Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0211 Acta Polytechnica 62(1):211–221, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague TIME-DEPENDENT MASS OSCILLATORS: CONSTANTS OF MOTION AND SEMICLASICAL STATES Kevin Zelaya Czech Academy of Science, Nuclear Physics Institute, 250 68 Řež, Czech Republic correspondence: kdzelaya@fis.cinvestav.mx Abstract. This work reports the construction of constants of motion for a family of time-dependent mass oscillators, achieved by implementing the formalism of form-preserving point transformations. The latter allows obtaining a spectral problem for each constant of motion, one of which leads to a non-orthogonal set of eigensolutions that are, in turn, coherent states. That is, eigensolutions whose wavepacket follows a classical trajectory and saturate, in this case, the Schrödinger-Robertson uncertainty relationship. Results obtained in this form are relatively general, and some particular examples are considered to illustrate the results further. Notably, a regularized Caldirola-Kanai mass term is introduced in an attempt to amend some of the unusual features found in the conventional Caldirola-Kanai case. Keywords: Time-dependent mass oscillators, Caldirola-Kanai oscillator, quantum invariants, coherent states, semiclassical dynamics. 1. Introduction The search for exact solutions for time-dependent (nonstationary) quantum models is challenging task as compared to the stationary (time-independent) coun- terpart. In the stationary case, the dynamical law (Schrödinger equation) reduces to an eigenvalue equa- tion associated with the energy observable, the Hamil- tonian, for which several methods can be implemented to obtain exact solutions. Particularly, new exactly solvable models can be constructed from previously known ones through Darboux transformations [1] (also known as SUSY-QM). In the nonstationary case, it is still possible to recover an eigenvalue problem for the Hamiltonian if one restricts to the adiabatic ap- proximation [2, 3]. However, in general, the latter is not feasible, and other workarounds should be imple- mented. Despite all these challenges, time-dependent phenomena find exciting applications in physical sys- tems such as electromagnetic traps of charged particles and plasma physics [4–8]. The parametric oscillator is perhaps the most well- known exactly solvable nonstationary model in quan- tum mechanics. A straightforward method to solve such a problem was introduced by Lews and Riesen- feld [9] by noticing that the appropriate constant of motion (quantum invariant) admits a nonstationary eigenvalue equation with time-dependent solutions and constant eigenvalues. In this form, nonstationary models can be addressed similarly to their station- ary counterparts. This paved the way to solve other time-dependent problems [10–14]. Recently, the Darboux transformation has been adapted into the quantum invariant scheme to con- struct new time-dependent Hamiltonians, together with the corresponding quantum invariant and the set of solutions [15–17]. Alternatively, other meth- ods exist to build new time-dependent models, such as the modified Darboux transfomation introduced by Bagrov et al. [18], which relies on a differential operator that intertwines a known Schrödinger equa- tion with an unknown one. This has led to new re- sults in the nonstationary Hermitian regime [19–21]. A non-Hermitian PT-symmetric extension has been discussed in [22], and some further models were re- ported in [23, 24]. On the other hand, the point transformations for- malism [25] has been proved useful to construct and solve time-dependent oscillators. This was achieved by implementing a geometrical deformation that trans- forms the stationary oscillator Schrödinger equa- tion into one with time-dependent frequency and mass [26, 27]. This allows obtaining further infor- mation such as the constants of motion, which are preserved throughout the point transformation [25], leading to a straightforward way to get such constants of motion without imposing any ansatz. A further extension for non-Hermitian systems was introduced in [28], whereas a non-Hermitian extension of the generalized Caldirola-Kanai oscillator was discussed in [29]. In this work, the point transformation formalism is exploited to construct and study the dynamics of semiclassical states associated with time-dependent mass oscillators. This is achieved by using the