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arXiv:1012.4857

Statistical deviation from stationary action to Schrödinger equation

Agus Budiyono

Published Dec 22, 2010• 0 citations
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Abstract

We discuss the dynamics of single particle by laying a hypothesis that the Hamilton's principle of stationary action is not exact. We then postulate that the deviation of the action with sufficiently short time interval from the stationary action is distributed along a sufficiently long trajectory according to an exponential law. We show that the dynamics of the ensemble of trajectories satisfies the Schrödinger equation with Born interpretation of wave function if the average deviation is given by $\hbar/2$ and if two opposite signs of deviation occur equally probably. The particle thus behaves as if it is guided by a virtual wave satisfying the Schrödinger equation.

DOI
10.48550/arxiv.1012.4857

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We discuss the dynamics of single particle by laying a hypothesis that the Hamilton's principle of stationary action is not exact. We then postulate that the deviation of the action with sufficiently short time interval from the stationary action is distributed along a sufficiently long trajectory according to an exponential law. We show that the dynamics of the ensemble of trajectories satisfies the Schrödinger equation with Born interpretation of wave function if the average deviation is given by ℏ/2\hbar/2ℏ/2 and if two opposite signs of deviation occur equally probably. The particle thus behaves as if it is guided by a virtual wave satisfying the Schrödinger equation.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.
A few years ago, one of the former Editors of this journal launched “a call to action” (E. F. Taylor, Am. J. Phys. 71, 423–425 (2003)) for a revision of teaching methods in physics in order to emphasize the importance of the principle of least action. In response, we suggest the use of Hamilton's principle of stationary action to introduce the Schrödinger equation. When considering the geometric interpretation of the Hamilton–Jacobi theory, the real part of the action S defines the phase of the wave function exp iS/ℏ, and requiring the Hamilton–Jacobi wave function to obey wave-front propagation (i.e., Re(S) is a constant of the motion) yields the Schrödinger equation.
I present a reconstruction of general Hamiltonian action mechanics that eliminates all foundational problems of quantum mechanics. The key advance is the completion of Hamiltonian mechanics to the universal mechanics of particles based on action-waves, consistent with the inclusive validity of the principle of stationary action. It is found that irreducible indeterminism is intrinsic and universal at all scales of dynamics. The new action-wave equation is the complete description of single dynamical histories, dissolving the classical-quantum divide. The statistical theory of quantum mechanics emerges as the ensemble average of modified action dynamics. The ensemble average of the new action mechanics leads to a hybrid function consisting of the action-waves and the probability density of the ensemble. This hybrid wavefunction obeys the Schrödinger equation, which is not a single particle dynamical equation. The reconstructed mechanics without matter waves is free of the cardinal problem known as the collapse of the wavefunction and with that, the vexing issue of quantum measurement is resolved. Another significant advance is the correct decoding of quantum entanglement and purging of nonseparability and nonlocality in quantum correlations. The action-waves do not carry the burden of divergent zero-point energy. The reconstructed mechanics is in complete agreement with all empirical requirements and in harmony with credible physical ontology.
The classical action: Integral (0,T) dt1 L(v(t1), x(t1),t1) may be varied to find a stationary solution which yields Newton’s second law, a standard procedure in classical mechanics. Is it possible to obtain quantum mechanical motion from this same classical action? We argue that one may write the action for a relativistic/nonrelativistic particle moving at constant speed in terms of X and T i.e. v=velocity=constant=X/T. Varying with respect to X and T yields: d/dT Action = -E and d/dX Action = p. Thus, changing X by delta X yields a change in the action of p delta X which holds for all X because of the constant velocity. If one envisions a periodic function such as exp(i Action), this behaves as exp(ipx) in terms of -id/dX. Thus, there is an intrinsic wavelength related to 1/p. This follows directly from small changes of the Action. Alternatively, one may create a differential equation such as the Klein-Gordon or Schrodinger equation from classical conservation of energy-momentum using id/dT exp(iAction) = E exp(i Action) and -id/dX exp(iAction) = p exp(i Action). We note that exp(ipx) which describes the fluctuational behaviour of the classical Action may have positive and negative values showing that fluctuations may be both positive and negative These negative and positive fluctuations are, however, periodic so there is some systematic behaviour occurring, it seems.. We argue that exp(ipx) is statistical as its modulus is 1, indicating that a free particle has equal probability to be at any x point. For a system with a potential V(x), one may create an ensemble describing fluctuations of all possible p values i.e. W(x)=wavefunciton= Sum over p a(p) exp(ipx). The fluctuations are regular positive and negative, but there is different wavelength behaviour for each p i.e. 1/p so there is interference to yield an average fluctuation pattern i.e. the wavefunction.
Abstract In terms of its time trajectory, a physical experiment is comprised of a preparation time, a start time, a time evolution (duration), and an end time (point at which the measurement actually occurs). In non-relativistic quantum mechanics the start of the experiment is described by the wave function at time t = 0 taking into account the starting conditions, the evolution is described by the wave function according to the Schrödinger equation and the measurement is described by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that renders quantum mechanics stochastic with all its consequences it entails. Still, the nature of the Born rule is unknown yet it is useful since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is therefore only necessary in order to account for the unknown preparation time and thus starting time of the experiment because only the experimental duration (time between start time and end time) can be known.
A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schrödinger equation and the measurement by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit necessary since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is necessary in order to eliminate the unknown initial and end time coordinate of the experiment in reference to the beginning of the universe.