Generalized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space
aa r X i v : . [ phy s i c s . g e n - ph ] J un Generalized Lewis-Riesenfeld invariance fordynamical effective mass in jammed granullar mediaunder a potential well in non-commutative space
Kalpana Biswas
Department of Physics, University of Kalyani, West Bengal, India-741235Department of Physics, Sree Chaitanya College, Habra, North 24 Parganas, WestBengal-743268E-mail: [email protected]
Jyoti Prasad Saha
Department of Physics, University of Kalyani, West Bengal, India-741235E-mail: [email protected]
Pinaki Patra
Department of Physics, Brahmananda Keshab Chandra College, Kolkata,India-700108E-mail: [email protected]
Abstract.
Consideration of the asteroid belt (Kuiper belt) as a jammed-granularmedia establishes a bridge between condensed matter physics and astrophysics. Itopens up an experimental possibility to determine the deformation parameters fornoncommutative space-time. Dynamics of the Kuiper belt can be simplified asdynamics of a dynamical effective mass for a jammed granular media under agravitational well. Alongside, if one considers the space-time to be noncommutative,then an experimental model for the determination of the deformation parametersfor noncommutative space-time can be done. The construction of eigenfunctionsand invariance for this model is in general a tricky problem. We have utilized theLewis-Riesenfeld invariant method to determine the invariance for this time-dependentquantum system. In this article, we have shown that a class of generalized time-dependent Lewis-Riesenfeld invariant operators exist for the system with dynamicaleffective mass in jammed granular media under a potential well in noncommutativespace. To keep the discussion fairly general, we have considered both position-position and momentum-momentum noncommutativity. Since, up to a time-dependentphase-factor, the eigenfunctions of the invariant operator will satisfy the time-dependent Schr¨odinger equation for the time-dependent Hamiltonian of the system,the construction of the invariant operator fairly solve the problem mathematically, theresults of which can be utilized to demonstrate an experiment.
Keywords: Lewis-Riesenfeld invariant; jammed-granular media; Kuiper belt;noncommutative geometry eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space PACS numbers: eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space
1. Introduction
There is a consensus that the fundamental concept of space-time is mostly com-patible with quantum theory in noncommutative space. It has become a Gospelthat the physics in Planck-scale will exhibit the noncommutative structure of space[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, lack of any direct experi-mental evidence is the major criticism for the unified quantum theory of gravity. Themain difficulty of experimenting with the validity of theory regarding noncommutativespace lies mainly in the limitation of the present-day experimental capacity of attain-able energy. Because, most theories of quantum gravity appear to predict departuresfrom classical relativity only at energy scales on the order of 10 GeV (By way ofcomparison, the LHC was designed to run at a maximum collision energy of 1 . × GeV [17]). Several proposals are there to deal with the scope to overcome the issueof energy limitation and perform passive experiments with the help of interferometry[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25]. To best of knowledgeof present authors, although none of them had yet been performed in an actual experi-ment.In this article, we wish to propose a new scheme of the experiment to determine the NC-parameters, which has never been reported so far in the literature. We are advocatingthat a successful experiment within the purview of present-day engineering advancementrequires some bridging between condensed matter experiments and cosmology. To doso, we have utilized the fact that the asteroid belt (Kuiper belt) can be considered asjammed-granular media [26, 27, 28, 29, 30]. Since the inner edge of