Phenomenological derivation of the Schrodinger equation.
The evolution of the wave function in single-particle quantum systems is described by the Schrodinger equation
[[??].sup.2]/2m [[nabla].sup.2] [psi] + V[psi] = i[??][partial derivative][psi]/ [partial derivative]t, (1)
where m is the mass and V is a potential. The complex wave function is generally presented in its polar form
[psi] = [square root of P] exp(iS /h), (2)
where P = [[absolute value of [psi]].sup.2] is the probability density, and S/[??] is a phase. Substituting (2) into (1) results in two equations
[partial derivative]P/[partial derivative]t + [nable] x (P [nabla]S/m) = 0, (3)
[partial derivative]S/[partial derivative]t + [([nabla]S).sup.2]/2m + V + Q = 0, (4)
Q = [[??].sup.2]/4m [[[nabla].sup.2]P/P - 1/2 [([nabla]P).sup.2]/[P.sup.2]] (5)
is known as quantum potential. At the classical limit (h [right arrow] 0) Q vanishes and (4) reduces to the Hamilton-Jacobi equation. For this reason, Bohm  suggested that S is the classical action function, which relates to the actual velocity, v = [nabla]S /m, of the particle. In this way (3) simply expresses the conservation of probability.
This alternative way of writing the Schrodinger equation presents advantages as regards its interpretation in terms of classical variables. However, the problem of ignoring the path followed by the particle persists. And more, we have an obvious increase in complexity: The Schrodinger equation is a single function and quite simple, on the other hand, the equation (4) is somewhat complicated - and still requires the continuity equation to account local activities. And above all, thinking that the quantum revolution, highly non-classical, has its origin in a classical equation with an additional potential is not very easy. In reality, Q is not a traditional potential, but part of the description of the motion, that is, P is playing the role of a dynamical variable at the same footing as S. Thus S and P can be said to codetermine each other. However, in approximate schemes to get information about quantum systems it can be used as a potential .
Equation (4) is referred as stochastic Hamilton-Jacobi-Bohm equation. Despite the fact that P is unique for a given quantum system, it is interpreted as a differential equation describing an ensemble of trajectories. This is grounded in the fact that the action S was originally defined as a field variable related with a set of potential trajectories .
It is paid much attention to equation (4) and less concern about (3). From a dynamical point of view, the emergence of the continuity equation is the most remarkable result: It highlights the local loss of determinism ([partial derivative]P/[partial derivative]t [not equal to] 0), is valid for one-particle systems (it was obtained in this way), and contains inherently the multiple path aspect of quantum systems , exactly how is assigned to equation (4).
Fundamentally, to have [partial derivative]P/[partial derivative]t [not equal to] 0 (change of probability at a given position), and thus to justify the emergence of the equation (3), it is necessary that the particle runs local random spatial fluctuations. Otherwise, there are local preferences, and these combined with the dynamics that emanates from the potential V (deterministic) results in a classical trajectory. Therefore these fluctuations require the presence of external forces with special features. Indeed, these forces exist and are related with the zero-point field (ZPF). They are formally treated in the context of the stochastic electrodynamics [5,6], and under certain conditions they may be measured [7, 8]. However, their definition is outside the scope of this work; just let's enumerate its indispensable characteristics to justify the continuity equation in the context of one-particle dynamics.
The above rewriting of the Schrodinger equation starts from propositions valid within the quantum formalism and arrives at seemingly classical equations. What will be done in the present paper is to follow an inverse path. The starting point is the fact that the local changes of the probability density --associated with isotropic random spatial fluctuations impressed by the ZPF--must be related with an energy.
2 The multi path aspect of the motion
Suppose a particle of mass m performing a motion with velocity v. If the associated probability density P is a continuous function of the coordinates and time, then its dynamical evolution along the trajectory is given by
dP/dt = [partial derivative]P/[partial derivative]t + v x [nabla]P, (6)
where [partial derivative]P/[partial derivative]t refers to the change of probability at a given position, and the second term accounts for the spatial changes. As P is a probability, then we cannot precise the angle between [nabla]P and v. Moreover, in principle, [nabla]P can show an isotropic distribution around each position. Indeed, as P is a conserved quantity, then the change of the probability density inside a given volume Q (arbitrary), containing the instantaneous position of the particle, must be equal to the probability flux through a surface A surrounding this volume. Formally, we have
[partial derivative]/ [partial derivative]t [[integral].sub.[OMEGA]] Pd[OMEGA] = - [[integral].sub.A] P[??] x dA, (7)
where [??] is a velocity, and the vector field P[??] represents all possible probability currents that cross the surface A. Obviously, if the particle is inside this volume, it emerges following one of these possibilities. In accordance with the properties of the ZPF, the field P[??] must present an isotropic distribution, however, as the velocity of the particle is dictated by the dynamics of the system as a whole, then there are some privileged probability currents (the resulting motion is not a random walk). According to Green's theorem and equation (7), each one of the possible currents obeys
[partial derivative]P/[partial derivative]t + [nabla] x (P[??]) = 0. (8)
As this process is repeated at all positions where the particle can be found, linking the successive probability currents, according to which the particle emerges from each volume [OMEGA], is defined a path described by the velocity
v = [nabla]S/m, (9)
where S is the Hamilton-Jacobi function of one possible path [3, see p. 36]. Therefore equation (8) must be written as (3).
