Quantum Damped Mechanical Oscillator

Abstract

The exact solutions of the Schrödinger equation for quantum damped oscillator with modified Caldirola-Kanai Hamiltonian are evaluated. We also investigate the cases of under-, over-, and critical damping.

1. Introduction

Quantum theory has been shown to be the fundamental law of nature and presently is the most correct theory of all microscopic and macroscopic systems [1–3]. Quantum effects usually manifested themselves at the microscopic level. The quantum theory is, however, governed by the Schrödinger equation whereas classical theory is governed by the Hamilton equation, for instance [3]. Dissipation is usually ascribed as having a microscopic nature. There have been attempts to understand dissipation at a more fundamental level [1–26]. The simplest model for dissipation is damped oscillators with one or two degrees of freedom. In the canonical approach, two different Hamiltonian representations have been introduced for these damped oscillators. One representation of the damped system is the Caldirola-kanai (CK) oscillator which is a one-dimensional system with an exponentially increasing mass [1–3, 18–20, 23–26]. Another representation is the Bateman-Feshbach-Tikochinsky (BFT) oscillator, which consists of a damped and an amplified oscillator [3, 23–26]. The CK, on one hand, is an open system whose parameters such as mass and frequency are all time dependent [1, 2],and, on the other hand, the BFT is a closed system whose total energy is conserved and the dissipated energy from the damped oscillator is transferred to amplified [3, 26]. Recently, Tarasov has evaluated the quantization and classical distribution for dissipative systems [27, 28]. The aim of this paper is to evaluate the damped harmonic mechanical oscillator. The damping is here considered in the form of Caldirola-Kanai Model [1, 2] and the recently developed model [4]. However, the problem of quantum oscillator with time-varying frequency had been solved [5–12]. The Hamiltonian of this model is usually quadratic in coordinates and momenta operators [4–7]. Our primary goals will be to construct the Lagrangian for this simple damped system and use the constructed Lagrangian to evaluate the equation of motion for the damped Harmonic oscillators and also evaluate the minimum uncertain relation for each damping regime.

2. Review of Bateman-Feshbach-Tikochinsky Oscillator

In a genuine dissipative system, the energy of the damped subsystem of the system must be dissipated away and transferred to another subsystem. This invariably means that the damped oscillator is described by a two-dimensional system; one subsystem of which dissipates the energy and the other of which gets amplified by the transferred energy. This kind of model has been suggested long ago by Bateman [23] and later by Morse and Feshbach [25] and Feshbach and Tikochinsky [24]. The equation of motion of the one-dimensional damped harmonic oscillator is 𝑚𝑞̈+𝛾𝑞̇+𝑘𝑞=0, (1) where the parameters 𝑚, 𝛾, 𝑘 are time independent. However, since the system in (1) is dissipative, a straightforward Lagrangian description leading to a consistent canonical quantization is not available [29]. In order to develop a canonical formalism, one requires (1) alongside its reversed image counterpart [29]: 𝑚𝑦̈−𝛾𝑦̇+𝑘𝑦=0. (2) so that the composite system is conservative. The BFT oscillator is now described by the Lagrangian of systems in (1) and (2) as follows: 𝑞̇̇𝛾𝐿=𝑚𝑦+2(𝑞𝑦̇−𝑞𝑦̇)−𝑘𝑞𝑦, (3) where 𝑞 is the damped harmonic oscillator coordinate and 𝑦 corresponds to the time-reversed counterpart. The Hamiltonian is given by 𝐻_𝛾=1𝑚𝑃_𝑞𝑃_𝑦+𝛾2𝑚(𝑦𝑃_𝑦−𝑞𝑃_𝑞)+𝑚𝜔²𝑞𝑦, (4) where 𝑃_𝑞=(𝑚𝑦̇−𝛾𝑞2),𝑃_𝑦=(𝑚𝑞̇−𝛾𝑞2). (5) The BFT oscillator is just the sum of two-decoupled oscillator with opposite signs in the limit of zero dissipation (𝑟=0): 𝐻₀=1𝑃2𝑚^2𝜉+𝑘2𝜉²−1𝑃2𝑚^2𝑜−𝑘2𝜂², (6) where we have introduced the hyperbolic coordinates 𝑥 and 𝑦as 1𝜉=√21(𝑞+𝑦),𝜂=√2(𝑞−𝑦). (7)