Kuiper-belt (KB)begins at about 30 AU from the Sun and the outer edge of the main part continuesoutward to nearly 1000 AU from the Sun, the overall toroidal shape of the Kuiper-beltcan be approximated as a cylinder with negligible height (a disc). Despite the distortedorbit, almost all the studied elements of KB had shown to have a periodic rotationaround the Sun, except for those who ended their life by crash landing as a meteoroidon the planets including the Earth. Therefore, we can consider that KB has an overallaverage velocity in which it rotates around the Sun. That means, the entire systemcan be thought of as a rotating cylinder containing loosely packed jammed granularmedia (JGM). The overall system experiences a huge gravitational pull towards thenearer gas-giant the Neptune or towards the hypothetical planet in the outer regionof the Solar System, named Planet Nine. Our proposal is to utilize this structure forthe experimental purpose, even for the measurement of noncommutative space param-eters. In particular, the present article deals with the proposition of utilizing the KBas a rotating JGM in noncommutative space (NC) and study the deviation from thecommutative counterpart. This will enable us to estimate the values of NC-parameters.Our toy model under consideration consists of a particle with dynamical effective masssubject to NC gravitational quantum well [31, 32, 33, 34, 35, 36, 37]. For example, thequantum well represents the gravitational pull of the Neptune or the Planet Nine.Although the exact system is much complicated than that have been mentioend in eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space H ( t ). Up to a time-dependent phase factor,the eigenstates of IO ( ˆ I ( t )) are also the eigenstates of ˆ H ( t ), although they will notbe isospectral in general. Specifically, the eigenvalues of ( ˆ I ( t )) are time-independent,whereas the same may be time-dependent for ˆ H ( t ). Once one has the LR-invariantoperator, it is not difficult to obtain the exact quantum states of the system and utilizeit to obtain the measurable quantities.The organization of the article is the following. At first, a brief description of thesystem under consideration is given. Then a generalized Lewis-Riesenfeld phase-spaceinvariant method is outlined briefly. After that, we have constructed a generalizedinvariant operator for dynamical effective mass in jammed granular media. Futureaspects and experimental viability are mentioned in discussion.
2. System under consideration: Free falling under gravity innoncommutative space
Since we are dealing with the free-falling under gravity in noncommutative space (NCS),without loss of generality we can confine ourselves in two spatial dimensions in NCS,namely x ′ and y ′ . Let us choose our x ′ axis in the direction of the attraction due togravity. The dynamics for the y ′ direction in NCS remains something like free-particle.We aim to write down the coherent state structure for this system.Better or worse, we are adopting the usual technique of writing the quantum version ofa theory corresponding to a known classical dynamical system with the aid of the Bohr-correspondence principle. One may wonder why most of the computations of quantumtheory can not stand on its ground without the help of a corresponding known classicaldynamical system. One may even demand the justification of the applicability of thedirect promotion of classical theory to its quantum version for NCS. Keeping aside thesefoundational issues, let us concentrate on the aims and scope of the present article andwrite down the energy operator (Hamiltonian) in NCS as follows.ˆ H nc = ˆ p ′ x m ( t ) + ˆ p ′ y m ( t ) + m ( t ) g ( t )ˆ x ′ (1) eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space m ( t ) and g ( t ) respectively. Acceleration due to gravity g ( t ) is allowed to be time dependent, so that the model can be utilized even for thelarge height for which g no longer can be considered as a constant. Moreover, thismodel can be utilized for homogeneous time dependent electric field, which is relativelyeasy to develop for experimental purpose. ˆ p ′ x and ˆ p ′ y are the conjugate momentumoperators corresponding to ˆ x ′ and ˆ y ′ respectively. We are considering both position-position and momentum-momentum noncommutativity to keep our discussion fairlygeneral. Following commutation relations for variables (cid:8) x ′ , y ′ , p ′ x , p ′ y (cid:9) in NCS are utilizedin present article.