afore- mentioned preservation of constants of motion and identifying their corresponding spectral problem. No- tably, it is shown that one constant of motion leads to an orthogonal set of solutions, whereas a differ- ent one leads to nonorthogonal solutions that behave like semiclassical states. That is, Gaussian wavepack- ets whose maximum point follows the corresponding classical trajectory and minimize, in this case, the 211 https://doi.org/10.14311/AP.2022.62.0211 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Kevin Zelaya Acta Polytechnica Schrödinger-Robertson uncertainty principle. Two particular examples are considered to illustrate the usefulness of the approach further. 2. Materials and methods Throughout this manuscript, the time-dependent mass m(t) and frequency Ω2(t) oscillator subjected to an external driving force F (t) is considered. Such a model is characterized by the time-dependent Hamiltonian Ĥck(t) = p̂2 2m(t) + m(t)Ω2(t) 2 x̂2 + F(t)x̂, (1) with x̂ and p̂x the canonical position and momentum operators, respectively, with [x̂, p̂x] = iℏI. Henceforth, the identity operator I is omitted each time it mul- tiplies a constant or a function. The corresponding Schrödinger equation iℏ ∂ψ ∂t = − ℏ2 2m(t) ∂2ψ ∂x2 + m(t)Ω2(t)x2 2 ψ +F(t)xψ, (2) is recovered by using the coordinate representation px ≡ −iℏ ∂∂x and x̂ ≡ x ∈ R. The solutions of Eq. (2) have been discussed by several authors, see [27, 30–32]. Here, a brief sum- mary of the point transformation approach discussed in [26, 27] is provided. This eases the discussion of semiclassical states and dynamics to be presented later in Section 3. 2.1. Point transformations In general, the method of form-preserving point trans- formations relies on a geometrical deformation that maps an initial differential equation with variable coefficients into another one of the same form but with different coefficients. To illustrate this, be the stationary oscillator Hamiltonian Ĥosc = p̂2y 2m0 + m0w 2 0ŷ 2 2 , [ŷ, p̂y ] = iℏ, (3) with ŷ and p̂y another couple of canonical position and momentum observables, respectively. The corre- sponding Schrödinger equation iℏ ∂Ψ ∂τ = − ℏ2 2m0 ∂2Ψ ∂y2 + m0w 2 0y 2 2 Ψ, (4) admits the well-known solutions [2] Ψn(y,τ) = e−iw0(n+ 1 2 )τ Φn(y), (5) where Φn(y) = √ 1 2nn! √ m0w0 πℏ e − m0 w02ℏ y 2 Hn (√ m0w0 ℏ y ) , (6) with Hn(z) the Hermite polynomials [33], fulfills the stationary eigenvalue problem HoscΦn(y) = E(osc)n Φn(y), E (osc) n = ℏw0 ( n + 1 2 ) , (7) with Hosc the coordinate representation of Ĥosc, i.e., a second-order differential operator that admits a Sturm-Liouville problem. To implement the point transformation, one im- poses a set of relationships between the coordinates, time paramaters, and solutions of both systems in consideration [25]. In general one has y(x,t), τ(x,t), Ψ(y(x,t),τ(x,t)) ≡ G(x,t; ψ), (8) where G(x,t; ψ) is a reparametrization of Ψ as an explicit function of x, t, and ψ. In the case under consideration, some further con- ditions are required to preserve the linearity and the Hermiticity of Ĥosc and Ĥck(t). A detailed discussion on the matter can be found in [27]. Here, the final form of the point transformation is used, leading to y(x,t) = µ(t)x + γ(t) σ(t) , τ(t) = ∫ t dt′ σ2(t′) , (9) and Ψ(y(x,t),τ(t)) ≡ G(x,t; ψ) = A(x,t)ψ(x,t), (10) with m(t) = µ2(t), together with σ(t) and γ(t) some real-valued functions to be determined. By substituting (9) into the Schrödinger equa- tion (4), and after some calculations, one arrives to a new partial differential equation for ψ(x,t) that takes the exact form in (2). The latter allows obtain- ing A(x,t) = √ σ µ exp A(x,t), A(x,t) := ( i m0w0 ℏ µ σ ( Wµ 2 x2 + Wγx ) + iη ) , (11) where A(x,t) is a local time-dependent complex-phase and [27] η(t) := m0 2ℏ γ(t)Wγ (t) σ(t) − 1 2ℏ ∫ t dt′ F(t′) µ(t′) , Wµ(t) = σ(t)µ̇(t) − σ̇(t)µ(t), Wγ (t) = σ(t)γ̇(t) − σ̇(t)γ(t), (12) with ḟ(t) ≡ df (t) dt a short-hand notation for the time derivative. In the latter, σ(t) and γ(t) fulfill the nonlinear Ermakov equation σ̈(t) + ( Ω2(t) − µ̈(t) µ(t) ) σ(t) = w20 σ3(t) , (13) and non-homogeneous equation γ̈(t) + ( Ω2(t) − µ̈(t) µ(t) ) γ(t) = F(t) m0µ(t) , (14) The solutions of the Ermakov equation are well- known [34–36] and computed from two linearly inde- pendent solutions of the associated linear equation q̈j (t) + ( Ω2(t) − µ̈(t) µ(t) ) qj (t) = 0, j = 1, 2, (15) 212 vol. 62 no. 1/2022 TD mass oscillators: constants of motion and