3. Caldirola-Kanai and Modified Caldirola-Kanai Oscillators

The CK oscillator with a variable mass 𝑚(𝑡)=𝑚𝑒^{𝛾𝑡/𝑚} has Hamiltonian of the form [1–3] 𝐻_ck=12𝑚𝑒^{𝛾𝑡/𝑚}𝑝²+𝑚𝜔²𝑒^{𝛾𝑡/𝑚}2𝑞², (8) or in the form [1, 2, 8] 𝐻_ck=12𝑚𝑒^2𝛾𝑡𝜔²(𝑡)𝑞²+1𝑒2𝑚^−2𝛾𝑡𝑝², (9) where m is the mass of the oscillator, γ is the damping coefficient, 𝑞 and 𝑝 are the coordinate and momentum operators, and 𝑤(𝑡) is time-dependent frequency of the oscillator. The Lagrangians associated with (8) and (9) are given as ̇𝑚𝐿(𝑞,𝑞,𝑡)=2𝑒^{𝛾𝑡/𝑚}[𝑞̇²+𝛾𝑚𝑞̇+𝜔²̇𝑚(𝑡)𝑞],𝐿(𝑞,𝑞,𝑡)=2𝑒^𝛾𝑡[𝑞̇²+2𝛾𝑞̇+𝜔²(𝑡)𝑞], (10)(11) respectively. The equations of motion for the classical coordinate 𝑞 and momentum 𝑝 of (10) and (11) are of the forms ̈𝛾𝑞(𝑡)+𝑚𝑞̇+𝜔²̈̇(𝑡)𝑞=0,𝑞(𝑡)+2𝛾𝑞+𝜔²(𝑡)𝑞=0. (12)(13)

We write the modified Caldirola-Kanai model [4] 𝐻_ck=1𝑒2𝑚^{−sin𝛽𝛾𝑡}𝑝²+12𝑚𝜔²(𝑡)𝑒^{sin𝛽𝛾𝑡}𝑞², (14) where (8)-(9) are obtained from (14) when sin𝛽𝛾𝑡 is expanded to first order in increasing power of 𝛽𝛾𝑡 with variable parameter 𝛽 being set to 1. The Lagrangian of this modified Caldirola-Kanai Oscillator becomes ̇𝑚𝐿(𝑞,𝑞,𝑡)=2𝑒^{sin𝛽𝛾𝑡}[𝑞̇²+𝛾𝛽cos𝛾𝑡𝑞̇+𝜔²(𝑡)𝑞], (15) and its equation of motion for the classical coordinate 𝑞 and momentum 𝑝 takes the form 𝑞(̈𝑡)+𝛽𝛾cos(𝛽𝛾𝑡)𝑞(̇𝑡)+𝜔²(𝑡)𝑞=0. (16) The solution of (16) is 𝑞(𝑡)=𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}[𝐴𝑒^𝑖Ω𝑡+𝐵𝑒^{−𝑖Ω𝑡}], (17) where Ω(𝑡)=4𝜔²(𝑡)−𝛽²𝛾²cos²𝛽𝛾𝑡 and approximating cos²𝛽𝛾𝑡≃1 for small damping yields Ω(𝑡)=4𝜔²(𝑡)−𝛽²𝛾². (18) Substituting (18) into (17) results in𝑞(𝑡)=𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}×[𝐴𝑒^{𝑖𝜔𝑡1−(𝛽2𝛾2/4𝜔2)}+𝐵𝑒^{−𝑖𝜔𝑡1−(𝛽2𝛾2/4𝜔2)}]. (19) We summarized the general solution of (16) for the over-damped (OD), critically damped (CD), and under-damped (UD) as 𝑞(𝑡)=𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}[𝐴coshΩ𝑡+𝐵sinhΩ𝑡],𝑞(𝑡)=𝑒^{−(𝛽𝛾𝑡/2)cos𝛽𝛾𝑡}[𝐴+𝐵𝑡],𝑞(𝑡)=𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}[𝐴cosΩ𝑡+𝐵sinΩ𝑡], (20)(21)(22) respectively.

