On the Quantization of One-Dimensional Conservative Systems with Variable Mass

The Hamiltonian associated to the mass variable system is constructed from first principles through finding a constant of motion of the system. A comparison is made of the classical motion of a body with its mass position depending in the (x,v) space and (x,p) space which are defined by the constant of motion and the Hamiltonian, for a particular model of mass variation. As one could expected, these motion looks different on these spaces. The quantization of the harmonic oscillator with this mass variation is done, and a comparison is made by using the usual Hamiltonian approach with the proposed quantization of the constant of motion approach. This comparison is done at first order in perturbation theory, and one sees a difference between both approaches which can, in principle, be measured.


Introduction
Mass variation problems for classical mechanics has a long history [1] and important applications have been studied on the dynamics of the Universe as black hole formation [2,3], where they have been known as Gylden-Meshcherskii problems [4][5][6][7][8][9][10][11].These types of problems are not free from controversy in the way they must be formulated and their relation with Galileo's transformation [12].In addition, these systems are becoming more and more important in quantum mechanics systems since the discovery of the neutrinos mass oscillations problem [13] and [14], the kinetic theory of dusty plasma [15], propagation of electromagnetic waves in a dispersivenonlinear media [16], and other possible applications in fluid dynamics [17].
The approach used so far to study these systems follows guessed Lagrangian or Hamiltonian with include the mass variation of the system [18][19][20].Above all, there is not right mathematical justification for this approach.Therefore, one may considered that to find the Hamiltonian for a mass variation system, one must do it from first principles, Newtonian's mechanics, and this can be done by using a well known approach for one dimensional autonomous system to construct the associated Lagrangian and Hamiltonian of the system [21][22][23].Once these expressions are gotten, one can proceed to make the quantization of the system [24].On the other hand, there has been a proposed extension for the non relativis-tic quantum mechanics for autonomous systems based on the use of the constant of motion instead of the Hamiltonian in the Schrödinger equation [25].Therefore, in this paper the Hamiltonian and the constant of motion are found for several conservative systems with position depending mass, using a model for the mass variation.The trajectories of the motion on the (x,v) and (x,p) spaces are shown to see the difference of this description.The quantization of the mass position depending systems is analyzed using the constant of motion and the Hamiltonian approaches, and finally, the spectrum of the harmonic oscillator are calculate at first order perturbation theory.

Constant of Motion and Hamiltonian
Consider a one-dimensional motion of a body with mass position depending,   = m m x , and which is affected by a conservative force   F x .This system is governed by Newton's equation of motion [1]  where v represents the velocity of the body.Since one has the expression where one has defined = d d m x x m , the equation of motion can be written as the following system which in turns, this equation can be written as the following autonomous dynamical system where  has defined as This autonomous system is dissipative for and anti dissipative for because of the quadratic term in the velocity.A constant of motion of this system is a function K x v which satisfies the first order partial differential equation The general solution of this equation is given by , where G is an arbitrary function, and C is the characteristic curve given by Suppose of the form , where one demands that .In addition, assuming that , one gets the usual constant of motion of the conservative system with constant mass (so called "Energy of the System").Then, one can select the functionality of G of the form such that under the mentioned conditions for g, one gets the energy of the system as a particular limit.The resulting constant of motion of the system is Given the constant of motion of a one-dimensional autonomous system, the Lagrangian for this system is determine through the well known expression ( [21][22][23]) Using this expression with Equation (9), it follows that The generalized linear momentum, , is given by (12) H vp L and the Hamiltonian of the system,  , is deduced as Since from Equation ( 5) one has that The constant of motion, Lagrangian, generalized linear momentum, and Hamiltonian are written as and Defining the effective potential as the constant of motion and Hamiltonian have the following form and To be able to continue with analysis, one needs a model for the variation of mass, .Let us assume that       As one can see from these plots, the effect of the first terms in Equations ( 20) and ( 21) is clearly marked on these trajectories.The constant of motion brings about some "regular" behavior meanwhile the Hamiltonian brings about some type of little odd behavior due to expression (17).
From Equations ( 23), ( 20) and ( 21) one sees that the constant of motion and the Hamiltonian can be written as and where the terms and This form of writing the constant of motion and the Hamiltonian is suitable for quantization studies.Before leaving this classical part, it is necessary to make some observations about the motion of the body and its mass position dependence.First, in our mass position dependence model, Equation (22), it has been assumed that the ptotic (   < 0 m x for any position "x", the damping-antidamping effect will appear.

