Quantization and Stable Attractors in a DissipativeOrbital Motion

We present a method for determining the motion of an electron in a hydrogen atom, which starts from a field Lagrangean foundation for non-conservative systems that can exhibit chaotic behavior. As a consequence, the problem of the formation of the atom becomes the problem of finding the possible stable orbital attractors and the associated transition paths through which the electron mechanical energy varies continuously until a stable energy state is reached.


Introduction
In this paper we present a new method for dealing with quantization problems which is based, on the one hand, on the concept of a stable attractor associated with a non-linear differential equation from the usual chaos theory and, on the other hand, on the variational formulation of Quantum Mechanics introduced by E. Schrödinger in 1926 [1].That is, our approach is not based in the current and well-known method of phase space representation in the semi-classical limit of quantum mechanics, usually known as "quantum chaos".
The theory of quantum chaos was pioneered by Einstein through his 1917 [2] paper, in which he made a connection between classical and (old) quantum mechanics.This theory was further improved by many authors, among which the works of Gutzwiller [3][4][5][6][7] and Ozorio de Almeida [8][9][10][11] have made major contributions.In particular, Gutzwiller obtained in 1967 [3] the exact wave functions for the bound states of the hydrogen atom, by performing a very complicated calculation using a phase integral approximation of a Green's function in momentum space.
We follow an alternative approach in this work, in which we show that it is possible to obtain the exact energy of the bound states of the hydrogen atom by searching for stable orbital attractors in a non-conservative Hamilton-Jacobi dynamics [12][13][14].Thus, the quantiza-tion problem is solved by selecting, from all of the possible electron paths in which energy is dissipated, those that tend to stable closed paths in which bound states of motion are reached in the limit as time approaches infinity, that is, to stable orbital attractors.This is done in Section 2 where we are conducted from the well-known linear Schrödinger equation to a non-linear momentum equation.This equation will be shown to generate the dissipative dynamics and allow the existence of a set of stable attractors which prevent the collapse of the system.In Section 3 we solve the equations for the hydrogen atom obtaining the form of the dissipative energy function along the electron trajectory, in which the mechanical energy varies continuously until a stable attractor is reached, when it becomes finally constant.

The Equations of Motion
We start by considering the Hamiltonian function for an electron which is considered as an ordinary charged particle, whose motion is caused by a scalar potential energy function 2 V e   r and also that no vector potential is present, i.e., Of course, since in general the system irradiates continuously, the Hamiltonian (1) cannot be a constant of the motion.Therefore, at first sight, the motion is always dissipative, the electron tending to fall into the proton, whose position may be called a collapsing attractor of the process.Hence, we shall search in this work for the possibility of non-collapsing stable attractor paths toward which the electron motion can tend asymptotically as time goes to infinity.
Since is a spherically symmetric potential, this implies the conservation of the angular momentum vector which defines a conservative or Hamiltonian system in Classical Mechanics.It can immediately be seen that in this case ( 2) is a non-linear equation in both and , so that Classical Mechanics is in its deepest grounds a non-linear theory.The closed paths that are solutions of Equation ( 2) are elliptic orbits which may be obtained by integrating it , as a function of the polar angle  .Alternatively we may also observe that the only way that Equation (2) may be satisfied for any value of the continuous variable  is by a composition of periodic sinusoidal functions of the form which substituted into Equation (2), and with the help of the definition of the angular momentum, results respectively in the following inverse average radius and eccentricity formulae A third way to address the Hamiltonian problem, which is followed in usual Lagrangian Classical Mechanics, is to make use of a variational procedure to transform the non-linear quadratic form given in (2) into an ordinary linear second-order differential equation in 1 r , whose solution is given exactly by the function in (3) (see chap.2 of Ref. [15]).We shall follow a similar approach in this work.
It is a well known fact in Classical Mechanics that the motion in any path corresponding to (3) is unstable against energy loss by radiation, so that the electron in fact follows a decreasing spiral motion toward the proton position.In order to look for a stable orbital attractor, that is, an orbit in which the motion can be stable against a loss or gain of energy by emission or absorption of radiation, we allow the Hamiltonian function in (1) to vary along a virtual path and try to get a special state of motion in which a loss in energy in a region of space may be compensated by the absorption of energy in another region, producing a self-restoration effect in the Hamiltonian, so that, no net loss of energy occurs overall and, therefore, the system becomes dynamically stable.
Thus, we consider along the path given by the classical linear momentum p, which is the solution of ( 1) with

H E
 .This variation momentum must satisfy the asymptotic limit , as , that define the stable attractors which we are looking for.At this limit, we get back to the p path at a matching point  , but with specific values for the parameters and 0 which determine the specific ellipses that make the system stable or self-restoring.