[ x ′ , y ′ ] = iθ, (cid:2) p ′ x , p ′ y (cid:3) = iη, (2) (cid:2) x ′ i , p ′ j (cid:3) = i ~ eff δ ij . Where ~ eff = (1 + ζ ) ~ , (3)with ζ = θη ~ .δ ij are Kronecker delta with the properties δ ij = ( i = j i = j . (4)One can recover the structure of classical commutative space for quantum mechanics bysetting the parameters θ and η to zero. If one wishes to be confined in only position-position noncommutativity then η has to be set to zero.Since our usual notion of calculus are mentally and practically settled in commutativespace, it will be a wise decision if we could transform the whole problem tosome equivalent commutative space structure. This can be done by the followingtransformation of co-ordinates. x ′ y ′ p ′ x p ′ y = − θ ~ θ ~ η ~ − η ~ xyp x p y . (5) { x, y, p x , p y } are usual co-ordinates in classical commutative space in which thecommutation relations are given by[ x, y ] = [ p x , p y ] = 0 , [ x i , p j ] = i ~ δ ij . (6)For computational purpose, with the help of 5 we can now write down the equivalentquantum hamiltonian for the system under consideration (1) in terms of classical eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space H c = ˆ p x m ( t ) + ˆ p y m ( t ) + η m ( t ) ~ (cid:0) ˆ x + ˆ y (cid:1) + η m ( t ) ~ (ˆ y ˆ p x − ˆ x ˆ p y ) + m ( t ) g ( t ) (cid:18) ˆ x − θ ~ ˆ p y (cid:19) . (7)For the purpose of computational convenience, we can rewrite 7 as follows.ˆ H c = ˆ H ( t ) + ˆ V ( t ) . (8)With ˆ H ( t ) = ~ ω ( t ) h ˆ a † ˆ a + ˆ a † ˆ a + i (cid:16) ˆ a † ˆ a − ˆ a † ˆ a (cid:17) + 1 i . (9)And ˆ V ( t ) = g ( t ) √ η ω ( t ) (cid:20) ˆ a † + ˆ a − i θη ~ (cid:16) ˆ a † − ˆ a (cid:17)(cid:21) . (10)Where the annihilation operators are defined byˆ a i = r m ( t ) ω ( t )2 ~ (cid:18) ˆ x i + im ( t ) ω ( t ) ˆ p x i (cid:19) , i = 1 , . (11) i = 1 stands for x = x and i = 2 stands for x = y . The creation operators ˆ a † i are thecorresponding adjoint operators of ˆ a i . The time dependent frequency ω ( t ) is defined by ω ( t ) = η m ( t ) ~ . (12)The commutation relations among the annihilation and creation operators are given by[ˆ a i , ˆ a † j ] = δ ij = ( i = j, , (13)and [ˆ a i , ˆ a j ] = [ˆ a † i , ˆ a † j ] = 0 . (14)The frequency ( ν ) dependent effective mass, ˜ m ( ν ), of jammed granular materials whichoccupy a rigid cavity to a specific filling fraction (the remaining volume being air ofnormal room condition or controlled humidity) is given by˜ m ( ν ) = πR √ ρKν tan( qL ) . (15)Where ρ is the density of the medium, L is the length of the fluid column, R is the radiusof the cup (container), K is the bulk-modulus of the medium and q = ν p ρK is the wavevector. It is worth noting that q and as well as K can be complex also. However, forcomplex K the medium becomes lossy. We have considered loss-less medium in ourdiscussion. Hence, we have considered only real valued of K and q . The generalizationfor complex valued K is straightforward.Now we propose a model for the time -varying depth of the granular medium. Inparticular, we are proposing the model for continously filling the granular material from eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space ν . The rate of filling up the cup is small enough such that the length of the materialcolumn can be approximated as L ( t ) = L (1 + νt ) . (16)Where L is the constant length at t = 0 and ν << ν , other than linear, in our calculation. Under thisassumption, we can approximate 15 as m ( t ) = m ( q + q νt ) . (17)Where q = tan( qL ) , (18) q = qL sec ( qL ) , (19) m = 1 ν πR p ρK. (20)In the next section we have constructed an invariant operator which is utilized toconstruct the coherent state structure of the system.
