Regularization, dynamics and spectra of central forces with damping in 2-dimensions
aa r X i v : . [ m a t h - ph ] F e b Regularization, dynamics and spectra of central forces with dampingin 2-dimensions
E. Harikumar ∗ and Suman Kumar Panja † School of Physics, University of Hyderabad,Central University P.O, Hyderabad-500046, Telangana, IndiaPartha Guha ‡ Department of MathematicsKhalifa University of Science and Technology P.O. Box 127788, Abu Dhabi, UAEFebruary 15, 2021
Abstract
Two damped, central force systems are investigated and equivalent, undamped systems are ob-tained. The dynamics of a particle moving in r potential and subjected to a damping force is shownto be regularized a la Levi-Civita. This mapping is then elevated to the corresponding quantum me-chanical systems and using it, the energy spectrum of the former is calculated. Mapping of a particlemoving in a harmonic potential subjected to damping to an undamped system is then derived usingBohlin-Sudman transformation, for both classical and quantum regime. Dissipation being unavoidable in natural systems, is of intrinsic interest. The investigation is of suchsystems has a long history and continues to be an active area of research [1–3]. Damped harmonicoscillator is one of the systems that has been studied vigorously as a prototype of dissipative systems.Various approaches such as (i) coupling the system to a heat bath, (ii) use of Bateman-Caldirola-Kanai(BCK) [4–6] model which uses a time dependent Lagrangian/Hamiltonian have been developedand adopted for studying different aspects of dissipative systems. Each of these methods though havingunique advantages, has some unsatisfactory features [7–10].BCK model, where the equation of motion for a damped harmonic oscillator is derived as Euler-Lagrange equation from a time dependent Lagrangian, has been studied in detail and it has been shownthat the quantization of this model leads to conflict with Heisenberg’s uncertainty relation. This modelhas been shown to be plagued by difficulties in interpretation even at the classical level, as the timedependent BCK-Lagrangian/Hamiltonian leading to correct damped equation of motion is shown todescribe a variable mass system rather than a truly damped oscillator [7, 9]. It has been argued thatthe equivalence between BCK model and damped harmonic oscillator is not valid globally( i.e., not forall times) and they are equivalent only for finite time scales [10].Many attempts to overcome above mentioned shortcomings were made with various degrees ofsuccess. These include, Schr¨odinger-Langevin approach [11], modification of BCK Hamiltonian [7], afirst order formalism which is locally equivalent to BCK model [10]. Using Madelung-deBroglie-Bohm ∗ [email protected] † [email protected] ‡ [email protected] r potential subjectedto damping, and (ii) particle in 2-dimensions moving under the influence of r potential subjected todamping. We study the regularization of collision orbits of the particle moving in r which is subjectedto a damping force also. We start with a Lagrangian (which can be related to a BCK type, timedependent Lagrangian) whose equation of motion describes a particle subjected to Kepler potential,in addition to a velocity dependent damping. We first map these equations using a time dependentpoint transformation such that the transformed equations follow as Euler-Lagrange equations froma time independent Lagrangian. This allows us to construct conserved energy, which in turn allowthe implementation of the mapping of dynamics on a constant energy surface. We apply Levi-Civitamap to these equations, after re-expressing them in terms of complex co-ordinates. The equations interms of the new complex co-ordinates and time parameter is shown to describe a harmonic oscillatoraugmented by an inverted sextic potential. We then generalize this mapping to the quantum mechanicallevel by establishing a transformation between the corresponding Schr¨odinger equations. After solvingthe Schr¨odinger equation for harmonic oscillator with an inverted sextic potential, using perturbationtheory, we use the mapping implied by Levi-Civita scheme to obtain the eigen values and eigenfunctionsof Scrh¨odinger equation corresponding to the starting Hamiltonian (which is obtained from the timedependent Lagrangian whose equations of motion describes damped Kepler motion in 2-dimensions).This is the main result of this paper. We then start with the equation describing damped harmonicoscillator in 2-dimensions and by applying a time dependent co-ordinate transformation, map themto equations of a shifted harmonic oscillator. As earlier, this allows us to define conserved energywhich facilitate the mapping of dynamics from a constant energy surface. This equation, after re-expressing in terms of complex co-ordinates, are mapped to that of 2-dim H-atom by Bohlin-Sudmantransformation [23]. We then generalize this equivalence to the case of corresponding Schr¨odingerequations. This allows us to obtain the mapping between the eigen values and eigenfunctions of thesetwo models. As