semiclasical states through the nonlinear combination σ(t) = [ aq21 (t) + bq1(t)q2(t) + cq 2 2 (t) ]1 2 , (16) with b2 − 4ac = −4 w 2 0 W 20 and W0 = Wr(q1(t),q2(t)) ̸= 0 the Wronskian of two linearly independent solutions of (15), which is in general a time-independent com- plex constant. The previous constraint on a, b, and c guarantees that σ(t) is different from zero [26] for t ∈ R. In this form, one obtains a set of solutions {ψn(x,t)}∞n=0 to the Schrödinger equation (2), where ψn(x,t) = √ µ(t) σ(t) [A(x,t)]−1e−iw0(n+ 1 2 )τ (t) × e− m0 w0 2ℏ ( µ(t)x+γ(t) σ(t) )2 Hn (√ m0w0 ℏ µ(t)x + γ(t) σ(t) ) . (17) From (10)-(11) it follows that (ψm,ψn) := ∫ R dxψ∗m(x,t)ψn(x,t) = ∫ R dyΨ∗m(y,τ)Ψn(y,τ) = δn,m, (18) with z∗ the complex conjugate of z. That is, the inner product is preserved and thus the set {ψn(x,t)}∞n=0 is orthonormal in L2(R,dx). The expressions presented so far are general, and specific result may be obtained once the time- dependent mass and frequency terms are specified. This is discussed in the following sections. Before concluding, an explicit expression for τ(t) can be determined in terms of the two linearly inde- pendent solutions q1(t) and q2(t) as well. One gets τ(t) = w−10 arctan [ W0 2w0 ( b + 2c q2(t) q1(t) )] . (19) 3. Results: Constants of motion and semiclassical states Additional information can be extracted from the sta- tionary oscillator into the time-dependent model. Par- ticularly, point transformations preserve first-integrals of the initial equation [25]. In the context of the Schrödinger equation, such first-integrals correspond to constants of motion, also known as quantum in- variant, associated with the physical models under consideration. From the stationary oscillator, it is straightforward to realize that the Hamiltonian Ĥosc is a constant of motion that characterize the energy observable. In the time-dependent case, Ĥck(t) is no longer a constant of motion, as dĤck(t) dt ≠ 0. This implies that an eigenvalue problem associated with Ĥck is not possible1. 1One can still link an eigenvalue problem with Ĥck(t) under the adiabatic approximation [3]. This work focuses on exact solutions and such an approach will be disregarded. On the other hand, an orthonormal set of solutions {ψn(x,t)}∞n=0 has been already identified, and it is still unclear the eigenvalue problem that such a set solves. This problem was addressed by Lewis-Riesenfeld [9] while solving the dynamics of the parametric oscillator. They notice that even in the time-dependent regime, there may be a constant of motion Î0(t) that admits a spectral problem Î0(t)ϕ(x,t) = λϕ(x,t), (20) where the eigenvalues λ are time-independent. The existence and uniqueness of such a quantum invariant is not necessarily ensured. Still, for the parametric oscillator, Lewis and Riesenfeld managed to find the quantum invariant and solve the related spectral prob- lem. Here, some quantum invariants associated with Ĥck can be found through point transformations. First, notice that the point transformation was implemented in the Schrödinger equation to get the time-dependent counterpart. The same transformation can be applied to a constant of motion of the harmonic oscillator to get the corresponding one on the time-dependent model. Particularly, by consider the eigenvalue prob- lem (7), and after some calculations, one gets a first quantum invariant of the form Î1(t) := σ2(t) 2m0µ2(t) p̂2x + m0 2 ( W 2µ (t) + w 2 0 µ2(t) σ2(t) ) x̂2 + σWµ(t) 2µ(t) (x̂p̂x + p̂xx̂) + σWγ (t) µ(t) p̂x + m0 ( Wγ (t)Wµ(t) + w20 µ(t)γ(t) σ2(t) ) x̂ + ( m0 2 W 2γ (t) + γ2(t) σ2(t) ) . (21) It is straightforward to show that Î1(t) is indeed a quantum invariant, i ℏ [Ĥck, Î1(t)] + ∂Î1(t) ∂t = 0. (22) Moreover, I1(t), the coordinate representation of Î1(t), defines a Sturm-Liouville problem with time- dependent coefficients, I1(t)ψn(x,t) = ℏw0 ( n + 1 2 ) ψn(x,t), (23) which justifies the existence of the orthogonal set of solutions found in Section 2. Note that orthogonal- ity has been alternatively proved in (18) using the preservation of the inner product. Remarkably, there are still more quantum invari- ants to be exploited. To see this, let us consider the operators â = √ m0w0 2ℏ ŷ + i p̂y√ 2m0ℏw0 , ↠= √ m0w0 2ℏ ŷ − i p̂y√ 2m0ℏw0 , (24) 213 Kevin Zelaya Acta Polytechnica which factorize the stationary oscillator Hamiltonian as Ĥosc = ℏw0(â†â + 12 ) and fulfill the commutation relationship [â, â†] = 1. Although â and ↠are not