Abstract
In preparation for the chapters that follow, this one offers an introduction to the study of quantum mechanics as a stochastic process, in the form of a basic phenomenological theory knownNelson theory as stochastic (or Nelson) mechanics, or Stochastic quantum mechanicsstochastic quantum mechanics. The kinematics of the process is developed first, using a description of the motion of particles that involves finite time intervals \(\Delta t\) and up to second-order derivatives; this kinematics is made in terms of two different velocities and four accelerations. An extended ‘Newton’s Second Law’ involving a linear combination of these accelerations, leads under appropriate conditions to two equations, one being equivalent to the Continuity equationcontinuity equation and the second one representing a general dynamical law. The Schrödinger equationDetailed energy balance!and Schrödinger equation follows from this set of equations for a particular selection of the parameters. By ignoringCetto, A. M. the source of the stochasticity, this theory does not provide the elements to derive the value of the parameters; this is shown to be one of its main limitations. It has, however, the value of providing an intuitive physical image of the stochastic process underlying quantum mechanics. Moreover, by showing that the Brownian process implies an altogether different choice of the parameters, it establishes a neat distinction between classical and quantum stochastic processesQuantum stochastic process.
Some physicists, among them myself, cannot believe that we must abandon, actually and forever, the idea of direct representation of physical reality in space and time...
A. Einstein (1954)
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- 1.
Not quite independently of the interpretation, strictly speaking. For example, Bell (1987, article 19) argues that “\(\ldots \)the reversibility of the SchrödingerDetailed balance!and Schrödinger equation equation strongly suggests that quantum mechanics is not fundamentally stochastic in nature.” This sentence sounds tempting\(\ldots \) At this stage, how would you respond to it? (An answer is given at the end of the chapter.)
- 2.
Randomness plays an important role in several interpretations of qm, in addition to the one developed in this book. A case of major interest is Griffiths’ theory of consistent historiesConsistent histories, according to which most of the evolution is due to randomness. See Omnès (1994, 1999a, b), Griffiths (1996), Griffiths and Omnés (1999).
- 3.
- 4.
As noted earlier, Nelson Nelson theorycalls the theoryBergmann, P. G. simply stochastic mechanics Stochastic mechanics. His work is that of a mathematician and should be of major interest to the more mathematically inclined readers. There is another entirely different theory that goes under the same name stochastic quantum mechanics, pioneered by Prugovečki (1984, 1995)Engliš, M. Prugovečki, E. [see also Ali and Engliš (2005); Ali and Prugovečki (1986)]. It represents an attemptAli, S. T. to unify physics into a rigorous quantum structure that considers a quantum spacetime and a universe which on the microscopic level follows a stochastic rather than deterministic evolution. Further, it should be noted that some authors speak of stochastic quantum mechanics Stochastic quantum mechanics while referring to Bohm’s theorySrivastava, Y. N. Feligioni, L. (see e.g. Feligioni et al. 2005).
- 5.
By writing
$$\begin{aligned} \frac{g(\varvec{x};t^{\prime \prime })-g(\varvec{x};t^{\prime })}{2\Delta t}=\frac{1}{2\Delta t}\int \nolimits _{t^{\prime }}^{t^{\prime }+2\Delta t} \frac{\partial g(\varvec{x};s)}{\partial s}ds \end{aligned}$$it becomes clear that this expression is a coarse-grained time-derivative obtained by time-averaging the derivative \(\partial g/\partial t.\) This procedure mimicks the time smoothing produced by an observation, which is always extended in time. Such smoothing is particularly appropriate to deal with highly irregular (and even non-differentiable) functions.
- 6.
The moving average \(x_{\Delta t}(t)\) of \(x(t)\) is defined as \(x_{\Delta t}(t)=(1/\Delta t)\int _{t}^{t+\Delta t}x(\tau )d\tau .\)
- 7.
To reproduce the above results it is required that the second moment \( \left\langle \left( \delta \varvec{x}_{+}\right) ^{2}\right\rangle \) be proportional to \(\Delta t,\) so that \(\left\langle (\Delta _{+}\varvec{x})^{2}\right\rangle /\Delta t\) acquires a finite value [as demanded by Eq. (2.6)]. This is a characteristic feature of Brownian motion (or rather, of a white noiseWhite noise), and explains the extended reference to theories as the present one as ‘Brownian-motion theories’.
- 8.
In the literature it is possible to find the velocity \(\varvec{u}\) defined with the sign reversed. Equation (2.27) can be recast into the form \(\varvec{j}_{\text {diff}}\equiv \varvec{u}\rho =D \varvec{\nabla }\rho ,\) known as Fick’s law (with due allowance for the reversed sign).
- 9.
- 10.
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- 12.
For example, when solving the Heisenberg, W.Heisenberg equations of motion for \(x\) and \(p,\) the initial conditions are given by matrices, which guarantees that the Heisenberg inequalitiesHeisenberg inequalities are satisfied starting from \(t=0.\)
- 13.
- 14.
Some of these matters are discussed in Vasudevan Vasudevan, R. Gopalan, M. N. Beckmann, M. J. Parthasarathy, K. V.et al. (2008). For the relativistic case see also Ramanathan Ramanathan, R. (1997). Extensive and Complementaritycomplementary lists of references to earlier work can be found in Jammer, M.Jammer (1974), Guerra, F.Guerra (1981 Waldenfels, W. V., 1984, 1988), Blanchard, Ph.Blanchard et al. (1987), de la Peña and Cetto Cetto, A. M.(1991)Reichl, L. E., and The Dice.
- 15.
The formula for the acceleration \(\varvec{a}_{B}\) for classical (Brownian) particles is of course as nonlocal as the quantum acceleration, but nobody denies the usefulness of the Brownian-motion theory of Einstein and Smoluchowski within its domain of applicability. It even played a most important and historic role in the empirical demonstration of the reality Realityof molecules at the beginning of the 20th century! Such description of the Brownian case is admittedly not a fundamental one. In the quantum case a problem arises when interpreting it as a fundamental theory, since a fundamental expression for the acceleration must be local.
- 16.
Recently, Nelson has attempted to apply stochastic mechanics Stochastic mechanicsto relativistic fields, hoping to avoid the above mentioned nonlocality features, and aiming to develop useful technical tools in constructive field theory (see Nelson 2013).
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de la Peña, L., Cetto, A.M., Valdés Hernández, A. (2015). The Phenomenological Stochastic Approach: A Short Route to Quantum Mechanics. In: The Emerging Quantum. Springer, Cham. https://doi.org/10.1007/978-3-319-07893-9_2
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