in the undamped case [24], here to, we see that only the parity even states of shiftedoscillator are mapped to the states of H-atom.We show that the damped system we consider here and the equivalent system (obtained by a time-dependent point transformations) are related by a non-inertial co-ordiante transformation. We alsoshow that the corresponding Schr¨odinger equations are related by the same transformations.This paper is organized as follows. In the next section, after a brief introduction to contact manifolds,2e first show the Levi-Civita mapping of the Kepler’s equations with damping in 2-dimesnions toequations of motion of a harmonic oscillator with an additional, inverted, sextic potential. This isdone by first mapping the damped Kepler equations to an equivalent set of equations and then theseequations are mapped by Levi-Civita regularization map to equations describing a harmonic oscillator,augmented with inverted, sextic potential. This is followed with a derivation of mapping between thecorresponding Schr¨odinger equations. Thereafter, we calculate the spectrum of harmonic oscillatorwith an additional, inverted, sextic potential using first order perturbation scheme. Using the mappingbetween the Schr¨odinger equations, we then obtain the eigenfunctions and eigen values of the systemwhich is equivalent to damped Kepler problem. In section.3, we first map the damped harmonic oscillatorequations in 2-dimensions. to a shifted harmonic oscillator whose frequency depends on the dampingco-efficient. Then using Bohlin-Sudman transformation, these equations are mapped to that of Keplerproblem. We then use these results to obtain the mapping between the corresponding Schr¨odingerequations and their eigenfunctions and eigen values. Our concluding remarks are given in the lastsection. In the appendix A, we show that the system described in Eqn.(2.11) and Eqn.(2.12) below, isrelated to the damped Kepler motion described in Eqn.(2.9) below, by a canonical transformation andthen we generalise this to quantum mechanical level. In appendix B, we show that these two systemsare related by a non-inertial co-ordinate transformation by mapping their Schr¨odinger equations. r potential with damping The best way to tackle dissipative systems is to use contact mechanics. Contact geometry is the odddimensional counterpart of symplectic geometry. We begin our section with a brief description of theLagrangian formulation of contact mechanics. Let us start with the definition of contact manifold.Let M be a (2 n + 1) dimensional manifold endowed with a 1-form η that satisfies η ∧ ( dη ) n = 0at every point. We define a unique vector field on a contact manifold ( M, η ), the Reeb vector field R ∈ X ( M ) satisfying i R η = 1 , i R dη = 0 . (2.1)There is a Darboux theorem for contact manifold, locally we can define Darboux coordinates ( x i , p i , z )such that these satisfy η = dz = n X i =1 p i dx i , R = ∂∂z . (2.2)Let Q = R \{ } be the configuration space of central force equation. Let L : T Q × R → R be aregular contact Lagrangian function, such that L = π ∗ L − λz ∈ C ∞ ( T Q × R ), where π : T Q × R → T Q .Let us introduce coordinates on
T Q × R denoted by ( x i , ˙ x i , z ), where ( x i ) are coordinates in Q , ( x i , ˙ x i )are the induced bundle coordinates in T Q and z is a global coordinate in R .The 1-form η L is T Q × R is given by η L = dz − ∂L∂ ˙ x i dx i , (2.3)and the Reeb vector field is given by R L = ∂∂z − W ij ∂ L∂ ˙ x i ∂z d ˙ x j , (2.4)where ( W ij ) is the inverse of the Hessian matrix ( W ij ) = ( ∂ L∂ ˙ x i ∂ ˙ x j ). Let us denote by ♭ L : T ( T Q × R ) → T ∗ ( T Q × R )the vector bundle isomorphism, defined as ♭ L ( X ) = i X ( dη L ) + ( i X η L ) η L . (2.5)3onsider contact Lagrangian vector field X L = ˙ x i ∂∂x i + F i ∂∂ ˙ x i + L ∂∂z . (2.6)A direct calculations shows that if ( x i ( t ) , ˙ x i ( t ) , z ( t )) is integral curve of the vector field X L we obtainthe generalized Euler-Lagrange equations considered by G.Herglotz in 1930 ddt ( ∂L∂ ˙ x i ) − ∂L∂x i = ∂L∂ ˙ x i ∂L∂z . (2.7)This yields Lagrangian picture of contact systems. One must note that Herglotz equation is not linearin Lagrangian. The Legendre transformation for contact Lagrangian systems can be defined via thefiber derivative F L of L : T Q × R → R , given in terms of the following coordinate expression F L : T Q × R → T ∗ Q × R ; ( x i , ˙ x i , z ) ( x i , ∂L∂ ˙ x i , z ) . Consider the time-dependent Lagrangian function L : T Q × R → R given by L = 12 m ˙ x i − C ( t ) r − λz, (2.8)where C ( t ) = ke − λt . Substituting this into Herglotz equation (2.7) we obtain equations of motion m ¨ x i + λm ˙ x i + ke − λt r x i = 0 , i = 1 , , (2.9)where ˙ x i = dx i dt and ¨ x i = d x i dt , describing motion of a particle of mass m under the influence ofa gravitational potential of another body of mass m as well as a damping force, in 2-dimensions.Here when λ −→