3. Construction of Lewis-Riesenfeld invariant operator
We shall utilize the method described in [53]. Let us assume that we know the eigen-functions of ˆ H ( t ) in 9. Then, for any complex time-dependent parameter µ ( t ), theeigenvalue equation of the following operator is known.ˆ O ( t ) = e µ ( t ) ˆ H ( t ) . (21)However, 21 is not an invariant operator associated with ˆ H ( t ). Indeed, one can see that˙ˆ O ( t ) = ∂ ˆ O ( t ) ∂t + 1 i ~ h ˆ O ( t ) , ˆ H i = ˆΘ( t ) . (22)Where, ˆΘ( t ) is some time-dependent operator. Here dot ( . ) denotes total derivativewith respect to time. We shall use this shorthand notation throughout this articleunless otherwise specified.Now, using the time -dependent Schr¨odinger equationˆ H ( t ) | ψ ( t ) i = i ~ ∂∂t | ψ ( t ) i , (23)one can note that i ~ ∂∂t h ˆ O ( t ) | ψ ( t ) i i = (cid:16) ˆ H ( t ) + i ~ ˆΘ( t ) ˆ O − ( t ) (cid:17) h ˆ O ( t ) | ψ ( t ) i i . (24)That means, ˆ O ( t ) | ψ ( t ) i is the eigen-function of the deformed hamiltonianˆ˜ H ( t ) = ˆ H ( t ) + i ~ ˆΘ( t ) ˆ O − ( t ) . (25) eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space t ) which relates the eigen-functions of ˆ H ( t ) and ˆ˜ H ( t ). One can readily identify thatˆ H ( t ) h Λ( t ) ˆ O ( t ) | ψ ( t ) i i = i ~ ∂∂t h Λ( t ) ˆ O ( t ) | ψ ( t ) i i , (26)with ˆ H ( t ) = ˆΛ ˆ H ˆΛ − + i ~ ˆΛΘ ˆ O − + ∂ ˆΛ ∂t ! ˆΛ − . (27)Therefore, Λ( t ) ˆ O ( t ) | ψ ( t ) i is the eigen-function of ˆ H ( t ). Now we impose the restrictionon ˆΛ( t ) such thatˆ H ( t ) = ˆ H ( t ) , (28)which immediately leads to ∂∂t ˆΛ( t ) + 1 i ~ h ˆΛ( t ) , ˆ H ( t ) i = − ˆΛ( t ) ˆΘ( t ) ˆ O − ( t ) . (29)Comparing 29 with 22, one can conclude that ∂∂t h ˆΛ( t ) ˆ O ( t ) i + 1 i ~ h ˆΛ( t ) ˆ O ( t ) , ˆ H ( t ) i = 0 . (30)Thus, we have obtained a desired time-dependent Lewis-Riesenfeld invariant operator(LRIO) ˆ I ( t ) = ˆΛ( t ) ˆ O ( t ) (31)corresponding to the hamiltonian ˆ H ( t ). If we can construct the eigen-functions of ˆ I ( t ),then we shall also have the eigen-functions of ˆ H ( t ). However, they will not be iso-spectral in general. From the discussion so far, it is clear that in order to construct theLRIO, we shall have to proceed according to the following algorithm. • Split the hamiltonian in terms of known part ˆ H ( t ) and unknown part ˆ V ( t ) asfollows ˆ H ( t ) = ˆ H ( t ) + ˆ V ( t ) . (32) • Construct the time-dependent operator as specified in 21, i.e, ˆ O ( t ) = e µ ( t ) ˆ H ( t ) . • Construct ˆΘ( t ) according to 22, i.e, ˆΘ( t ) = ∂ ˆ O ( t ) ∂t + i ~ h ˆ O ( t ) , ˆ H i .Since, this step involves a computation of a commutator of two operators one ofwhich is the exponential of an operator, it will be convenient to use Kubo’s identitywhich are given as follows.[ ˆ A, e − β ˆ H ] = − Z β e − ( β − u ) ˆ H [ ˆ A, ˆ H ] e − u ˆ H du. (33)The integrand can be easily determiend by the Baker-Campbell-Hausdorff formula e ˆ A ˆ Be − ˆ A = ˆ B + [ ˆ A, ˆ B ] + 12! [ ˆ A, [ ˆ A, ˆ B ]] + ........ (34) eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space • Next is the most crucial step : choose an unitary transformation ˆΛ( t ). There isno consensus for the choice of ˆΛ( t ). However, it is convenient to choose ˆΛ( t ) = e ˆ π ,where ˆ π is some anti-hermitian operator. Sometime, an anti-hermitian operatorconstructed by a linear combination of the basic constituent operators of ˆ V ( t )serves the purpose. For example, in this present article this scheme has shown tobe useful. • Final step is to construct the invariant operator ˆ I ( t ) and determine its eigen-functions which are the eigen-functions of our desired time dependent hamiltonianˆ H ( t ).Now for our system we have ˆ H ( t ) is given by 9 and ˆ V ( t ) is given by 10. On the way todetermine ˆΘ( t ) we observe thatˆ C = [ ˆ V , ˆ H ] = g (cid:16) ˆ b − ˆ b † (cid:17) . (35)Where ˆ b = ˆ a + i ˆ a . (36) g ( t ) = 12 ~ g ( t ) √ η (cid:18) θη ~ (cid:19) . (37)It is worth noting that[ ˆ H , ˆ C ] = ( − ~ ω ) g (cid:16) ˆ b + ˆ b † (cid:17) . (38) h ˆ H , h ˆ H , ˆ C ii = ( − ~ ω ) ˆ C . (39)This leads to the use of Baker-Campbell-Hausdorff formula to obtain the following.ˆ G = e u ˆ H ˆ C e − u ˆ H = g (cid:16) e − u ˆ b − e u ˆ b † (cid:17) . (40)Where u = 2 ~ ωu. (41)Now one can apply the Kubo’s identity and write the following.[ ˆ O , ˆ H ] = g ~ ω e µ ˆ H h(cid:0) − e ~ ωµ (cid:1) ˆ b + (cid:0) − e − ~ ωµ (cid:1) ˆ b † i . (42)Another term in 22 involves partial derivative of e µ ( t ) ˆ H with respect to time. To performthis derivative let us examine the system once again according to the what follows. Onecan rewrite ˆ a i and ˆ a † i as the followings.ˆ a i = M (cid:18) ˆ x i + i M ~ ˆ p x i (cid:19) ; i = 1 , . (43)ˆ a † i = M (cid:18) ˆ x i − i M ~ ˆ p x i (cid:19) ; i = 1 , . (44)The time derivative of them are given by˙ˆ a i = ˙ MM ˆ a † i ; i = 1 , . (45)˙ˆ a † i = ˙ MM ˆ a i ; i = 1 , . (46) eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space M = r mω ~ . (47)And ˆ H = ~ ω (cid:16) ˆ N + ˆ N + i ˆ N + 1 (cid:17) . (48)Where ˆ N i = ˆ a † i ˆ a i ; i = 1 , . (49)ˆ N = ˆ a † ˆ a − ˆ a † ˆ a . (50)One can identify that the following commutation relations hold. h ˆ N i , ˙ˆ N i = 0 ; i = 1 , . (51) h ˆ N i , ˙ˆ N j i = 2 ˙ MM ˆ A ij δ ij ; i, j = 1 , . (52) h ˆ N , ˙ˆ N i = − h ˆ N , ˙ˆ N i = 2 ˙ MM (ˆ a † ˆ a † + ˆ a ˆ a ) . (53)Where ˆ A ij = ˆ a † i ˆ a † j − ˆ a i ˆ a j . (54)Using 51,right commutationndot2 and 53, one can see that[ µ ˆ H , ∂∂t ( µ ˆ H )] = ( ~ µ ) ωm ddt ( mω )( ˆ A + ˆ A ) . (55)Since mω = η ~ (eq.12) is a constant, we can conclude[ µ ˆ H , ∂∂t ( µ ˆ H )] = 0 . (56)One can ask the relevance of the steps from 43 to 56. Since, 11 along with 12 clearlyindicates that ˆ a i ’s and their corresponding adjoints are time