constants of motion of Ĥosc, one can introduce a new pair of operators â := eiw0τ â, ↠:= e−iw0τ â†, (25) where the straightforward calculations show that i ℏ [Ĥosc, â] + ∂â ∂τ = 0, and similarly for â†. That is, a and a† are quantum invariants of Ĥosc. The latter can now be mapped into the time- dependent model, leading straightforwardly to new quantum invariants of Ĥck(t) of the form Îa(t) = eiw0τ (t) [ i √ 2m0ℏw0 σ(t) µ(t) p̂x + (√ m0w0 2ℏ µ(t) σ(t) + i √ m0 2ℏw0 Wµ(t) ) x̂ + (√ m0w0 2ℏ γ(t) σ(t) + i √ m0 2ℏw0 Wγ (t) )] , (26) and its adjoint Άa(t). Before proceeding, it is worth to recalling that two arbitrary quantum invariants Î(t) and ˆ̃I(t) of a given Hamiltonian Ĥ(t) can be used to construct further invariants. This follows from the fact that the linear combination ℓÎ(t) + ℓ̃ˆ̃I(t) and the product ℓÎ(t)ˆ̃I(t) of quantum invariants are also quantum invariants of the same Hamiltonian Ĥ(t), for ℓ, ℓ̃, and ℓ time- independent coefficients. In this form, Îa(t) and Î † a(t) generate Î1(t) through Î1(t) = ℏw0 ( Άa(t)Îa(t) + 1 2 ) , (27) which is analogous to the factorization of the station- ary oscillator. Similarly, the commutation relationship [â, â†] = 1 of the stationary oscillator is preserved. One thus get [Îa(t), Î † a(t)] = 1 together with [Î1(t), Îa(t)] = −ℏw0Îa(t), [Î1(t), Άa (t)] = ℏw0Î † a (t), (28) which means that Îa(t) and Î † a(t) are annihilation and creation operators, respectively, for the eigensolutions of Î1(t). The latter leads to Îa(t)ψn+1(x,t) = √ ℏw0(n + 1)ψn(x,t), Άa(t)ψn(x,t) = √ ℏw0(n + 1)ψn+1(x,t), (29) for n = 0, . . . . On the other hand, the orthonormal set {ψn(x,t)}∞n=0 can be used as a basis to expand any arbitrary solution ψ(x,t) of (2) through ψ(x,t) = ∞∑ n=0 Cnψn(x,t), Cn := (ψn(x,t),ψ(x,t)). (30) Now, from the above results, one may investigate the spectral problem related to the remaining quan- tum invariants Îa(t) and Î † a(t). By considering the annihilation operator Îa(t), one obtains the eigenvalue problem Îa(t)ξα(x,t) = αξα(x,t), (31) where the eigensolution ξα(x,t) can be expanded as the linear combination ξα(x,t) = ∞∑ n=0 C̃n(α)ψn(x,t), α ∈ C. (32) This corresponds to the construction of coherent states using the Barut-Girardelo approach [37]. The com- plex coefficients C̃n(α) are determined by using the action of the ladder operators (29) and exploiting the orthonormality of the set {ψn(x,t)}∞n=0. After substituting the linear combination ξα(x,t) into the corresponding eigenvalue problem in (31), one obtains the one-parameter normalized eigensolutions ξα(x,t) = exp ( − |α|2 2ℏw0 ) ∞∑ n=0 ( α √ ℏw0 )n ψn(x,t)√ n! . (33) Henceforth, the latter are called time-dependent co- herent states or semiclassical states interchangeably. Similar to Glauber coherent states [38], the eigenso- lutions of the annihilation operator Îa are not orthog- onal among themselves. This follows from the overlap between two solutions with different eigenvalues, let say α and β, leading to |(ξβ,ξα)|2 = exp ( − |α|2 + |β|2 − 2 Re(α∗β) ℏw0 ) , (34) which is different from zero for every α,β ∈ C, with the inner product defined in (18). Interestingly, the eigensolution ξα(x,t) can be brought into an alterna- tive and handy expression by using the explicit form of ψn(x,t) given in (17), together with the well-known summation rules for the Hermite polynomials. By doing so one gets ξα(x,t) ≡ √ µ(t) σ(t) √ m0w0 πℏ [A(x,t)]−1e−i w0 τ (t) 2 × exp [ i √ 2m0w0 ℏ ( µ(t)x + γ(t) σ(t) ) Im α̃(t) ] ×exp [ − m0w0 2ℏ ( µ(t)x + γ(t) σ(t) − √ 2ℏ m0w0 Re α̃(t) )2] , (35) with α̃(t) = αe−iw0τ (t). Thus, ξα(x,t) is a normal- ized Gaussian wavepacket with time-dependent width. The complex constants α plays the role of the initial conditions of the wavepacket at a given time t0. See the discussion in the following section. Despite the lack of orthogonality in the elements of the set {ξα(x,t)}α∈C, they can be still used as a non- orthogonal basis so that any arbitrary solution of (2) can be constructed through the appropriate linear superposition. That is, a given solution ψ(x,t) of (2) expands as ψ(x,t) = ∫ C d2α πℏ C(α)ξα(x,t), (36) 214 vol. 62 no. 1/2022 TD mass oscillators: constants of motion and semiclasical states where C(α) = (ξα(x,t),ψ(x,t)). So far, the spectral problem related to the quantum invariants Î1(t) and Îa(t) has led to a discrete and a continuous representation, respectively, in which any solution of (2) can be expanded. Although the eigenvalue problem related to the quantum invariant Άa(t) can be established, it leads to non-finite norm solutions and is thus discarded. 