0, above Eqn.2.9 becomes equation of motion for well known Kepler problem in 2-dimensions . We now re-write these equation of motion in terms a new set of co-ordinates ( X , X )which are related to old co-ordinates through time dependent transformation given by X = x e λt ,X = x e λt . (2.10)In terms of these co-ordinates, Eqn.(2.9) become m ¨ X − mλ X + kX ( X + X ) = 0 , (2.11) m ¨ X − mλ X + kX ( X + X ) = 0 . (2.12)It is easy to see that these are Euler-Lagrange equations following from the Lagrangian L = m X + ˙ X ) + mλ X + X ) − mλ X ˙ X + X ˙ X ) + kr (2.13)Note that the third term can be re-written as − mλ ddt ( X + X ), a total derivative and thus willnot contribute to equations of motion. Following the standard procedure, we calculate the canonicalconjugate momenta corresponding to X and Y as P X = m ˙ X − λ mX (2.14) P X = m ˙ X − λ mX (2.15) We note that the above equation of motion can be derived as Euler-Lagrange equation from a BCK type Lagrangiangiven by L = e λt (cid:20) m ( ˙ x + ˙ x ) + ke − λt r (cid:21) where r = p x + x , m = m + m m m and k = Gm m . H = ( P X + P X )2 m + λ X P X + X P X ) − kr , (2.16)where we now use r = p X + X . This Hamiltonian does not have any explicit time dependence andis invariant under time translations and hence total energy of this system is conserved. We note thatthe angular momentum L = X P X − X P X is also conserved, as in the usual Kepler problem. Forlater purposes, we express (negative of) this Hamiltonian, which is a conserved quantity, in terms ofvelocities and co-ordinates, viz: −E = m X + ˙ X ) − λ m ( X + X ) − kr . (2.17)We re-express the Eqn.(2.11) and Eqn.(2.12) in terms of complex variable Z given by Z = X + iX as m ¨ Z + k | Z | Z = mλ Z, (2.18)where | Z | = p X + X = r . Note that the derivative with respect to the time variable t is representedby ‘overdot’, in the above equations.We now apply a re-parametrisation of time and demand ddt = cr ddτ . (2.19)Note here that the c appearing in the above equation is a proportionality constant(and not the velocityof light). Applying this re-parametrisation, we find dZdt = cr dZdτ (2.20) d Zdt = c r d Zdτ − c r drdτ dZdτ . (2.21)Using these, we re-express Eqn.(2.18) as c ( r d Zdτ − drdτ dZdτ ) + µZ = λ r Z, (2.22)where µ = k/m .Next we apply a co-ordinate transformation(Levi-Civita regularisation) and re-write above equationin terms new complex co-ordinate U = U + iU which is related to Z as Z = γU (2.23)where γ is constant having dimensions of inverse length(without lose of generality, we set this γ tobe one from now onwards). This sets r = | Z | = U ¯ U = | U | , r = ¯ ZZ . We also find dZdτ = 2 U dUdτ , d Zdτ = 2( dUdτ ) + 2 U d Udτ and drdτ = ¯ U dUdτ + d ¯ Udτ U , using which, the Eqn.(2.22) becomes2 c U ¯ U U d Udτ + (cid:18) dUdτ (cid:19) ! − c (cid:18) ¯ U dUdτ + d ¯ Udτ U (cid:19) U dUdτ + µU = λ U ( ¯ U U ) . (2.24)This equation, after straight forward algebra gives2 r d Udτ − (cid:12)(cid:12)(cid:12)(cid:12) dUdτ (cid:12)(cid:12)(cid:12)(cid:12) U − λ c r U + µc U = 0 . (2.25) Conserved angular momentum in damped Kepler problem and perturbed harmonic oscillator (seeEqn.(2.13),Eqn.(2.32)) allows us to equate these numbers and this leads to the relation in Eqn.(2.19). −E in terms of dUdτ . For this, we first re-express −E in terms of Z and dZdt as −E = m d ¯ Zdt dZdt − mλ ZZ − k | Z | (2.26)and apply re-parametrisation to replace derivative with respect to t with derivative with respect to τ and finally, re-express these terms using the derivative of U . This gives −E = m c r d ¯ Zdτ dZdτ − mλ r − kr (2.27) ≡ mc r (cid:12)(cid:12)(cid:12)(cid:12) dUdτ (cid:12)(cid:12)(cid:12)(cid:12) − mλ r − kr . (2.28)Using this, we re-write (cid:12)(cid:12) dUdτ (cid:12)(cid:12) in Eqn.(2.25) and find d Udτ + (cid:18) E mc − λ c r (cid:19) U = 0 , (2.29)where r = | U | . This equation describes a perturbed harmonic oscillator for λ <<
1. To see thisclearly, we re-write the above equation in U and U as U ′′ + 12 c (cid:18) E m − λ ( U + U ) (cid:19) U = 0 (2.30) U ′′ + 12 c (cid:18) E m − λ ( U + U ) (cid:19) U = 0 (2.31)where a ‘prime’ over U stands for derivative with respect to the new time variable, τ . It is easy to seethat these equations follow from the Lagrangian L = m U ′ + U ′ ) − E c ( U + U ) + 132 mλ c (cid:0) U + U (cid:1) . (2.32)We now see that the system described by the above Lagrangian is oscillator in 2-dimensions with invertedsextic potential(i.e., with coefficients of U and U are negative) and couplings. Note that, for small λ ,we can treat this system as uncoupled harmonic oscillators with perturbations involving couplings andinverted sextic potentail .The Hamiltonian following from the above Lagrangian is given by H = P U m + P U m + E c ( U + U ) − mλ c (cid:0) U + U (cid:1) . (2.33)Note that the λ dependent term is with negative coefficient.Thus we have shown that the equations of motion following from the Lagrangian in Eqn.(2.13) aremapped by Levi-Civita map to equations following from the Lagrangian in Eqn. (2.32). This generalizesthe equivalence of the Kepler problem to the harmonic oscillator in 2-dimensions to the equivalence ofthe Kepler problem in presence of a damping force to a perturbed harmonic oscillator. This is shown byfirst mapping the Eqn.(2.9) describing the damped Kepler equations in 2-dimensions to the equationsgiven in Eqns.(2.10) (which follow from the Lagrangian given in Eqn.(2.13)), which are then mapped tothat of harmonic oscillator in 2-dimensions with specific couplings and inverted sextic potential. Thismapping is obtained for the surface defined by the constant value of E . From Eqn.(2.26), we note thatthis conserved quantity reduces to energy of the Kepler system in the limit λ → λ →