4. Investigation of the Under-Damped (UD), Over-Damped (OD), and Critical Damped (CD) Oscillators

4.1. The Under-Damped Oscillator

We consider the quantum damped oscillator with time-dependent varying frequency given by (17). Subjecting (17) to continuity conditions [8], we obtain the arbitrary constants A and B as 𝐴_𝑘=(1−𝑖𝛾𝐵2Ω),_𝑘=𝑖𝛾,2Ω (23) with 𝑘=(0,1) corresponding to delta kick, and the classical trajectory becomes 𝑞(𝑡)=𝑒^{−(𝛽𝛾𝑡/2)cos𝛽𝛾𝑡}×[(1−𝑖𝛾2Ω)cosΩ𝑡+𝑖𝛾2ΩsinΩ𝑡]. (24)

The wave functions of (8)-(9) and (14) are determined by different methods [13–15], and for the latest review see [16]. An invariant operator for the general time-dependent oscillator whose eigenfunction is an exact quantum state up to a time-dependent phase factor had been introduced by Lewis and Riesenfeld [17]. We introduce a pair of operators first order in position and momentum [3, 8, 18–20] as follows: 𝑎(𝑡)=𝑖[𝜀^∗(𝑡)𝑝−𝜀^∗̇𝑎(𝑡)𝑞],_𝑡(𝑡)=−𝑖[𝜀(𝑡)𝑝−𝜀(̇𝑡)𝑞], (25) and they are required to satisfy the quantum Liouville-von Neumann equations defined as 𝜕𝑖ℏ𝐻𝜕𝑡𝑎(𝑡)+[𝑎(𝑡),_ck𝜕(𝑡)]=0,𝑖ℏ𝑎𝜕𝑡⁺(𝑡)+[𝑎⁺(𝑡),𝐻_ck(𝑡)]=0, (26) where 𝜀(𝑡) in (23) must satisfy the classical damped equation of (16).

The operator in (24) and its Hermitian conjugate satisfy at any time t the boson commutation relation [8], and 𝜀(𝑡) must also satisfy the Wronskian condition 𝑒^{(𝑑/𝑑𝛽)sin𝛽𝛾𝑡}[𝜀^∗̇(𝑡)𝜀(𝑡)−𝜀(̇𝑡)𝜀^∗(𝑡)]=𝑖, (27) where Ω²(0)=4𝜔²(0)−𝛾². The number operator defined by 𝑁(𝑡)=𝑎⁺(𝑡)𝑎(𝑡) (28) also satisfies (25) [3] such that each number state 𝑁(𝑡)|𝑛,𝑡⟩=𝑛|𝑛,𝑡⟩ (29) is also an exact quantum state of the time-dependent Schrödinger equation 𝜕𝑖ℏ𝐻𝜕𝑡Ψ(𝑥,𝑡)=_ck(𝑡)Ψ(𝑥,𝑡), (30) where 𝐻_ck(𝑡) is the modified Caldirola-Kanai Hamiltonian of (14). The wave function that satisfies (14) can be written as [22] 𝜓_𝛼(𝑞,𝑡)=(𝜋𝜀²ℏ²)𝑚Ω(0)^−1/4×exp[𝑖𝜀exp(sin𝛽𝛾𝑡)𝑚Ω(0)𝑞2𝜀(𝑡)Ω(0)²+√2𝛼(𝜀(𝑡)𝑚Ω(0)ℏ)^1/2𝑞−𝜀^∗(𝑡)𝛼²−2𝜀(𝑡)|𝛼|²2𝜓],_𝑛(𝑞,𝑡)=(𝜋𝜀²(𝑡)ℏ)𝑚Ω(0)^−1/4(𝜀^∗(𝑡))2𝜀(𝑡)^𝑛/21√𝑖̇𝑛!×exp[𝜀(𝑡)𝑚Ω(0)𝑒^{sin𝛽𝛾𝑡}𝑞2ℏ𝜀(𝑡)²+]𝐻_𝑛(𝑞√𝜀𝜀^∗), (31)(32) where 𝛼 in (31) is a complex number and the wave function in co-ordinate representations of (31)-(32) are Gaussian packets with time-dependent coefficients in quadratic form under the exponential function [8].