3.1. Semiclassical dynamics With the time-dependent coherent states already con- structed, one can now study the evolution on time of such state and its relation with physical observables such as position and momentum x̂ and p̂x, respec- tively. To this end, note that the quantum invariants obtained through point transformations preserve the commutation relation (28) of the corresponding op- erators of the stationary oscillator. That is, the set {Îa(t), Î † a(t), Î † a(t)Îa(t)} fulfill the Weyl-Heisenberg al- gebra [39]. This allows the construction of a unitary displacement operator of the form [39] D(α; t) = eαÎ † a(t)−α ∗Îa(t) = e− |α| 2 eαÎ † ae−α ∗Îa, α ∈ C, (37) so that D†(α,t)Îa(t)D(α,t) = Îa(t) + α, D†(α; t)Άa(t)D(α; t) = Î † a(t) + α ∗. (38) It follows that the action of the first relationship acted on ψ0(x,t) leads to Îa(t)D(α,t)ψ0(x,t) = αD(α,t)ψ0(x,t), from which one recovers the eigen- value equation previously analyzed in (31) by identi- fying ξα(x,t) = D(α,t)ψ0(x,t). This corresponds to the coherent states construction of Perelomov [39]. So far, two different and equivalent ways to con- struct the solutions ξα(x,t) have been identified, a property akin to Glauber coherent states. To further explore the time-dependent coherent states, one can take the unitary transformations (38) and combine them with the relationship between the ladder opera- tors and the physical position x̂ and momentum p̂x observables presented in (26). After some calculations one obtains ⟨x̂⟩α(t) = √ 2ℏ m0w0 σ(t) µ(t) r cos (w0τ(t) − θ) − γ(t) µ(t) , (39) where α = reiθ. By using (19) and some elemen- tary trigonometric identities, one recovers an explicit expression in terms of q1(t) and q2(t) as ⟨x̂⟩α(t) = − γ(t) µ(t) + √ 2ℏw0 m0c r W0 [( cos θ + W0 2w0 c sin θ ) q1(t) µ(t) + W0 w0 c sin θ q2(t) µ(t) ] . (40) Similarly, the calculations for the momentum observ- able leads to ⟨p̂x⟩α(t) = −m0 µ(t) σ(t) (Wµ(t)⟨x̂⟩α(t) + Wγ (t)) − √ 2m0ℏw0 µ(t) σ(t) r sin (w0τ(t) − θ)). (41) In the latter, ⟨Ô⟩α(t) ≡ (ξα(x,t), Ôξα(x,t)) stands for the average value of the observable Ô computed through the time-dependent coherent state ξα(x,t). The expectation value of the momentum (41) can be further simplified so that it simply rewrites as ⟨p̂x⟩α(t) = m(t) d dt ⟨x̂⟩α(t), m(t) = m0µ2(t), (42) which is an analogous relation to that obtained from the canonical equations of motion of the corresponding classical Hamiltonian. This is also consequence of the quadratic nature of the time-dependent Hamiltonian Ĥck(t) and the Ehrenfest theorem. From the expectation values obtained in (39)-(42), a relationship between the complex parameter α = reiθ and the expectation values at a given initial time t = t0 can be established. The straightforward calcu- lations lead to ( Re α̃t0 Im α̃t0 ) = (√m0w0 2ℏ γt0 σt0√ m0 2ℏw0 Wγt0 ) + ( √m0w0 2ℏ µt0 σt0 0√ m0 2ℏw0 Wµt0 1√ 2m0ℏw0 σt0 µt0 )( ⟨x̂⟩t0 ⟨p̂x⟩t0 ) , (43) with α̃t0 = αe−iw0τt0 = rei(θ−w0τt0 ), τt0 = τ(t0), σt0 = σ(t0), γt0 = γ(t0), Wγt0 = Wγ (t0), Wµt0 = Wµ(t0), ⟨x̂⟩t0 = ⟨x̂⟩α(t0), and ⟨p̂x⟩t0 = ⟨p̂x⟩α(t0). On the other hand, one can write the probability density associated with the time-dependent coherent state in terms of ⟨x̂⟩α(t) through Pα(x,t) := |ξα(x,t)|2 = √ m0w0 πℏ µ(t) σ(t) × exp [ − m0w0 2ℏ µ2(t) σ2(t) (x − ⟨x̂⟩α(t)) 2 ] , (44) which is a Gaussian wavepacket whose maximum follows the classical trajectory. That is, the time- dependent coherent state is considered as a semiclas- sical state. Before concluding this section, it is worth explor- ing the corresponding uncertainty relations associ- ated with the canonical observables, which can be computed by using (26), (31), and (38). After some calculations one gets (∆x̂)2α = ℏ 2m0w0 σ2(t) µ2(t) , (∆p̂x) 2 α = m0ℏw0 2 µ2(t) σ2(t) ( 1 + σ2(t)W 2µ (t) w20µ 2(t) ) , (45) 215 Kevin Zelaya Acta Polytechnica from which the uncertainty relation reduces to (∆x̂)2α (∆p̂x) 2 α = ℏ2 4 ( 1 + σ2(t)W 2µ (t) w20µ 2(t) ) , (46) where it is clear that, in general, ξα(x,t) does not min- imize the Heisenberg uncertainty relationship, except for those times t′ at which Wµ(t′) = 0. The latter follows from the fact that σ ̸= 0 for t ∈ R. Still, there are two special cases for which Eq. (46) minimizes at all times. • For µ(t) = µ0 and Ω(t) = w1, one can always find a constant solution σ4(t) = w20/w21 so that Wµ = 0. The uncertainty