0, we get back the well known equivalence between Kepler motion and harmonic oscillations in2-dimensions, on the constant energy surface. Inserting γ back into above equation, we find the coefficient of the last term in the Lagrangian is mλ γ c . .2 Mapping of Schr¨odinger Equation for r potential with damping In this section, we obtain the spectrum of the the Schr¨odinger equation corresponding to the systemdescribed by the Hamiltonian in Eqn.(2.16), using its equivalence to that of harmonic oscillator withinverted sextic potential and couplings. This is obtained by mapping the Schr¨odinger equation corre-sponding to the Hamiltonian in Eqn.(2.16) to the Schr¨odinger equation corresponding to the harmonicoscillator with inverted sextic potential and couplings and its solution to the Schr¨odinger equation. Forthis, we start with the Hamiltonian H = ( P X + P X )2 m + λ X P X + P X X ) + λ X P X + P X X ) − kr , (2.34)where we have symmetrised the X i P X i terms. After re-expressing above Hamiltonian asˆ H = ( P X + mλX ) m + ( P X + mλX ) m − λ m X + X ) − k p X + X (2.35)we set up the Schr¨odinger equation ˆ H Φ = E Φ . (2.36)We now apply a transformation η ˆ H η − η Φ = Eη Φ to above equation and redefine η ˆ H η − = H and η Φ = ψ . Taking η = exp (cid:18) imλ ( X + X )4¯ h (cid:19) (2.37)and noting η ( P X i + mλX i ) n η − = P nX i , we find the transformed Hamiltonian to be H = ( P X + P X )2 m − λ m X + X ) − kr . (2.38)This Hamiltonian describes a 2-dimensional H-atom with an additional inverted harmonic potential. Inthe polar coordinates, this Hamiltonain becomes H = 12 m ( P r + P θ ) − λ mr − kr (2.39)and we will now find solution to the Schr¨odinger equation, in polar coordinates, i.e., (cid:20) − ¯ h m (cid:18) ∂ ∂r + 1 r ∂∂r + 1 r ∂ ∂θ (cid:19) − λ m r − kr (cid:21) ψ ( r, θ ) = Eψ ( r, θ ) . (2.40)where we have used the realisation P r = − i ¯ h (cid:18) ∂∂r + N − r (cid:19) (2.41) P θ = − i ¯ h ∂∂θ . (2.42)Note that, in our case, N = 2. in the above.To find the solution to above equation, Eqn.(2.40), we first generalize the equivalence of abovesystem to that described by the Schr¨odinger equation corresponding to the Hamiltonian in Eqn.(2.33)and use the solution of the latter, obtained using perturbative approach. For this we re-express theHamiltonain in Eqn.(2.33) in polar coordinates, viz:˜ H = P ρ m + P φ m + m ρ − λ m c ρ (2.43) Here the Hamiltonian in Eqn.(2.34), in polar co-ordinate is ˆ H = m ( P r + P θ ) − λ ( P r r + rP r ) − kr . We note that with η = exp (cid:16) imλr h (cid:17) , we get H = η ˆ Hη − , where H is same as the one given in the Eqn.(2.39). Thus, the transformationfrom ˆ H in Eqn.(2.35) to H in Eqn.(2.38) can be implemented in polar co-ordinates, directly giving the Hamiltonian inEqn.(2.39). m Ω = E c . The corresponding Schr¨odinger equation is given by (cid:20) − ¯ h m (cid:18) ∂ ∂ρ + 1 ρ ∂∂ρ + 1 ρ ∂ ∂φ (cid:19) + m ρ − λ m c ρ (cid:21) ψ ( ρ, φ ) = ˜ Eψ ( ρ, φ ) . (2.44)To see the equivalence of the Schr¨odinger equations given in Eqn.(2.44) and Eqn.(2.40), we haveto apply the coordinate transformations corresponding to the Levi-Civita map in Eqn.(2.23) used inrelating these two systems, classically . That is, we apply the relations r = ρ , θ = 2 φ, and choose c = 14 (2.45)and by a change of variables, map above Hamiltonian operators(and thus the Schr¨odinger equations),where we make further identifications E = − m Ω , and 4 k = ˜ E. (2.46)We now solve the Schr¨odinger equation in Eqn.(2.44), by treating the λ dependent term as aperturbation. After writing ˜ E = ˜ E + λ g as the unperturbed energy, we readily find the energy eigenvalue and eigenfunction as [25]˜ E n ρ ,l = ¯ h Ω (2 n ρ + | l | +1) , (2.47) ψ n ρ ,l ( ρ, φ ) = 1 √ π e − αρ ρ | l | F ( − n ρ ; | l | +1; αρ ) e ilφ , (2.48)respectively. In the above, we have defined α = m Ω / ¯ h . Using the perturbative scheme, we find thefirst order correction to the eigen value to be˜ E ′ n ρ ,l = − λ m c α (cid:18) n ρ + | l | n ρ (cid:19) (cid:18) n ρ − n ρ (cid:19) Γ( | l | +4) F ( − n ρ , | l | +4 , | l | +1 , − n ρ ; 1) A n ρ ,l , (2.49)where A n ρ ,l = (cid:16) n ρ ! | l | !( n ρ + | l | )! (cid:17) α | l | and the first order correction to the eigenfunction is ψ ′ n ρ ,l ( ρ, φ ) = X n ρ = n ′ ρ AB ( ˜ E n ρ − ˜ E n ′ ρ ) ψ n ′ ρ ,l ′ ( ρ, φ ) (2.50)where A = − λ m c α | l | n ρ ! | l | !( n ρ + | l | )! n ′ ρ ! | l | !