We obtain the quantum dispersion coordinate in the form ⟨𝑞²⟩=ℏ²𝜀^∗=(𝑡)𝜀(𝑡)ℏ𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡}𝛽2𝑚Ω[1+²𝛾²Ω²sin²Ω𝑡+𝛽𝛾Ωsin2Ω𝑡], (33) and the uncertainty in momentum is⟨𝑝²⟩=ℏ²𝑚^′2(𝑡)𝜀^∗̇̇=(𝑡)𝜀(𝑡)ℏ𝑚𝜔²𝑒2Ω^{(2−𝛽)𝛾𝑡cos𝛽𝛾𝑡ℎ}×[1+𝛽𝛾²Ω²sin²Ω𝑡+(𝜎(𝑡)Ω)²×(𝛽𝛾2Ωcos2Ω𝑡−𝛽𝛾Ω1sin2Ω𝑡−2(𝛽𝛾Ω)²+−1)𝛽𝛾𝜎(𝑡)Ω²(𝛽𝛾Ωsin2Ω𝑡−2cos2Ω𝑡)+𝛽𝛾Ωsin2Ω𝑡], (34) where the quantity 𝜎(𝑡) is given as 𝜎(𝑡)=𝛽𝛾2(cos𝛽𝛾𝑡+𝛽𝛾𝑡sin𝛽𝛾𝑡), (35) and the reduced mass 𝑚^′(𝑡) of the modified oscillator is defined as [4] 𝑚^′(𝑡)=𝑚𝑒^{(𝑑/𝑑𝛽)(sin𝛽𝛾𝑡)}. (36) These results show that the spreading of the wave packet is suppressed by the appearance of dissipation [21]. However, the generalized uncertainty relation has the valueΔ𝑞Δ𝑝=ℏ𝜔𝑒2Ω^{(1−𝛽)𝛾𝑡cos𝛽𝛾𝑡}[1+Λ(𝑡)]^1/2, (37a) where 𝛽Λ(𝑡)=(1+²𝛾²Ω²sin²Ω𝑡+𝛽𝛾Ω𝛽sin2Ω𝑡)×{1+²𝛾²Ω²sin²Ω𝑡+(𝜎(𝑡)Ω)²1×[2(𝛽𝛾Ω)²cos2Ω𝑡−𝛽𝛾Ω1sin2Ω𝑡−2(𝛽𝛾Ω)²+−1]𝛽𝛾𝜎(𝑡)Ω²[𝛽𝛾Ωsin2Ω𝑡−2cos2Ω𝑡]+𝛾𝛽Ωsin2Ω𝑡}. (37b)

Equation (37a)-(37b) is a generalized uncertainty relation and it satisfies the Heisenberg Uncertainty relation when the variable parameter β is set to unity. Figure 1 shows the uncertainty in coordinate for 𝛾=0, 0.2Ω, 0.4Ω, 0.6Ω, 0.8Ω, and Ω. The products of (32) and (33) give a generalized Heisenberg relation which reduces to the exact when the variable parameter 𝛽 is set to unity and the damping coefficient 𝛾 is set to zero.

Figure 1 

The uncertainties in position for the under-damped oscillator as a function of Ω𝑡 are governed by the modified Caldirola-Kanai Hamiltonian with dissipation coefficients 𝛾=0, 0.2Ω, 0.4Ω, 0.6Ω, 0.8Ω, and Ω, respectively.

