On the Hamilton-Jacobi method in classical and quantum nonconservative systems

In this work we show how to complete some Hamilton-Jacobi solutions of linear, nonconservative classical oscillatory systems which appeared in the literature and we extend these complete solutions to the quantum mechanical case. In addition, we get the solution of the quantum Hamilton-Jacobi equation for an electric charge in an oscillating pulsing magnetic field. We also argue that for the case where a charged particle is under the action of an oscillating magnetic field, one can apply nuclear magnetic resonance techniques in order to find experimental results regarding this problem. We obtain all results analytically, showing that the quantum Hamilton-Jacobi formalism is a powerful tool to describe quantum mechanics.


I. INTRODUCTION
In 1924 the physicist Max Born put forward for the first time the name "quantum mechanics" in the literature [1]. In that work, quantum mechanics denoted a theoretical framework of atomic and electronic motion, which was understood in the same level of generality and consistency of the classical mechanics laws. Approximately one year after that work, in 1925, the historical paper presented by Heisenberg and entitled "Quantum-theoretical reinterpretation of kinematic and mechanical relations" [2] has shown a new quantum-theoretical quantity which contains information about the measurable line spectrum of an atom. Motivated by Heisenberg's work, Born, Jordan and Heisenberg published the articles "On quantum mechanics" [3] and "On quantum mechanics II" [4], which were the first comprehensive explanations of quantum mechanics. It is worth mentioning that those works have been performed using the matrix framework.
On the other hand, Dirac formulated independently a consistent algebraic framework of quantum mechanics [5], where the equations were obtained with no use of matrix theory.
However, it was only in 1926 that the Schrödinger formalism (SF) appeared in the literature. Since then, day after day, several problems linked to quantum mechanics have been analyzed rigorously in the literature [6][7][8][9][10]. Formal developments have been arisen, in particular to deepen the comprehension regarding quantum fields. Quantum canonical transformations have attracted interest since the incipient development of the theory about one century ago.
Although the SF is a prevailing framework, alternative formalisms emerged. For instance, the path integral formulation plays a prominent role in quantum field theory [12].
The basic postulates of a third version for the study of quantum mechanics have also been proposed, namely a quantum version of the Hamilton-Jacobi formalism [13], where a better understanding of the quantum HamiIton-Jacobi theory and its consequences was presented.
Moreover, in that work the authors have shown applications of the quantum Hamilton-Jacobi formalism (QHJF) for the calculation of the propagators of the harmonic oscillator potential and of the same potential with time-dependent parameters. Here, it is important to highlight that Leacock and Padgett (LP) [14] and independently Gozzi [15] are a few names who have worked this formalism out. For instance, LP developed the QHJF for the case of conservative systems, where the main feature of their theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamic equation [14]. On the other hand, Castro and Dutra (CD) have obtained the QHJF through basic postulates similar to the case of the Heisenberg picture [13]. An important feature in CD's work is the straightforward equivalence of the QHJF with both the Feynman and Schrödinger formalisms.
Currently we can find in several areas of physics a considerable amount of works dedicated to the studies of the QHJF. Among the different research areas, we can find an interesting connection of quantum Hamilton-Jacobi theory with supersymmetric quantum mechanics (SUSYQM) [16,17]. In this case, the quantum momenta of supersymmetric partner potentials are connected via linear fractional transformations. Moreover, in the SUSYQM context, it has been shown by Dauod and Kibler a connection between fractional and ordinary SUSYQM [18]. Another line of investigation comes from one-dimensional scattering problems in the framework of the QHJF [19]. In addition, Roncadelli and Schulman solved the quantum Hamilton-Jacobi equation, by a prescription based upon the propagator of the Schrödinger equation [20]. It provided the use of quantum Hamilton-Jacobi theory, developing an unexpected relation between operator ordering and the density of paths around a semiclassical trajectory. Related to it, black hole tunnelling procedures have been placed as prominent methods to calculate the temperature of black holes using the Hamilton Jacobi technique in the Wentzel, Kramers, and Brillouin (WKB) approximation [21][22][23]. Various types of black holes have been studied in the context of tunnelling of fermions and bosons as well [21][22][23][24]. Tunnelling procedures are quite well used to investigate black holes radiation, by taking into account classically forbidden paths that particles go through, from the inside to the outside of black holes. Moreover, quantum WKB approaches were employed to calculate corrections to the Bekenstein-Hawking entropy for the Schwarzschild black hole [25].
As we can see in [26], the problem of the electron quantum dynamics in hydrogen atom has been modeled exactly by QHJF, where the quantization of energy, angular momentum, and the action variable are originated from the electron complex motion. In addition, the shell structure observed in hydrogen atom arises from the structure of the complex quantum potential, from which the quantum forces acting upon the electron can be uniquely determined.
Moreover, much has been learned regarding the QHJF in the last years, when several developments have been accomplished in the literature. These include the definability of time parameterization of trajectories [27], corrections for any soliton equation for which action-angle variables are known [28], lattice theories [29], gauge invariance in loop quantum cosmology [30], treatment of the relativistic double ring-shaped Kratzer potential [31], shape invariant potentials in higher dimensions [32], application to the photodissociation dynamics of NOCl [33], and Dirac-Klein-Gordon systems [34].
Furthermore, Vujanovic and Strauss [35] developed a series of calculations using the classical Hamilton-Jacobi method to study linear nonconservative systems. In order to obtain solutions for the cases studied, the authors used an expression for the classical action that contains only the quadratic term, which reads: Despite this term does not alter the classical solution, here we shall show that it does not hold for the quantum mechanical case. In fact, when quantum systems are approached, we shall study the Hamilton's principal function S given by a polynomial of x, which is written in the form In fact, the linear term is necessary for the development of the quantum propagator. Hence, this term can not be neglected when quantum solutions are regarded. In addition, in order to deal with a more interesting application from the point of view of QHJF, we will study the problem of an electric charge in an oscillating pulsed magnetic field [38,39].
This paper is organized as follows. In the next section, we present a complete review about the QHJF and its basic postulates. In Sec. III, we show an illustration of the QHJF to the standard case of the harmonic oscillator. In Sec. IV, we apply the ideas to analyze the driven oscillator case. Section V is devoted to the resonance example. In Sec. VI, we show an application of the Hamilton-Jacobi formalism to the problem related with the quantum dynamics of an electric charge in an oscillating pulsing magnetic field. We end up with some general remarks and conclusions in Sec. VII.