( n ′ ρ + | l | )! (2.51) B = (cid:18) n ′ ρ + | l | n ρ (cid:19) (cid:18) n ρ − n ρ (cid:19) Γ( | l | +4) F ( − n ′ ρ , | l | +4 , | l | +1 , − n ρ ; 1) . (2.52)In calculating these corrections we have used certain identities given in [26]. Under the map, Z = U ,(or equivalently, r = ρ , θ = 2 φ ), conserved angular momentum J H of the modified H-atom and thatof perturbed oscillator J O are related as 2 J H = J O and this, apart from fixing C = (see Eqn.(2.45)),relates the angular momentum quantum numbers of modified H-atom, ˜ l and that of the perturbedoscillator l as ˜ l = l . (2.53) One also need to implement the reparametrisation of time variables, but this will not affect the relations − i ¯ h ∂ψ ( r,θ ) ∂t = Eψ ( r, θ ) and − i ¯ h ∂ψ ( ρ,φ ) ∂τ = ˜ Eψ ( ρ, φ ). Under the map, Z = U the conserved angular momentum J H of the modified H-atom and that of perturbed oscillator J O are related as 2 J H = J O and this fixes C = . n r = n ρ i.e., when n ρ ∈ Z + (i.e., even integer), n r ∈ Z and when n ρ ∈ Z − (i.e., oddinteger), n r ∈ Z ⇒ n r ∼ half integer, which is not allowed. Thus, we see that the eigenfunctions andeigen values corresponding to even values of n ρ are mapped to the eigenfunctions and eigen values ofthe damped Kepler problem. We will comment on the possible mapping of eigenfunctions and eigenvalues corresponding to odd integer values of n ρ in the concluding remarks.Now, using r = ρ , θ = 2 φ , l = 2˜ l and α = 2 β in Eqns.(2.47,2.48,2.49,2.50), we find the eigen valueand eigenfunction corresponding to the Schr¨odinger equation in Eqn.(2.40) E = E n r , ˜ l + E ′ n r , ˜ l (2.54) ψ n r , ˜ l = ψ n r , ˜ l + ψ ′ n r , ˜ l . (2.55)where E n r , ˜ l = − mk h ( n r + | ˜ l | + ) (2.56) E ′ n r , ˜ l = − λ m β ) B n r , ˜ l (cid:18) n r + 2 | ˜ l | n r (cid:19) (cid:18) n r − n r (cid:19) Γ(2 | ˜ l | +4) F (2.57)where F = F ( − n r , | ˜ l | +4 ,
4; 2 | ˜ l | +1 , − n r ; 1) (2.58)and B n r | ˜ l | = (2 β ) − | ˜ l | n r ! (2 | ˜ l | )!( n r + 2 | ˜ l | )! ! (2.59)and similarly, we find ψ n r , ˜ l = 1 √ π e i ˜ lθ r | ˜ l | e − βr F ( − n r ; 2 | ˜ l | +1; 2 βr ) (2.60) ψ ′ n r , ˜ l = X n r = n ′ r ˜ B (cid:18) n ′ r + 2 | ˜ l | n r (cid:19) (cid:18) n r − n r (cid:19) Γ(2 | ˜ l | +4) ˜ F ψ n ′ r , ˜ l E n r , ˜ l − E n ′ r , ˜ l (2.61)where ˜ B = − λ m β ) | ˜ l | n r ! (2 | ˜ l | )!( n r + 2 | ˜ l | )! n ′ r ! (2 | ˜ l | )!( n ′ r + 2 | ˜ l | )! (2.62)˜ F = F ( − n ′ r , | ˜ l | +4 ,
4; 2 | ˜ l | +1 , − n r ; 1) (2.63)Note here that the Hamiltonian in Equ.2.34 and one in Equ.2.38 are gauge equivalent. Thus, measuredquantities such as energy eigen values will be same for both Hamiltonian but energy eigen function willmodified. From the relation η Φ = ψ we get energy eigenfunction of the Hamiltonian in Eqn.(2.34) asΦ = exp (cid:18) − imλr h (cid:19) ( ψ n r , ˜ l + ψ ′ n r , ˜ l ) , (2.64)where ψ n r , ˜ l and ψ ′ n r , ˜ l are given in Eqn.2.60 and Eqn.2.61, respectively. r potential with damping In this section, we study the application of Bohlin-Sudman mapping to the equations of motion de-scribing a damped harmonic oscillator. After mapping these equations to that of a shifted harmonicoscillator by a time dependent point transformation, we re-express the equations in terms of complexco-ordinates. Then by applying a re-parametrisation of time followed by a co-ordinate change, we mapthese equations describing a dynamics on a constant energy surface to that of a particle moving in r potential. We then extend this mapping to corresponding Schr¨odinger equations.9 .1 Mapping of Harmonic oscillator with damping: Bohlin-Sudman Map We start from the equations of motions q ′′ i + λq ′ i + Ω q i , i = 1 , . Here q ′ i = dq i dτ and q ′′ i = d q i dτ . We now apply thetime-dependent co-ordinate transformation x i = q i e λτ , i = 1 , x ′′ + ˜Ω x = 0 , (3.3) x ′′ + ˜Ω x = 0 , (3.4)where ˜Ω = Ω − λ . These are Euler-Lagrange equations following from the Lagrangian L = m (cid:0) x ′ + x ′ (cid:1) − m ( x + x ) − mλ x x ′ + x x ′ ) . (3.5)We re-express these equations using complex co-ordinate ω = x + ix (3.6)as ω ′′ + ˜Ω ω = 0 . (3.7)We now apply Bohlin-Sudman transformation ω → Z = ω . (3.8)and also implement re-parametrisation of time using¯ Z dZdt = ¯ ω dωdτ (3.9)Using Eqn.(3.8) in Eqn.(3.9), we get ddt = 14¯ ωω ddτ (3.10)and using this we find ˙ Z = dZdt = 12¯ ω dωdτ (3.11)¨ Z = d Zdt = 18¯ ωω (cid:20) ω d ωdτ − ω (cid:18) d ¯ ωdτ (cid:19) (cid:18) dωdτ (cid:19)(cid:21) (3.12)From Eqn.(3.7), we have ω ′′ = − ˜Ω ω and using this, we re-write the second equation in the above as¨ Z = − ωω (cid:20) ω ˜Ω ω + 1¯ ω (cid:18) d ¯ ωdτ (cid:19) (cid:18) dωdτ (cid:19)(cid:21) (3.13)= − Z | Z | (cid:20)(cid:18) d ¯ ωdτ (cid:19) (cid:18) dωdτ (cid:19) + ˜Ω ¯ ωω (cid:21) (3.14) These equations follow from the BCK Lagrangian L = m (cid:0) ˙ q i − Ω q i (cid:1) e λτ , i = 1 , Bohlin-Sudman transformation has been applied to derive the mapping between 2-dim harmonic oscillator to 2-dimKepler problem. Here, angular momentum is conserved in both systems and demanding these constants of motion areproportional to each other, results in the relation between time parameters of these two systems. In the present case too,angular momentum is conserved for damped harmonic oscillator as well as (damped) Kepler problem in 2-dim.