E L
In order to accomplish this, we shall observe that, due to the emission of radiation, the finite difference must approach zero if H E  as t .Also, in the absorption process, Equation (5) must approach zero if In any case, (5) must be expressed as a quadratic form which is suitable for the variational procedure.This is made by introducing a variation function through a transformation proposed originally by Schrödinger [1]: where is the rationalized Planck's constant. The Lagrangian density function we need is then obtained by considering 2   times the difference between the Hamiltonian H and the energy attractor E. After substituting (6) into (5) and using (1) we obtain Here, is a quadratic form of the function  and its space derivatives, so that the variation of the volume integral of Q conducts to a partial linear differential equation, as expected in a Lagrange's variational problem.
We also need to impose the constraint that the function  must be square integrable or normalizable, because 2  stands for an averaging weight function where, for simplicity, the unity value for the normalization constant has been assumed.Let us consider now the calculation of the volume integral over all space.If there is a finite loss or gain of energy due to radiation, H E  is a finite quantity too.In order to avoid collapse of the system I must be a finite constant.In order to assure that, it is enough that Q is limited at the origin and tend to zero as the space volume tends to infinity.Therefore, in order to allow the existence of stable attractors, we shall impose that I must have an extreme value near zero 1 so that its variation vanishes, namely It will be seen in what follows that only trajectories which tend to a closed path as time goes to infinity will satisfy the variational problem, reaching a stable attractor path.Equation ( 9) cannot be satisfied if we consider either the free electron motion or the motion in a scattering process, since such motions are remarkably unbound, and therefore cannot satisfy (8).Now, by introducing the Lagrangian density, (7), into (9), we get In the calculation of the variations, usual integration by parts has been made and dF is the surface element vector.In order to satisfy (10), it is sufficient to require that the integrand in the volume integral and the surface integral vanish separately, namely and Equation ( 11) is the variational form of the Schrödinger equation and ( 12) is automatically satisfied by the requirement that the variation   vanishes at infinity, where the surface integral is calculated, although it would also be asked to vanish on a finite surface 2 .This is guaranteed by the constraint (8), which implies that  as well as   vanish at infinity.Now, from (11), we can obtain an equation for the varied linear momentum of the electron through (6) and the identity 2 which substituted into Equation ( 11) results in From this non-linear momentum variation equation for a non-conservative system, which is analogous to (2) for a conservative system, the electron trajectories resulting in the stable attractor mentioned above will be obtained.Thus, Equation ( 11) is the linear differential equation associated with the non-linear momentum equation, (14).

Determination of the Electron Path Functions in the Formation of a Hydrogen Atom
We can now perform the reduction of both the equations of motion, ( 11) and ( 14).Starting with the former, we consider a variation in path with a constant angular momentum 3 0 L   , so that .This value assigned to L is provisory because the actual value will come from the specific orbits to be found.Thus, depends only on the radius and hence (11) becomes where the numeric factor will be determined later. The radial variation may be obtained through (6) as Now, we remember that the divergence of a vector u in polar coordinates is through which we can reduce (14) to its radial form, namely This will be considered in a future work concerning discrete transitions. 3 We will consider a variable angular momentum in a future work in connection with the transition between energy levels. 1 It would be exactly zero for a Hamiltonian system, since in this case H = E always.
where is the above-mentioned parameter. Since ( 18) is non-linear, it is not a simple task to obtain its solution directly.Instead of this, we shall employ the radial solutions of the linear (15), and generate solutions for the (18), through (16).In its simplest form the solutions for (15) that are regular at the origin and at infinity can be written as [16]   which, after substitution into (15) and equating coefficients in the same power, yields where 2 2 2 13.6 eV 2 2 is the ionization or Bohr energy of the hydrogen atom.We see immediately that the only possible values for the parameter which make the above expression for the energy levels to agree with the experiment are the half-integers4 : Thus, the solution of (18) obtained through (19) becomes Therefore the simplest stable attractor condition is given by and the possible stable attractor radii are given by in which 0 assumes the values 0.5, 3.0 and 7.5 a.u., respectively.We note that for 0 , we have   so that the three curves correspond to decreasing spirals toward 0 .On the other hand, for 0 , we have , so that the three curves correspond to increasing spirals also toward 0 , therefore illustrating the workings of the self-restoring effect.Thus 0 is really an asymptotically stable orbital point, i.e. a stable attractor.
By considering now the solution (21) of the non-linear equation (18) and again comparing coefficients in the same power, we obtain the same energy levels specified by (20).And from the radii given in (22) we can calcu which, once inserted into Equation ( 18), yields the conservative form We see that, for has the same classical conservation form as (2), which is consequently the actual value of the angular momentum.Therefore, the resulting conservative orbit is described in phase space by the twofold function The plot of (25) together with the radial variation (21) are shown in Figure 2 for , from which we can note that 0 (= 0.5 a.u.) is the intersection point of the paths because it is their only common root.r  In addition, is not the only root of (24) or for the condition 0 r  0 r p  in (25); the other root being so that 0 and 0 r r  become respectively the semiminor and semimajor axes of the ellipse which satisfy (25) and, thus, it is the orbital stable attractor searched for, where 1 and The corresponding average electron position for a circular motion is given by the average value of the ellipse semi-axes, which agrees with the Bohr model: In order to obtain the actual electron trajectory we must integrate the varied radial linear momentum together with the constant angular momentum, from which we readily obtain the equation for the electron trajectory whose corresponding integration constant is undetermined, since the followed trajectory depends on the initial conditions.This means that if we consider for example , we would have and the path, Equation (31), would become a decreasing spiral.
On the other hand, for we would have

Conclusions
In this paper we have introduced a different chaotic approach in which we show that it is possible to obtain the exact energy of the bound states of the hydrogen atom by looking for stable orbital attractors in a non-conservative Hamilton-Jacobi dynamics.In it, a variable energy process tends asymptotically toward an energy stable attractor Copyright © 2011 SciRes.

m
are the electron and proton masses, respectively.

Figure 1 .
Figure 1.The radial varied linear momenta
and the path, (31), would become an increasing spiral.In both cases they converge toward the stable attractor o r r   .The plot of the ellipse, (27), and the spirals, (31), is shown in Figure 3 for 1 2   .We can immediately write the electron energy function, (1), in polar coordinates as  is in Ry and   r  is in a.u.The plot of   H  is shown in Figure4for

H
or first excited energy) of the hydrogen atom.For each value of , the self-restoring process of the stable attractor is quite clear: in the emission range, 0  first decreases from its starting energy to a point of minimum, and then it increases converging to the energy attractors if H E  as   .