II. A BRIEF REVIEW ON HAMILTON-JACOBI FORMALISM
In this section we will present a review about the QHJF and its basic postulates. We present a prescription for obtaining the QHJE from the classical one. At this point, it is important to remark that this approach is analogous to the Heisenberg prescription, which makes a link between the Poisson brackets and quantum commutation relations. Here, we follow the work presented by CD [13], and revisit the QHJF as well.
Let us start by remembering that the Hamilton principal function, or action, S cl , is a generating function of the canonical transformation ( r, p) → ( r′, p′), which generates new time-dependent variables r′ and p′ with null Hamiltonian. In this case, the classical Hamilton-Jacobi equation reads where ∇S cl = p. It is worthwhile to point out that the above classical Hamilton-Jacobi equation provides a successful form for establishing the equations of motion of a mechanical system.
Following the approach given in [13], where the authors used classical mechanics as a short wavelength limit of wave mechanics, and by taking into consideration the similarity with the electromagnetic quantities and their limits to geometrical optics, it was postulated that the quantum wave amplitude has the form where S is the quantum Hamilton's principal function, or complex action, ℏ represents the Planck constant and 2 −1/2 is a factor introduced for convenience. Therefore, the action S can be realized as a phase of the wave motion process. In order to accomplish the transition from the classical Hamilton-Jacobi equation to the quantum case, one defines the momentum in the operatorial form, given by Hence the classical momentum is obtained in the limit ℏ → 0, where the commutation relations are established. Thus, when the Hamiltonian has the standard form one can find, using (3) and (5), the following quantum Hamilton-Jacobi equation (QHJE): In the next sections we will show how linear, strictly nonconservative, oscillatory systems with one degree of freedom may be analyzed within the quantum Hamilton-Jacobi framework. The motivation for this study is that linear dissipative systems, possessing even one degree of freedom, have not been analyzed in the context of the quantum Hamilton-Jacobi method, despite of its practical, theoretical, and pedagogical interests.