10e now derive the conserved “energy” associated with the Lagrangian in Eqn.(3.5) and re-expressthe terms in the [ ] appearing in the above equation. To this end, we first obtain the conjugate momentacorresponding to x i as p i = mx ′ i − mλ x i , i = 1 , H = m (cid:0) x ′ + x ′ (cid:1) + m ˜Ω (cid:0) x + x (cid:1) (3.16)= m h ¯ ω ′ ω ′ + ˜Ω ¯ ωω i ≡ E. (3.17)Since E above is a constant, we use it to re-express Eqn.(3.14) as¨ Z = − E m Z | Z | . (3.18)We note that with the identification of conserved E with 4 k = m ˜Ω , the strength of Kepler potential, E k, (3.19)the Eqn.(3.18) is the Kepler’s equation in 2-dim, written in the complex co-ordinate Z = X + iX ,Thus we have mapped the equation of ‘damped’ harmonic oscillator in 2-dim, to the equation ofmotion corresponding to the (undamped) Kepler problem in 2-dim.We now start with the expression for energy of 2-dim Kepler system, described in terms of ¯ Z and Z and re-express it in terms of ¯ ω and ω , i.e., E Kepler = m d ¯ Zdt dZdt − k | Z | (3.20)= m (cid:20) ωω d ¯ ωdτ dωdτ (cid:21) − k ¯ ωω (3.21)Using above and Eqn.(3.19), we get − E Kepler = m (cid:18) Ω − λ (cid:19) . (3.22)Thus we find1. The equations of motion of 2-dim damped Harmonic oscillator in Eqn.(3.1) are first mapped toequations of a shifted harmonic oscillator given in Eqn.(3.4) which are then mapped to that of(undamped) Kepler problem in 2-dim.2. The strength of the Kepler potential is related to the conserved energy of the shifted harmonicoscillator. In the above, we have first mapped the equations of motion of a damped harmonic oscillator to that ofKepler problem via the equations of motion of a shifted harmonic oscillator. Now, we will derive therelation between the Schr¨odinger equations corresponding to these systems. Mapping of Schr¨odingerequations harmonic oscillator and H-atom has been derived in [24] and we adapt this results for theshifted oscillator we have studied in the previous section.The Hamiltonian corresponding to the system described by the Lagrangian in Eqn.(3.5) is given by H = 12 m ( p + p ) + m Ω x + x ) + λ x p + p x + x p + p x ) . (3.23)11s earlier, we transform this Hamiltonian to an equivalent one, by the transformation ˆ H = ηHη − where η = e i h mλ ( x + x ) and obtainˆ H = 12 m ( p + p ) + m ˜Ω2 ( x + x ) , (3.24)which is the Hamiltonian for harmonic oscillator with frequency ˜Ω = (Ω − λ ). The eigenfunctionsatisfying the corresponding Schr¨odinger equation, in polar co-ordinates (cid:20) − ¯ h m (cid:18) ∂ ∂ρ + 1 ρ ∂∂ρ + 1 ρ ∂ ∂φ (cid:19) + m ρ (cid:21) ψ = Eψ (3.25)is ψ = 1 √ π e ilφ ψ lE ( ρ ) (3.26)where ψ lE ( ρ ) = e − αρ ρ | l | F ( − n ρ , | l | +1 , αρ ) . (3.27)Here F ( a, b, c ) is Confluent Hypergeometric Function and we have used α = m ˜Ω / ¯ h . The eigen value isgiven by E = ¯ h ˜Ω(2 n ρ + | l | +1) (3.28)Next, we apply the Bohlin-Sudman map (see Eqn.(3.8)) to above equation, i,e., in polar co-ordinate,we apply ρ → r ; φ → θ α → β (3.29)and after simple algebra, get the eigen value equation to be " − ¯ h m (cid:18) ∂ ∂r + 1 r ∂∂r + 1 r ∂ ∂θ (cid:19) + m ˜Ω ψ ( r, θ ) = E r ψ ( r, θ ) (3.30)Substituting ψ ( r, θ ) = √ π e i ˜ lθ ψ ( r ), we re-write the above equation as " − ¯ h m (cid:18) ∂ ∂r + 1 r ∂∂r (cid:19) − ˜ l ¯ h mr − E r ψ ( r ) = − m ˜Ω ψ ( r ) (3.31)Now using Eqn.(3.22) and renaming E k as E H with the identification E/ k , we re-express aboveequation as " − ¯ h m (cid:18) ∂ ∂r + 1 r ∂∂r (cid:19) − ˜ l ¯ h mr − kr ψ ( r ) = E H ψ ( r ) . (3.32)We now easily identify this as the Schr¨odinger equation describing H-atom. Since we have 2 φ = θ (seeEqn.(3.29)), we find that the orbital quantum numbers are related as l = 2˜ l . Using this and Eqn.(3.29)in Eqn.(3.27) we find the eigenfunction ψ ( r ) of Eqn.(3.32) and thus obtain ψ ( r, θ ) = 1 √ π e i ˜ lθ e − βr r | ˜ l | F ( − n r , | ˜ l | +1 , βr ) (3.33)where β = m ˜Ω / h . From Eqn.(3.28), we also find the eigen value to be E H = − mk h n r + | ˜ l | + ) (3.34)Since ˜ l = l , we note that of the eigenfunctions in Eqn.(3.27) only those corresponding to even valuesof n ρ are mapped to the eigenfunctions ψ ( r, θ ) given in Eqn.(3.33), as in the undamped case [24].12 Conclusion