If the local activities are ignored (classical limit), then the function S is defined on a single trajectory. This also can be easily inferred making [partial derivative]P/[partial derivative]t = 0 in equation (7). In this case the probability flux that enters the volume [OMEGA] equals the one that emerges from it. This means that the particle has only one possibility (probability current) to leave each successive volume [OMEGA].
If the external field acts on the particle everywhere (homogeneously), without preferred directions (isotropic) and in a totally unpredictable (random) way, that is, like the ZPF, then we will have a local motion compatible with the continuity equation. Therefore, as the particle has several possibilities to leave each position (following one possible current P[??]), this assigns a multi path aspect to the motion. This means that the particle can travel on each one of them indiscriminately; there is no preferred path. Note, not having a preferred path means that all are equally probable. We realize that this fact is consistent with the formulation of quantum mechanics in terms of path integrals, where Feynman and Hibbs [4, see p. 28] begin with the following statement: "Now we can give the quantum-mechanical rule. We must say how much each trajectory contributes to the total amplitude to go from a to b. It is not that just the particular path of extreme action contributes; rather, it is that all the paths contribute. They contribute in equal amounts to the total amplitude, but contribute as different phases. The phase of the contribution from a given path is the action S for that path in units of the quantum of action [??]". Coincidently, this is a description of the evolution operator exp (iS/[??]) (unitary), present in (2), which is the core of the path integrals.
3 The main proposition
In a classical system, the particles are actuated by forces in such a way that they move along single predictable trajectories, and this leads to [partial derivative]P/[partial derivative]t = 0 everywhere (the local activities are ignored). By other side, if particles are being actuated by a field, with the characteristics pointed above, local exchange of energy between them occurs in such a way that [partial derivative]P/[partial derivative]t [not equal to] 0. Admitting that this is a fact, let's write an effective stationary action function [S.sub.eff] that, in addition to describing a path through the function S, also takes into account the local activities described in terms of probability density, that is,
[S.sub.eff] = S + [S.sub.l], (10)
where [S.sub.l] is a local action that depends only on P. Following the same formalism obeyed by the stationary Hamilton's function, the energy and momentum of the particle over a possible path are, respectively, written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
p = [nabla][S.sub.eff] = [nabla]S + F[nabla]P, (12)
where F = [partial derivative][S.sub.l]/ [partial derivative]P should be a function of P which must comply the dynamics of the system. Specifically, this function must obey the conservation of probability and the local conservation of energy (the particle cannot extract energy from the field indefinitely).
The motion equations of the system can be obtained in the following way: As S and P are taking values on a volume, then the average energy of the multi path system need to be written in the form
[bar.H] = [integral] [d.sup.3]rPH = [integral] [d.sup.3]rH, (13)
where the integral is taken over whole space. Here, H has the role of Hamiltonian density. With H given by (11) we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
As [bar.H], written in this way, is a functional of the functions S and P, taking the functional derivatives with respect to these functions, according to the well known rules
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
where [x.sub.[alpha]] = x, y, z, t and [xi] = S or P, we obtain respectively
[delta][bar.H]/[delta]S = [partial derivative]P/[partial derivative]t (16)
[delta][bar.H]/[delta]P = [partial derivative]S/[partial derivative]t (17)
This shows that the proposition (10) preserves the shapes of the canonical equations, where S and P behave as dynamical conjugate variables of the canonically transformed Hamiltonian [bar.H] .
Taking into account the momentum (12), the energy (11) can be expressed by
H = [[absolute value of [nabla]S + F[nabla]P].sup.2]/2m + V, (18)
then (13) can also be written as
[bar.H] = [integral][d.sup.3]rP([[absolute value of [nabla]S + F[nabla]P].sup.2]/2m + V). ,19)
and, consequently, the canonical equation (16) takes the form
[[partial derivative]P/[partial derivative]t + [nabla] x (P[nabla]S/m)] + (F + PF') [([nabla]P).sup.2]/m + PF [[nabla].sup.2]P/m = 0, (20)
where F' = [partial derivative]F / [partial derivative]P. The first term, being the continuity equation, is zero, and the trivial solution of the resulting equation gives simultaneously F = cte/P and F = 0. However, if this trivial solution is valid, F is not defined in the field of real numbers.