relationship (46) is minimized, and the time-dependent Hamiltonian becomes Ĥck(t) = p̂2x 2m0µ20 + m0µ 2 0w 2 1 2 x̂2 + F(t)x̂, (47) which is nothing but a stationary oscillator with an external time-dependent driving force F(t)2. Thus, the uncertainty relation gets minimized in the sta- tionary limit, as expected. • For Ω2(t) = w20µ−4(t), there is a solution σ(t) = µ(t) for which Wµ = 0. This leads to a Hamiltonian of the form Ĥck(t) = 1 µ2(t) ( p̂2x 2m0 + m0w 2 0 2 x̂2 + µ2(t)F(t)x̂ ) . (48) Although the solutions ξα(x,t) minimize the Heisen- berg uncertainty relation only on some restricted cases, one can still explore the Schrödinger-Robertson in- equality [40, 41]. This is defined for a pair of observ- ables  and B̂ through( ∆ )2 ( ∆B̂ )2 ≥ |⟨[Â,B̂]⟩|2 4 + σ2A,B , (49) where σA,B := 12 ⟨ÂB̂ + B̂Â⟩ − ⟨Â⟩⟨B̂⟩ stands for the correlation function. For the canonical position x̂ and momentum p̂x observables one gets σ2x,px = ℏ2 4w20 σ2(t)W 2µ (t) µ2(t) , (50) when computed through ξα(x,t). Thus, the semiclassi- cal states ξα(x,t) minimize the Schrödinger-Robertson relationship for t ∈ R. 4. Discussion: Conventional and regularized Caldirola-Kanai oscillators So far, the most general setup has been addressed for a time-dependent mass oscillator. Two particular 2The Hamiltonian (47) is essentially stationary, for the term F (t) can be absorbed through an appropriate reparametrization of the canonical coordinate. examples are considered in this section to further illus- trate the usefulness and behavior of the so-constructed solutions and coherent states. Henceforth, all calcula- tions are carried on by working in units of ℏ = 1 to simplify the ongoing discussion. Throughout the rest of this manuscript, the following two time-dependent masses are considered: µck(t) = e−κt, κ ≥ 0, (51a) µrck(t) = e−κt + µ0, κ,µ0 ≥ 0. (51b) The first one corresponds to the well-known Caldirola-Kanai oscillator [42, 43], which contains a mass-term that asymptotically approaches to zero. This is a rather unrealistic scenario in the context of the Schrödinger equation. Still, one can study the dynamics on a given time range, let say t ∈ [0,T], where T denotes the time spent by the mass to re- duce its initial value in a factor e−1. In other words, T = κ−1 is equivalent to the lifetime of a decay- ing system. One thus may disregard the dynamics for t > T. To amend such issue, the second mass term µrck(t) has been introduced, which transits from µrck(0) = 1 to µck(t → ∞) = µ0. Thus, there is no need to introduce any artificial truncation on the time domain. The Hamiltonian associated with this mass-term will be called regularized Caldirola-Kanai oscillator. Despite the apparent advantages of the regularized system, analytic expressions for σ(t) are significantly more complicated with respect to those obtained from µck(t). Still, exact result can be ob- tained. The discussion is thus divided for each case separately. 4.1. Caldirola-Kanai case The so-called Caldirola-Kanai system is another well- known nonstationary problem, characterized by time- dependent mass decaying exponentially on time. It was independently introduced by Caldirola [42] and Kanai [43] in an attempt to describe the quantum counterpart of a damped oscillator. This model has been addressed by different means, such using a Fourier transform to map the map it into a para- metric oscillator [32], and using the quantum Arnold transformation [30]. For this particular case, a constant frequency ω2(t) = w21 and a driven force F(t) = A0 cos(νt), for ν, A0 ∈ R, are considered. This leads to a forced Caldirola-Kanai oscillator Hamiltonian [10, 44] of the form Ĥck(t) = e2κt 2m0 p̂2x +e −2κt m0w 2 1 2 x̂2 + A0 cos(νt)x̂. (52) From the results obtained in previous sections, one gets the solutions to the Ermakov and non- homogeneous equations as σ(t) = ( aq21 (t) + bq1(t)q2(t) + cq 2 2 (t) )1 2 , γ(t) = γ1q1(t) + γ2q2(t) + γp(t), (53) 216 vol. 62 no. 1/2022 TD mass oscillators: constants of motion and semiclasical states (a) . Wµ(t) (b). Figure 1. (A) Wµ(t) = σ(t)µ̇(t) − σ̇(t)µ(t) for the Caldirola-Kanai mass term µck(t). (B) Variances (∆x̂)2α (solid- blue), (∆p̂x)2α (dashed-red), the Schrödinger-Robertson uncertainty minimum (dotted-green), and the Heisenberg uncertainty minimum (thick-solid-black) associated with the coherent states ξα(x,t) and the mass term µck(t). The parameters have been fixed as a = c = w0 = 1, w1 = 2, and κ = 0.5. (a) . n = 0 (b) . n = 1 (c). Figure 2. Probability density Pn = |ψn(x,t)|2 for n = 0 (A), n = 1 (B), and Pα = |ψn(x,t)|2 (C) associated with the Caldirola-Kanai mass term µck(t). For simplicity, the external force F (t) and γ(t) have been fixed to zero. The rest of parameters have been fixed as w0 = a = b = 1, w1 = 2, and κ = 0.5. respectively, with γ1 and γ2 arbitrary real constants, b2 − 4ac = −16 w 2 0 w21 −κ 2 , and q1(t) = cos( √ w2 − κ2 t), q2(t) = sin( √ w2 − κ2 t), γp(t) = A0e−kt (w21 − ν2) cos(νt) − 2κν sin(νt) (w21 + ν2)2 − 4ν2(w21 − κ2) . (54) In the sequel, κ = 0.5 is consider so that the Caldirola-Kanai oscillator is constrained to the time interval t ∈ [0, 2]. Further discussions concerning the dynamics will be restricted to such a time interval. It is worth recalling that the zeros of Wµ(t) corre- spond to the times for which the Heisenberg uncer- tainty relationship saturates. Although the expression for Wµ(t) is rather simple in this case, determining the zeros consist of solving a transcendental equation. Thus, to get further insight, one may analyze Fig- ure 1a, which depicts the behavior of such a function for µck(t) (solid-blue). From the latter, one can see that zeroes do exist indeed, and thus one should expect points in time for which the Heisenberg inequality sat- urates. Despite the latter, the Schrödinger-Robertson inequality saturates at all times. In Figure 1b, one can see the behavior of the vari- ances, from which it is clear that the variance in the position blows up at time pass by, whereas the mo- mentum variance squeezes indefinitely, approaching asymptotically to zero. This odd behavior results from a mass term that quickly decays, approaching zero but never converging to it. For those reasons, a truncation on the time interval was previously in- troduced in the form of a mean lifetime, which in this case becomes T = κ−1 = 2. In this form, one still has a realistic behavior for t ∈ (0, 2). The previous results can be verified by looking at the probability density associated with the solutions ψn(x,t) and the coherent state ξα(x,t), which is de- picted in Figure 2. From those probability densities, one may see the increase on the position variance (∆x̂)2α, for the wavepacket spreads rapidly on time, 217 Kevin Zelaya Acta Polytechnica (a) . Wµ(t) (b). Figure 3. (A) Wµ(t) = σ(t)µ̇(t) − σ̇(t)µ(t) for the regularized Caldirola-Kanai mass term µrck(t). (B) Variances (∆x̂)2α (solid-blue), (∆p̂x)2α (dashed-red), the Schrödinger-Robertson uncertainty minimum (dotted-green), and the Heisenberg uncertainty minimum (thick-solid-black) associated with the coherent states ξα(x,t) and the mass term µrck(t). The parameters have been fixed as a = c = w0 = 1, w1 = 2, µ0 = 0.3, and κ = 0.5. to the point that, for times t > 4 is almost indistin- guishable. For completeness, the classical trajectory is depicted as a dashed-black curve in Figure 2c, where the initial conditions ⟨x̂⟩t0 = 2 and ⟨p̂x⟩t0 = 0 have been used. 4.2. Regularized Caldirola-Kanai In this section, the regularized Caldirola-Kanai os- cillator is introduced so that it amends the diffi- culties found in the Caldirola-Kanai for t >> T. This model is characterized by a constant frequency Ω2(t) = w21 and a mass term µrck(t) = µ0e−κt + µ1, with w1,µ0,µ1,κ > 0. The mass term will converge at a constant value (different from zero) and the anoma- lies found in the conventional Caldirola-Kanai case will be fixed. The main consequence of the mass reg- ularization is that the classical equation of motion is not as trivial as in Section 4. In turn one has q̈(t) + ( w21 − κ2 1 + µ0eκt ) q(t) = 0. (55) Two linearly independent solutions to the correspond- ing linear equation (15) can be found as q1(t) = z(t) i w1 k 2F1  A1,A2 1 − 2i w1 k ∣∣∣∣∣∣ −1z(t)   , q2(t) = q ∗ 1(t), (56) where z(t) = µ0eκt, A1 = −iw1k −i √ w21 k2 − 1, and A2 = −iw1 k + i √ w21 k2 − 1. On the other hand, 2F1(a,b; c; Z) stands for the hypergeometric function [33], which converges in the complex unit-disk |Z| < 1. Given that z(t) : R → (1, ∞), the solutions q1,2(t) in (56) converge for t ∈ R. Since both solutions in (56) are complex-valued, with q2 = q ∗ 1, one can construct a real-valued solution to the Ermakov equation by taking q1 =Re(q1) and q2 =Im(q1). To simplify the ongoing discussion, the external force is considered null, F(t) = 0. One thus obtains σ2(t) = a Re[q1(t)]2 + b Re[q1(t)] Im[q1(t)] + c Im[q1(t)]2, (57) γ(t) = γ1 Re[q1(t)] + γ2 