independent for constant η and ~ and therefore 56 is trivial. However, we would like to emphasize that equationsfrom 43 to 56 are generally true even for time dependent η . In particular, although weare confining ourselves for time-independent background, the results outlined here canbe directly used for the time-dependent background. Now, 56 enables us to writeΘ( t ) = e µ ˆ H ∂∂t (cid:16) µ ˆ H (cid:17) + g i ~ ω e µ ˆ H [(1 − e ~ ωµ )ˆ b + (1 − e − ~ ωµ )ˆ b † ] . (57)Since the following relations hold[ ˆ H , ˆ b ] = − ~ ω ˆ b, (58) h ˆ H , ˆ b † i = 2 ~ ω ˆ b † , (59)we can write the followings. e µ ˆ H ˆ be − µ ˆ H = e − ~ ωµ ˆ b, (60) e µ ˆ H ˆ b † e − µ ˆ H = e ~ ωµ ˆ b † . (61) eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space O − = ∂∂t (cid:16) µ ˆ H (cid:17) + β ˆ b + β ˆ b † ] . (62)Where β = g i ~ ω ( e − ~ ωµ − , (63) β = g i ~ ω ( e ~ ωµ − . (64)Now our task is to define the unitary transformation ˆΛ( t ). Let us define it as thefollowing displacement operatorˆΛ( t ) = e ˆΠ , (65)with ˆΠ = v ˆ a − v ∗ ˆ a † + v ˆ a − v ∗ ˆ a † . (66)Here ∗ denotes the complex-conjugate. If we get some consistent time-dependentcomplex valued functions v i ( t ) , i = 1 ,
2, then we are done. To calculate ˆΛ ˆ H ˆΛ − , weobserve the followings.ˆ π = h ˆΠ , ˆ H i = ~ ω ( v ˆ a + iv ˆ a + v ∗ ˆ a † − iv ∗ ˆ a † ) . (67) h ˆΠ , ˆ H i = ˆ π + ~ ωv . (68) h ˆΠ , h ˆΠ , ˆ H ii = h ˆΠ , h ˆΠ , ˆ H ii = 2 ~ ω | v | . (69)Where v = v − iv . (70) v = v g + v ∗ g − iv g g + iv ∗ g g . (71) g = g √ η ~ ω , g = θη ~ ω . (72)It is clear from 69 that all the higher order commutators vanishes identically. HenceˆΛ ˆ H ˆΛ − = ˆ H + ˆ π + ~ ω ( v + | v | | ) . (73)We can further observe the followings. e ˆΠ ˆ be − ˆΠ = ˆ b + v ∗ . (74) e ˆΠ ˆ b † e − ˆΠ = ˆ b † + v . (75)Therefore ˆΛ ˆΘ ˆ O − ˆΛ − = 1 ω ( ˙ ωµ + ω ˙ µ )( ˆ H + ˆ π + ~ ω | v | ) + ( β ˆ b + β ˆ b † + β v ∗ + β v ) . (76)Since ˆΠ and ˙ˆΠ commutes to each other, we also have ∂ ˆΛ ∂t ˆΛ − = ˙ˆΠ . (77) eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space ddt ( ωµ ) = 0 . (78) ωv + iβ + i ˙ v = 0 . (79) ωv ∗ + iβ − i ˙ v ∗ = 0 . (80) iωv − β + i ˙ v = 0 . (81) − iωv ∗ + β − i ˙ v ∗ = 0 . (82) ω ( v + | v | ) + i ( β v ∗ + β v ) = 0 . (83)Taking complex conjugation of 79 and comparing with 80 (or equivalently 81 and 82 )indicates that β = − β ∗ . (84)This implies µ ( t ) is purely imaginary. The specific form of µ ( t ) can be obtained byutilizing 78 and 84 as follows. µ ( t ) = k iω ( t ) , k ∈ R . (85)Using 84 in 79 and 81, we get v ( t ) = − iv ( t ) , (86)˙ v − iω ( t ) v + iβ ( t ) = 0 . (87)Here we have considered the additive integration constant to be zero (since, we canalways redefine v otherwise). Using 85 in 63, we can rewrite 87 as follows ddt v r v i ! = − ω ( t )2 ω ( t ) 0 ! v r v i ! + g ~ ω sin ~ k cos ~ k ! sin ~ k . (88)where the real part and imaginary part of