III. HARMONIC OSCILLATOR
A particular important physical system is the harmonic oscillator. There exists a large number of important physical applications for it, such as the vibrations of the atoms of a molecule about their equilibrium position or even an electromagnetic field, for instance. In fact, whenever the behavior of a physical system in the neighborhood of a stable equilibrium position is studied, one obtains equations which, in the limit of small oscillations, are those of a harmonic oscillator.
Let us start our study with a straightforward example of the harmonic oscillator. The associated quantum Hamilton-Jacobi equation is provided by [13] ∂S ∂t The substitution of (2) into the QHJE (8) generates a polynomial equation leading to a system of first-order coupled differential equations for the arbitrary coefficients introduced in (2). The polynomial equations can be split into the following set of first-order non-linear differential equations:α (t) + α 2 (t) + ω 2 = 0, yielding the general solutions α(t) = −ω tan(ωt + c 1 ), where c 1 , c 2 and c 3 are arbitrary integration constants. Hence a complete solution of (1) is given by It is worth to emphasize that in the limit → 0, the classical Hamilton's principal function is reobtained. The general solution for the classical case of the Hamilton-Jacobi equation can be obtained from the constraint ∂S ∂c 1 = B, where B is a constant. Furthermore, it is straightforward to verify that the classical solution is given by By analyzing the classical case for Eq. (11), the solution can also be immediately determined by ∂S ∂c 2 = B. Thus, in this case the classical solution contains two integration constants, as it should be expected, since the equation of motion is a second-order one.
Moreover, by using Eq.(11), the solution for the problem consists in obtaining the quantum propagator, by imposing the following boundary condition [13] S(x, 0) = kx.
The concept of propagators is of great importance in quantum physics and in the Feynman's formulation, particularly. All the time evolution of a given system may be obtained through the propagators [13]. They are used mostly to calculate the probability amplitude for particle interactions using Feynman diagrams.
The propagator can be obtained by considering a physical wave packet where S k (x, t) denotes the quantum Hamilton's principal function if the boundary condition S k (x, 0) = kx is taken into account. Inserting the Fourier transform Φ(k) = dxΨ(x, 0) exp(−ikx) in Eq. (14) yields where the propagator reads We observe that the constant c 2 is related to the term that generates the quantum propagator. It is important to remark that this constant appears in the linear term of Eq.(2).
Hence we conclude that the linear term must also compose the principal Hamilton function, in order to construct the quantum propagator.
By substituting the solution and the initial conditions imposed to the expression of the propagator and integrating in k, one gets The quantum propagator can be alternatively constructed [13], by imposing that where S represents the quantum solution of the equation of Hamilton-Jacobi.
The propagator must satisfy the condition where δ(x −x) represents the Dirac delta function. For our purposes it is useful to employ the following representation: By using Eqs. (18 -20), we determine By substituting Eq.(21) in Eq.(18), the propagator is reduced to the form presented in (17).
We emphasize that the linear term in S is quite necessary. Hereon we are going to implement this approach in similar cases which, up to our knowledge, have not been taken into account in the literature, at least from the point of view of the quantum Hamilton-Jacobi formalism.

IV. DRIVEN OSCILLATOR
Driven harmonic oscillators are damped oscillators further affected by an externally applied force. The potential of a driven harmonic oscillator can describe many phenomena in physics, such as superconducting quantum-interference devices [36] and magnetohydrodynamics [37].

Its classical equation of motion reads
x + ωx 2 = h cos(Ωt), (22) and the corresponding Lagrangian can be written as where f (t) = h cos(Ωt) The following Hamiltonian is then derived: Hence, the Hamilton-Jacobi equation assumes the form and the principal Hamilton function is represented by By substituting Eq. (26) in Eq. (25), the quantum Hamilton's principal function reads The limit → 0 leads to the classical case, and the solution is obtained by imposing that ∂S ∂c 1 = B, implying that Our result can be led to the one in [35], with some mathematical manipulations. The above solution can also be obtained by imposing ∂S ∂c 2 = B. For the quantum case, once again the condition S(x, 0) = kx (29) shall be imposed, what implies that Remembering that f (t) = h cos(Ωt) , and imposing the described conditions in (29) and (30), the propagator reads From the initial condition of the second method, the propagator can be obtained if we choose which lead to the result in (31).