Im[q1(t)], (58) where the Wronskian of the two linearly independent solutions Re q1 and Im q1 becomes W0 = w1, leading to the constraint b2 − 4ac = −4 w 2 0 w21 . Similarly to the Caldirola-Kanai case, the Heisen- berg uncertainty relation saturates for times tm such that Wµ(tm) = 0. In this case, an analytic expression for such points is fairly complicated. Instead, one may look at the behavior of Wµ(t) depicted in Figure 3a, from which it is clear that such points exist. On the other hand, Figure 3b reveals that, in contradistinc- tion to the Caldirola-Kanai case, the position variance does not grow indefinitely in time. This is rather ex- pected as, for asymptotic times t >> 1, the mass term converges to a finite value different from zero. That is, the Hamiltonian becomes stationary for asymptotic values. Before conclude, the probability density for ψn(x,t) and ξα(x,t) are shown in Figure 4. In the latter, it can be verified that the width of the wavepackets oscillates in a bounded way for times t > 2. Partic- ularly, for the coherent state case of Figure 4c, the dynamics of the wavepacket can be identified clearly, where the maximum point follows the corresponding classical trajectory (dashed-black). Therefore, there is no need to introduce a truncation time T, for the mass converges to a constant value different from zero, remaining physically reasonable at all times. 5. Conclusions In this work, the class of form-preserving point trans- formations has been used to construct the constants of 218 vol. 62 no. 1/2022 TD mass oscillators: constants of motion and semiclasical states (a) . n = 0 (b) . n = 1 (c). Figure 4. Probability density Pn = |ψn(x,t)|2 for n = 0 (A), n = 1 (B), and Pα = |ψn(x,t)|2 (C) associated with the regularized Caldirola-Kanai mass term µrck (t). The rest of parameters have been fixed as w0 = a = b = 1, w1 = 2, µ0 = 0.3, and κ = 0.5. motion for the family of time-dependent mass oscilla- tors. This was achieved by exploiting the preservation of first-integrals on the initial stationary oscillator model. Since several constants of motion are already known for the initial system, the corresponding coun- terparts for the time-dependent model are straightfor- wardly constructed by implementing the appropriate mappings. Notably, three different constants of mo- tion were identified, one that admits an orthogonal set of eigensolutions, another that permits non-orthogonal eigensolutions, and the third one that does not admit finite-norm solutions. Interestingly, the non-orthogonal eigensolutions are actually coherent states, for they are constructed from the annihilation operator of the time-dependent os- cillator. Furthermore, by exploiting the underlying Weyl-Heisenberg algebra fulfilled by the quantum in- variants, it was possible to find exact expressions for the expectation values of the position and momentum observables. The latter revealed the coherent states are represented by Gaussian wavepacket whose maxi- mum follows the corresponding classical trajectory. Besides the latter properties, it was also found that, in general, the Schrödinger-Robertson uncertainty re- lation saturates for all times, whereas the Heisenberg one gets minimized only for some times. Still, two special time-dependent Hamiltonians exist so that the Heisenberg inequality saturates at all times, one of which is the stationary limit case, as expected. Remarkably, the newly introduced regularized Caldirola-Kanai mass term admits exact solutions that regularize the unusual behavior observed in the con- ventional Caldirola-Kanai case. More precisely, the variances become bounded as well as the expectation values. 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Quasi-coherent states for damped and forced harmonic oscillator. Journal of Mathematical Physics 54(9):092102, 2013. https://doi.org/10.1063/1.4819261. 221 https://doi.org/10.1007/BF01646483 https://doi.org/10.1103/PhysRev.131.2766 https://doi.org/10.1103/PhysRev.34.163 https://doi.org/10.1007/BF02960144 https://doi.org/10.1143/ptp/3.4.440 https://doi.org/10.1063/1.4819261 Acta Polytechnica 62(1):211–221, 2022 1 Introduction 2 Materials and methods 2.1 Point transformations 3 Results: Constants of motion and semiclassical states 3.1 Semiclassical dynamics 4 Discussion: Conventional and regularized Caldirola-Kanai oscillators 4.1 Caldirola-Kanai case 4.2 Regularized Caldirola-Kanai 5 Conclusions Acknowledgements References