 4 σ M ≥ (Γ t + χ ) ⇒ t ≤
1Γ (2 √ M σ − χ ) . (125)From Fig.(2), we see indeed the expectation value of energy h E i decays with time following powerlaw as expected for the rationally decaying solutions.4 Γ t 〈 Ε 〉 / ω Figure 2:
A study of the variation of expectation value of energy, scaled by ω ( h E i ω ) in order tomake it dimensionless, as we vary Γ t (again a dimensionless quantity). Here we consider massM=1, µ =1, ∆ = , σ = , ω = , χ = 1 and Γ =1 in natural units. The expectation value ofenergy h E i is calculated for rationally decaying Hamiltonian parameters. We consider f ( t ) = 1 and ω ( t ) = ω (Γ t + χ ) . For the elementary solution set, the expectation value of the energy reads, h E n, − n ( t ) i = 12 ( n + 1) (cid:20)(cid:18) ∆ µ + σµ (cid:19) t + χ ) + µ Γ σ (cid:21) + n " ω √ M σ − t + χ ) + 1(Γ t + χ ) s ∆ M (Γ t + χ ) − ω . (126)Further, the constraint relation ∆ µ = ξ σ results in the following form for the expectation valueof energy (setting ξ = 1), h E n, − n ( t ) i = 12 ( n + 1) (cid:20) σµ (Γ t + χ ) + µ Γ σ (cid:21) + n " ω √ M σ − t + χ ) + 1(Γ t + χ ) s ∆ M (Γ t + χ ) − ω . (127)This expression also provides an upper bound of the time limit above which the expectation valueof energy would become complex. This upper bound reads,∆ M (Γ t + χ ) ≥ ω ⇒ t ≤ " ω r ∆ M − χ . (128)In Fig.(3), we observe that the expectation value of energy again undergoes a power law decaywith time for the elementary solution.5 Γ t 〈 Ε 〉 / ω Figure 3:
A study of the variation of expectation value of energy, scaled by ω ( h E i ω ) in order tomake it dimensionless, as we vary Γ t (again a dimensionless quantity). Here we consider massM=1, µ =1, ∆ = , σ = , ω = , χ = 1 and Γ =1 in natural units. The expectation value ofenergy h E i is calculated for elementarily decaying Hamiltonian parameters. We consider f ( t ) = 1 and ω ( t ) = ω (Γ t + χ ) . We now summarize our results. In this paper we have considered a two-dimensional dampedharmonic oscillator in noncommutative space with time dependent noncommutative parameters.We map this system in terms of commutative variables by using a shift of variables connecting thenoncommutative and commutative space, known in the literature as Bopp-shift. We have thenobtained the exact solution of this time dependent system by using the well known Lewis invariantwhich in turn leads to a non-linear differential equation known as the Ermakov-Pinney equation.We first obtain the Lewis invariant in Cartesian coordinates. We then make a transformation topolar coordinates and write down our results in these coordinates. Doing so, we use the operatorapproach to obtain the eigenstates of the invariant. With this background in place, we makevarious choices of the parameters in the problem which in turn leads to solutions for the timedependent noncommutative parameters. We have considered three different sets of choices forwhich solutions have been obtained, namely, exponentially decaying solutions, rationally decayingsolutions and elementary solutions. Interestingly, the solutions obtained make it possible tointegrate the phase factor exactly thereby giving an exact solution for the eigenstates of theHamiltonian. We have then computed the matrix elements of operators raised to a finite integerpower in both the eigenstates of the Hamiltonian as well as the Lewis invariant. From these results,we are able to compute the expectation value of the Hamiltonian. Expectedly, the expectation6value of the energy varies with time. For the exponentially decaying solutions, we get threekinds of behaviour corresponding to the choices of the damping factor and the frequency ofthe oscillator. For the case where the damping factor is set to unity and the frequency of theoscillator decays with time, the expectation value of the energy first decreases with time andthen increases. The reason for this behaviour is due to the particular form of the solutions of theErmakov-Pinney equation which fixes the forms of the noncommutative parameters. It is thesetime dependent forms of the noncommutative parameters that results in the above mentionedbehaviour of the expectation value of the energy with time. In this case, we also observe thatthere is an upper bound of time above which the energy expectation value ceases to be real. Forthe case where the damping factor has a decaying part and the frequency of the oscillator is aconstant, we observe that the expectation value of the energy remarkably remains constant withtime. This must be the case because the effect of the exponentially decaying coefficient in theHamiltonian and the damping term gets balanced out by the exponentially increasing coefficientin the Hamiltonian. For the case where both the damping term as well as the frequency of theoscillator decays with time, we find an exponentially decaying behaviour of the expectation valueof the energy. For the rationally decaying and the elementary solution, we observe a power lawdecay of the energy expectation value with time together with an upper bound of time above whichthe energy expectation value ceases to be real. Investigating these cases of damped oscillators,we conclude that the behaviour corresponding to the exponentially decaying solution, where boththe frequency and damping term are decaying exponentially with time, is similar to a dampedoscillator in commutative space.
Acknowledgement
MD would like to thank Ms. Riddhi Chatterjee and Ms.Rituparna Mandal for their helpfulassistance to operate the software Mathematica.