We have derived the mapping between equations of motion of a particle subjected to central potentialswith damping in 2-dimensions. These mappings are obtained for both classical and quantum mechanicalsystems. The two systems we have investigated here are (i) r potential with damping, and (ii) r potential with damping.We have studied in the section 2, damped Kepler motion in 2-dimensions and first mapped it to aparticle moving under the combined influence of Kepler and harmonic potentials. This is achieved byusing time-dependent, point transformation. Here, the strength of the harmonic potential is related tothe damping constant and thus the system retains memory of the damping force. These equations arethen mapped to that of harmonic oscillator with an additional, inverted, sextic potential, using Levi-Civita regularization. We then showed the equivalence between the Schr¨odinger equation correspondingto r potential augmented with the harmonic potential(see Eqn.(2.35)) to the Schr¨odinger equationcorresponding to a particle moving under the influence of harmonic potential with an additional inverted,sextic potential. Using perturbative solution of the latter, we obtain the eigenfunctions and eigen valuesof the former where the additional harmonic potential’s strength is proportional to the damping co-efficient of the damped motion under the potential r . Our results reduce to the standard results of themapping between H-atom and harmonic oscillator in the vanishing limit of damping co-efficient, i.e., λ → ω = x + ix = ρe iφ intro-duced in Eqn.(3.6) is (4 ∂ ∂ ω ∂ ¯ ω + m ˜Ω2 ¯ ωω ) ψ ( ω, ¯ ω ) = Eψ ( ω, ¯ ω ). Under the parity transformation, ω → ± ω and we have z → z . Thus we see that under ω → ± ω , arg ( z ) changes by 2 π . Thus parity eigenstates ψ σ ( ω, ¯ ω ) = ψ σ ( ω, ¯ ω ) e iσarg ( ω ) , where σ = 0 for even and σ = for odd states, respectively, are mappedto ψ σ (¯ z, z ) e iσarg ( z ) . The parity even states, having zero phase factor will get mapped to the eigenfunc-tions of H-atom. The non-zero phase factor for the parity odd states will lead to the mapping of thesestates to that of charged magnetic vortex system as in the undamped case, [24]. This discussion isrelevant for the mapping of eigenfunctions and eigen values obtained in subsection 2.2. Here too, wehave seen that the eigenfunctions corresponding to the even eigen values of harmonic oscillator withinverted sextic potential are only mapped to that of system with combination of r potential and har-monic potential. It is thus natural to expect that the eigenfunctions corresponding to odd integer eigenvalues will be related to a damped H-atom in presence of a magnetic vortex.In the appendices we show the connection between the damped systems described by Eqn.(2.9) tothe system described by Eqns.(2.11,2.12) using canonical transformation as well as using a non-inertialtransformation of co-ordinates.In [27–29], we have studied the regularization of central potentials in non-commutative space-time.Thus it is of interest to see whether the results obtained here can be generalized to non-commutativespace-time. Acknowledgments
EH and SKP thank SERB, Govt. of India, for support through EMR/2015/000622. SKP also thankUGC, India for support through JRF scheme(id.191620059604).13 ppendix : A
In this appendix we show that the models described by damped equation given in Eqn.(2.9) and equa-tions given in Eqns.(2.11, 2.12) are related by a canonical transformation. We also show how thecorresponding Schr¨odinger equations are related.The Hamiltonian given in Eqn.2.16) H = ( P X + P X )2 m + λ X P X + X P X ) − k p X + X is mapped to the Hamiltonian H = e − λt ( p x + p x )2 m − ke − λt p x + x , (A.1)by the canonical transformation generated by F ( x i , P X i ) = x i e λt P X i , as H ( X i , P X i ) = H ( x i , p x i ) + ∂F ∂t .Here we have used the relations obtained from this generating function, p x i = e λt P X i , X i = x i e λt . (A.2)The corresponding Hamiltonian operators are related by [30] H ( ˆ X i , ˆ P X i ) = H ( ˆ x i , ˆ p x i ) + ∂F ( ˆ x i , ˆ P X i ) ∂t , (A.3)where we useˆindicate the operator nature explicitly. This can be expressed using an unitary operatoras (see [30]) H ( ˆ X i , ˆ P X i ) = H ( ˆ x i , ˆ p x i ) + i ¯ h ˆ U ∂ ˆ U † ∂t , (A.4)where ˆ U = e i h (cid:20) ( ˆ x i ˆ P Xi + ˆ x i ˆ P Xi ) e λt (cid:21) . Appendix : B