Quantiz n with Mass Position Depending Systems
The usual Schrödinger quantization approach is based on the association of an Hermitian op r to the Hamilton function, [27,28], and the solution of a linear complex partial differential equation for the wave function, is the Hermitian o associated to the generalized linear momentum, or associ d where v is the Hermitian operat ate to the velocity, , satisfying the obvious commutation These realtions can be assumed to be valid for position mass depending as Of course, for constant mass conse there is not difference at all between both approaches he nd the velocity r, as we have seen previously, for mass position depending systems this relation is not trivial any more, Equation (17).Therefore, to fin for the quantization of the constant of motion to make ph rs, and to see experimentally whether or not it makes sense.Mass position depending systems have indeed this property because of th tion (17).For autonomous systems, one does not need, of course, to fully solve the Equations ( 30) and (31) to see whether or not there is a difference on both approaches.To see this, it is enough to look at their spectra, and this spectra can e be calculated at first order in perturbation theory.Even more, one can see these spectra just a second order in the Taylor expansion of the mass position depending.Doing this with the Equations ( 25)-( 27), one gets and where the function    W x is defined as To associate a Hermitian operator to these functions, one knows that there is an ambiguity, not resolved yet by any experiment, on selecting a proper Hermitian ope to make the quantization.Although one could norm follow Weyl approach [29] and [30], it is easier to take wing approach by noticing the following for polynomials operators: given the Herm operators rator ally the follo itian A and B to the functions A and B rator , the ope  is an Hermitian operator for any mber "n ing H associated to the product of functions integer nu ".In this way, the follow ermitian operators can be ) Taking the following identifications ˆ= (39 A x and ˆ= B p or v , it follows that the Hermitian operators associated to the expressions (34) and (36) are given by Using the commutation relations   , = x p i I  and  gets the following expressions for the constant of motion and Hamiltonian operators and where the functions   x and   h x , and the operator f   , D x f v have been defined as Then, at first order perturbation theory, the eigenvalues would be given by and  , and ĉtation value of the I K and ˆI H represent the remaining terms of the Equations ( 43) and (44),

Harmonic Oscillator with Variable Mass
The harm 1 = 0 m in the Weyl algesis , ,   (54) onic oscillator with   † bra ba I a a [31] has the following characteristics with the following identifications Note that all the expectation values of monomial term of odd power have zero values.The expression for  W n by  up to fourth order in "x" is give Thus, using the expectation values given in the ap e terms appearing in Equations ( 53) and (54) can be calculated, resulting the following eigenvalues at first order and where one has used the definition = n n n   .Let us define the following parameter J as This parameter represents the relative variation of the eigenvalues of the constant of motion quantization and the Hamiltonian quantization approaches.the eigenv be relavely large.Thus, this suggest the it can be observable perimentally, opening up the possibility to see whether or not the quantization of constant of motion makes sense for mass variable quantum systems.

Conclusion
A mathematical consistent approach has been used to deduce the constant of motion, Lagrangian, and Hamiltonian for a mass position depending non relativistic classical systems.The trajectories on the spaces ( , alues for both approaches (J) could ti ex  d x v ).In addition, an study was made about the quantization of the constant of motion, as an extension approach of the usual Hamiltonian quantization approach The harmonic oscillator with mass position depending was used for this study.One observed that, already, at first order perturbation theory, a significant difference on the spectra of the constant of motion and Hamiltonian approaches can be significant, bringing about the possibility for this difference to to be observed experimentally.
the asymptotic distance where the mass would be the Hook's law (harmonic os )
 ,  is the Plank constant divided by 2π , having the confutation relation   , = x p i I  (with "I" the identity o basis perator) in the   , , I x p of the Weyl algebra.However, as a ion fun for the wave function, possible extension of this quantization approach, there is the proposal of using the constant of motion, instead of the Hamiltonian, in the Schrödinger equation, where an Hermitian operator is associated to the constant of mot ction, and the Schrödinger like equation is solved ven rvative systems, since t relation between the generalized linear momentum a of the body is really trivial, = o p m v. Howeve d out the possibility ysical sense, it is necessary to look for a system where both approaches diffe e Equa-
x p ) were given for constant force, Coulomb type force, and Hook force with a chosen model for the mass variation.The dependence of the generalized linear mo mentum w f the ody makes the plots in the space ( , ith respect the position and the velocity o x b p to look quite ifferent from those in the space ( ,