the complex valued function v ( t ) are v r ( t ) and v i ( t ) respectively. In particular v ( t ) = v r ( t ) + iv i ( t ) , v r , v i : R → R . (89)However, 83 provides following constraint on v r ( t ) and v i ( t ). v i ( g (1 + g ) ω + g ~ ω (cos(2 ~ k ) − v r g ~ ω sin(2 ~ k ) + 2 ω ( v r + v i ) = 0 . (90)Given the time-dependent mass m ( t ), we can solve v ( t ) with the help of equations88 and 90. Before goint to some special cases we can write the general form of theinvariant operator ˆ I ( t ) = ˆΛ( t ) ˆ O ( t ) with the help of BakerCampbellHausdorff formula,which reads If , e X e Y = e Z , (91)then, Z = X + Y + 12 [ X, Y ] + 112 ([ X, [ X, Y ]] + [ Y, [ Y, X ]]) − eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space Y, [ X, [ X, Y ]]] + 1120 ([ Y, [ X, [ Y, [ X, Y ]]]] + [ X, [ Y, [ X, [ Y, X ]]]]) +1360 ([ X, [ Y, [ Y, [ Y, X ]]]] + [ Y, [ X, [ X, [ X, Y ]]]]) − Y, [ Y, [ Y, [ Y, X ]]]] + [ X, [ X, [ X, [ X, Y ]]]]) + ...... (92)In our case, X = ˆΠ and Y = µ ( t ) ˆ H . Hence one can deduce that[ X, Y ] = − i ~ ωµ ( v ˆ b − v ∗ ˆ b † ) , (93)[ X, [ X, Y ]] = 8 ~ ωµ | v | , (94)[ Y, [ Y, X ]] = (2 ~ ωµ ) ˆΠ , (95)[ Y, [ Y, [ Y, X ]]] = (2 ~ ωµ ) i ( v ˆ b − v ∗ ˆ b † ) , (96)[ Y, [ Y, [ Y, [ Y, X ]]]] = (2 ~ ωµ ) ˆΠ , (97)[ X, [ Y, [ Y, [ Y, X ]]]] = − ~ ωµ ) | v | . (98)All other commutators vanish identically. Therefore our invariant operator is given byˆ I ( t ) = e ˆ Z , (99)where ˆ Z = (1 − ~ − ( ~ k )
45 ) ˆΠ + k ω i ˆ H + ~ k ( v ˆ b − v ∗ ˆ b † ) + 23 (1 + 2 ~ k ) ~ k i | v | . (100)Using 17 in 87, one can write v ( t ) = v (1 + ν t ) . (101)Where v = g m ( e − ~ k i −
1) tan ( qL ) iη − ~ ηνqm L sec ( qL ) , (102) ν = 4 qνL sin(2 qL ) . (103)
4. conclusion
We have shown that a class of time-dependent Lewis-Riesenfeld invariance exists forgravitational well in noncommutative space. Since, up to a time-dependent phase factor,the eigenstates of the invariant operator will satisfy the time-dependent Schr¨odingerequation, one can utilize the eigenstates of the invariant operators to construct theexpectation values of the observables. The model under consideration deals with thedynamics of a dynamical effective mass which is a common feature of jammed-granularmedia. If one considers the asteroid belt as a jammed-granular media, it will openup a possibility to utilize this astronomical system as a condensed matter system withdynamical effective mass for an experiment.We have partially solved the problem under consideration. Further experimental designsare required for the actual performance of an experiment. eneralized Lewis-Riesenfeld invariance for dynamical effective mass in jammed granullar media under a potential well in non-commutative space [1] Seiberg N and Witten E Journal of High Energy Physics JHEP09 (1999).[2] Douglas R M and Nekrasov A N
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