Generalizing the constant to complex numbers, the non zero solution is written as F = ([S.sub.1] + i[S.sub.0])/P, where S1 and [S.sub.0] are real constants (they have dimension of action). Thus, returning this complex shape of F into (19), from (16), results
[[partial derivative]P/ [partial derivative]t + [nabla] x (P[nabla]S/m)] + [S.sub.1] [[nabla].sup.2]P/m = 0, (21)
which shows that probability conservation is obeyed if F is a pure imaginary ([S.sub.1] = 0). As this occurs independently of the [P.sup.-1] functionality, then it only justifies the complex aspect of the trivial solution of (20).
Another evidence that F is pure imaginary comes from the fact that the momentum (12) is apparently incompatible with the actual velocity (9); it seems that we should have
v = [nabla]S/m + F [nabla]P/m. (22)
In reality, this behavior is not entirely unexpected, since, as we saw earlier, the actual velocity is the end result of the system dynamics as a whole, that is, S is also dictated by the local activities. Therefore, to reconcile these equations, F shall be such that (9) refers to the real part of (22).
The resulting apparent complex character of the energy (11) and the momentum (12) is only a stage of the calculations. In effect, the canonical equations (16) and (17) can also be obtained even making
[integral][d.sup.3]rP(-F [partial derivative]P/[partial derivative]t) =0 (23)
in Eq. (14), which makes the average energy (14) real. However, this implies that, on average, the exchange of energy between the particle and the field is zero, meaning that the energy provided by field is promptly returned to it in equal amount. This, besides constituting the desired local energy balance--it can be related with atomic stability --also puts some insight in the complex shape of the mentioned real quantities.
In fact, the local energy balance (23) is satisfied by the trivial solution of (20), expressed by
F = [partial derivative][S.sub.l]/ [partial derivative]P = [iota] [S.sub.o]/P , (24)
as can be easily verified from the normalization of P. So this proven the [P.sup.-1] functionality, which is not achieved only from probability conservation, as pointed above.
Substituting (24) into (19), results in
[bar.H] = [integral] [d.sup.3]rP([([nabla]S).sup.2]/2m + [S.sup.2.sub.o]/2m [([nabla]P/[P.sup.2]).sup.2] + V , (25)
which, with the canonical equations (16) and (17), reproduces the equations (3) and (4), respectively, if [S.sub.0] is identified with h/2. Therefore, to complete the classical derivation of the Schrodinger equation, the anzatz (2) must also be obtained in a classical context. This is the subject of the next section.
4 Parameterization of the equations
Knowing any one of the solutions (one of the paths) of the motion equations resulting from (25), the energy and momentum at each position, according to the equations (13), (12) and (24), are, respectively, given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
p = [nabla]S + [iota][S.sub.o]/P [nabla]P. (27)
Integrating these partial differential equations (minus a possible constant), we obtain the following dimensionless equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
as can be easily verified by following the inverse procedure. The upper limits of the integrals are the coordinates and time of the positions occupied by the particle along a possible path, therefore the left hand side of (28) is a complex function of these parameters, which will be defined in the following way:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
As both sides of (28) are independent of the path followed by the particle, we can write the following relation between S and P, valid for all paths:
ln [psi] = S/2[iota][S.sub.o] + In [square root of P], (30)
[psi] = [square root of P] exp(S/2[iota][S.sub.o]). (31)
This equation with [S.sub.0] = [??]/2 is in full agreement with (2). And more, for constant energy and momentum the function defined in (29) is a solution of the Schrodinger equation for a free particle.
Finally, let's re-write the equations obtained in this work in terms of [psi]. From (30) and its complex conjugate we obtain the following parametric shapes for S and P:
S = [iota][??]/2(lg [psi] - ln [psi]*) (32)
P = [psi]*[psi]. (33)
Consequently, the equations (25), (26) and (27) can be re-written, respectively, in the forms:
H = [integral][d.sup.3]r([h.sup.2]/2m [nabla] [psi]* x [nabla] [psi] + [psi]*V[psi]), (34)
[iota][??] [partial derivative][psi]/ [partial derivative]t = H[psi], (35)
-[iota][??][nabla] [psi] = p[psi]. (36)
Applying the divergence operator on both sides of equation (36), allied to fact that p is coordinate independent (it is independent of the followed path), gives
-[iota][??][nabla] x [nabla] [psi] = p x [nabla][psi], (37)
and expressing p in terms of the complex conjugate of (36), we obtain the equality
-[psi]* [[nabla].sup.2] [psi] = [nabla] [psi]* x [nabla] [psi]. (38)
Therefore the equation (34) can be written in the well known quantum form
[bar.H] = [integral] [d.sup.3] r[psi]*([[??].sup.2]/2m [[nabla].sup.2] + V) [psi]. (39)
The approach shows that the Schrodinger equation and its accessory are necessary and natural equations, parameterized shapes of the complicated--not to say unsolvable--equations resulting from a classical treatment including a special field with homogeneous, isotropic and random characteristics.
91220-480, Porto Alegre, RS, Brazil
Submitted on May 21, 2011 / Accepted on May 28, 2011
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|Publication:||Progress in Physics|
|Date:||Oct 1, 2011|
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