4.2. The Over-Damped Oscillator

The Over-damping occurs when the damping factor 𝛾>𝜔. When this happens, the solution to the classical trajectory takes the form and imposing the boundary conditions [8] leads to 𝑞(𝑡)=𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}[coshΩ𝑡+(𝑖+𝛽𝛾Ω)sinhΩ𝑡]. (38) We obtain the uncertainty in the coordinate as ⟨^⌢𝑞²⟩=ℏ𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}2𝑚Ω×[cosh2Ω𝑡+𝛽𝛾Ωsinh2Ω𝑡+(𝛽𝛾Ω)²sinh²Ω𝑡]. (39) Similarly, the dispersion of momentum takes the form ⟨𝑝²⟩=ℏ𝑚𝜔²𝑒2Ω^{(2−𝛽)𝛾𝑡cos𝛽𝛾𝑡}×[cosh2Ω𝑡+(𝜎(𝑡)Ω)²cosh2Ω𝑡−2(𝜎(𝑡)Ω𝛾)sinh2Ω]−(Ω)²[cosh2Ω𝑡+(𝜎(𝑡)Ω)cosh2Ω𝑡+2(𝜎(𝑡)Ω𝑟)sinh2Ω𝑡]−(Ω)²[sinh2Ω𝑡+(𝜎(𝑡)Ω)²sinh2Ω𝑡+2(𝜎(𝑡)Ω)cosh2Ω𝑡]. (40) However, since cosh2Ω𝑡≥1, in (39) and (40), then the dispersion cannot be less than ℏ𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡}/2𝑚Ω and (ℏ𝑚𝜔²/2Ω)𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡} in both equations, respectively. Figure 2 shows the uncertainties in the coordinate for the Over-damped Oscillator for various damping factors of 𝛾=0, 0.2Ω, 0.4Ω, 0.6Ω, 0.8Ω, and Ω, respectively.

Figure 2 

The uncertainties in position for the over-damped oscillator as a function of Ωt is governed by the modified Caldirola-Kanai Hamiltonian with dissipation coefficients 𝛾=0, 0.2Ω, 0.4Ω, 0.6Ω, 0.8Ω, and Ω, respectively.

4.3. The Critical Damped Oscillator

The equation for the critical damped oscillator is of the form, when 𝜔=0, 𝑞(𝑡)=𝐴_𝑘+𝐵_𝑘𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}. (41) Subjecting (41) to continuity condition [8], 𝑖𝑞(𝑡)=1+2(1−𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}). (42) The uncertainty in the coordinate in space is given by ⟨𝑞²(𝑡)⟩=ℏ²1[1+4(1−𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2})²], (43) and dispersion in the momentum counterpart is ⟨𝑝²ℏ(𝑡)⟩=4𝑒^{(2−𝛽)𝛾𝑡cos𝛽𝛾𝑡}𝜎(𝑡)²(𝑡). (44) We observed in (43) that 𝑒^{−𝛽𝛾𝑡cos𝛽𝛾𝑡/2}<1, so that the product of dispersion becomes ⟨Δ𝑝Δ𝑞⟩=ℏ𝛽𝛾2[cos²𝛽𝛾𝑡−2𝛽𝛾𝑡sin2𝛽𝛾𝑡+𝛾²+𝛽²𝛾²𝑡²sin²𝛽𝛾𝑡]^1/2. (45) When the variable parameter is set to unity, Figure 3 shows the variation of 𝜎²(𝑡) in (44) for various damping factors.

Figure 3 

Variation of 𝜎(𝑡) with 𝛾𝑡 for the critical damped oscillator with various damping factors of 0, 0.5Ω, and Ω.

5. Conclusion

We have evaluated within the frame of Caldirola-Kanai model the damped harmonic oscillator for different damping regimes. Here, we obtain the modified Caldirola-Kanai Hamiltonian and show that the undamped regime 𝛾<𝜔 satisfied the uncertainty relation with the chosen variable parameter 𝛽 being set to unity. In the region of strong and critical damping, Heisenberg uncertainty relation is violated even when this variable parameter is set to unity.

Acknowledgments

This work was supported by the Imienyong Nandy and Leabio Research Foundation under Grant no. INL-743-214.