V. RESONANCES
Resonance occurs when a given system is driven to oscillate by another vibrating system with greater amplitude at a specific preferential frequency. They occur with all types of waves, such as mechanical, electromagnetic and quantum wave functions.
The Lagrangian reads whereas the Hamiltonian is given by Hence the corresponding quantum Hamilton-Jacobi equation becomes Considering the Eq.(26), where f (t) = ht 2ω sin(ωt) , and applying it in Eq. (36), it follows that In the classical case we have the solutions By imposing the condition the quantum propagator for the resonance reads K(x, t;x, 0) = ω 2πi sin(ωt) On the other hand, if we try to construct the propagator from the initial conditions procedure, we find With these values, the propagator (18) is led to the form given by Eq.(40).

VI. ELECTRIC CHARGE IN AN OSCILLATING PULSED MAGNETIC FIELD
In this section we show an application of the Hamilton-Jacobi formalism to the problem related with the quantum dynamics of an electric charge in an oscillating pulsed magnetic field [38]. It becomes important then to analyze, through a parallel formalism, the validity of the solutions presented, since the systems can describe experimental measurements in nuclear magnetic resonance techniques [39].
We consider an electric charge e in an oscillating pulsed magnetic field given by The Lagrangian for a charge in an electromagnetic field reads where A = − 1 2 r × B and φ( r) denotes the scalar potential. The Hamiltonian is usually written as or explicitly, as where γ ≡ e m . By substituting p = ∇S + i and H = − ∂S ∂t , yields where the Hamilton principal function reads It is worth to realize that in the limit when goes to zero we obtain the respective classical Therefore, after resolving the corresponding set of non-linear differentials equations, the quantum Hamilton principal function reads The solution consists in obtaining the quantum propagator if we impose the following boundary condition [13]: Hence we obtain Now, using substituting the solution which is in accordance with the initial conditions imposed to the expression of the propagator and integrating in k, we arrive to K(x, y, z, t;x,ỹ,z, 0) = mω 2πi sin(ωt) It leads to a two-dimensional oscillator in the plane xy and a free particle in the direction 0z.
On the other hand, the problem of an electric charge in an oscillating pulsed magnetic field can be approached through SF. In fact, the Schrödinger equation reads where µ is a magnetic moment and is represented, according to the reference [39], by µ = γ L, where L represents the angular momentum. Now, we perform a rotation in the reference system where the z axis is stationary, namely x =x cos(δt) −ȳ sin(δt) , y =x sin(δt) +ȳ cos(δt) , Hence the Schrödinger equation reads For an effective static field, Therefore, the possibility suggested by the authors of the reference [38] is not valid for the studied system, although it is correct for a differential equation of first-order.
Rewriting the expression of the magnetic field (42) only with the part oscillating in the x direction it implies that and consequently we getv where ξ 2 = ω 2 0 −α 2 µ 2 . In this form the problem was transformed into a classical harmonic oscillator with time-dependent frequency. We can particularize this problem by requiring that ξ = const, thus obtaining the solution v = A cos(ηT + δ), so that and s = A −1 sec(ηT + δ).
Therefore, it is easy to check that the conditions (82) are true and, therefore, the problem is reduced when a particular case is required.

VII. CONCLUSION
We studied classical and quantum solutions for harmonic oscillator-like systems, further encompassing the driven case and with resonances as well, by using the Hamilton-Jacobi method. For the quantum case, the propagator allows to study the time evolution of the system, if we take into account the Hamilton's principal function with a linear term. This term is shown to be essential to obtain the respective quantum propagators of the systems studied. Therefore, it can be verified that the Hamilton-Jacobi quantum formalism is an alternative version for the quantum mechanical formulation, obtaining the classical limit when → 0.
After that, we computed, through this approach, the propagator for an electric charge in a oscillating magnetic field. Since we observed that the Schrödinger approach to this problem in the literature presents a technical flaw, we computed its solutions also through the SF.