In this appendix, we show that the non-inertial transformaton that relates the models described byEqn.(2.9) and Eqns.(2.11,2.12) also map the corresponding Schr¨odinger equations.The time dependent Schr¨odinger equation corresponding to the Hamiltonian in Eqn.(2.16) is " − ¯ h m ( ∂ ∂X i ) − i ¯ h λ X i ∂∂X i + ( ∂∂X i ) X i ) − k p X + X ψ ( X i , t ) = i ¯ h ∂ψ ( X i , t ) ∂t (B.1)Now we will apply non-inertial transfromation X i → x i = X i e − λt , t = ˜ t (B.2)the above Schr¨odinger equation, mapping it to " − ¯ h m ( e − λ ˜ t ∂ ∂x i ) − ke − λ ˜ t p x + x ˜ ψ ( x i , ˜ t ) = i ¯ h ∂ ˜ ψ ( x i , ˜ t ) ∂ ˜ t (B.3)where ˜ ψ ( x i , ˜ t ) = ψ ( x i e − λ ˜ t , ˜ t ). In obtaining above mapping, we have used [31–33] ∂∂t = ∂∂ ˜ t − λ ∂∂x i ; ∂∂X i = e − λ ˜ t ∂∂x i . (B.4)Note that this is the Schr¨odinger corresponding to the Hamiltonian given in Eqn.(A.1), showing themapping between the quantum systems under the above non-inertial transformation. which is derived from the BCK type Lagrangian given by L = e λt (cid:20) m ( ˙ x + ˙ x ) + ke − λt r (cid:21) where r = p x + x ,m = m + m m m and k = Gm m eferences [1] H. Dekker, Phys. Rep. (1981) 1.[2] R. W. Hasse, J. Math. Phys. (1975) 2005.[3] Quantum Dissipative Systems, U. Weiss, 4th Edition( World Scientific, 2008).[4] H. Bateman, Phys. Rev. (1931) 815.[5] P. Caldirola, Nuovo Cimento (1941) 393.[6] E. Kanai, Prog. Theor. Phys. (1950) 440.[7] D. M. Greenberger, J. Math. Phys. (1979) 626.[10] M. C. Baldiotti, R. Fresneda and D. M. Gitman, Phys. Lett. A375 (2011)1630.[11] M. D. Kostin, J. Chem. Phys. (1972)3589.[12] S. Garashchuk, V. Dixit, B. Gu and J. Mazzuca, J. Chem. Phys. (2013) 054107.[13] L. Herrera, L. Nunez, A. Patino and H. Rago, Am. J. Phys. (1986) 273.[14] M. Razavy, Cand. J. Phys. (1978) 311.[15] M. Razavy, Z. Phys. B26 (1977) 201.[16] G. Herglotz, Lectures at the University of G¨ottingen, G¨ottingen (1930); R. B. Guenther, C. M.Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Transformations and HamiltonianSystems, Lecture Notes in Non-linear Analysis, Vol.1, J. Schauder Center for non-linear Studies,Nicholas Copernicus university, Torun(1996).[17] A.Bravetti, Entropy, (12) (2017) 535.[18] J.F. Carinena and P. Guha, Int. J. Geo. Meth. Mod. Phys. (2019) 1940001.[19] M. de Le´on and M. L. Valcazar, J. Math. Phys. (2019) 102902.[20] M. Serhan,. M. Abusini , A. Al-Jamel, H. El-Nasser and E. M. Rabei, J. Math, Phys. (2018)082105;ibid J. Math. Phys. (2019) 094102; F. M. Fernandez, J. Math. Phys. (2019) 094101.[21] N. A. Lemos, Am. J. phys. (1979) 857.[22] T. Levi-Civita, Acta Math. (1920) 99; Linear and Regular Celestial Machanics, E. L. Stiefeland G. Scheifele, Springer- Verlag, 1971.[23] K. Bohlin, Bull. Astro. (1911) 144.[24] A. Nersessian, V. Ter-Antonyan and M. Tsulaia, Mod.Phys.Lett. A11 (1996) 1605.[25] X. Fu, IOP Conf. Ser.: Earth Environ. Sci (2019) 032042.[26] Table of Integrals, series and products, I. S. Gradshteyn and I. M. Ryzhik, 7th Edition, ElsevierAcademic Press, page 1001; L. Poh-aun, S-h Ong and H. M. Srivastava, Int. J. Computer Math. (2001) 303.[27] P. Guha, E. Harikumar, N. S. Zuhair, Int. J. Mod. Phys. A29 (2014) 1450187.1528] P. Guha, E. Harikumar and N. S. Zuhair, J. Math. Phys. (2016) 112501.[29] P. Guha, E. Harikumar and N. S. Zuhair, Eur. Phys. J. Plus. (2015) 205.[30] J. H. Kim and H. W. Lee, Can. J. Phys. (1999) 411.[31] S. Takagi. Quantum dynamics and non-inertial frames of references. I: Grnerality, Prog. Theor.Phys. (1991) 463.[32] S. Takagi. Quantum dynamics and non-inertial frames of references. II: Harmonic Oscillators, Prog.Theor. Phys. (1991) 723.[33] S. Takagi. Quantum dynamics and non-inertial frames of references. III: Charged Particle in Time-Dependent Uniform Electromagnetic Field, Prog. Theor. Phys.86