On different random processes of stochastic quantum mechanics

Giovanni Jona-Lasinio

Leon W. Cohen

We contrast standard stochastic process methods with those of quantum mechanics. While the end results of both methods are common quantities, such as expectation values and probabilities, the methods of calculation are dramatically different. We present a simple example of a nonstationary stochastic process that can be exactly solved in both standard and quantum mechanics. The example illustrates both approaches and crystallizes the similarities and differences between the methods.

Thomas Schuermann

Ying Oon Tan

Previously [J. Mod. Optics, 51, 1093, (2004)], we introduced an alternative stochastic mechanics in the form of a stochastic differential equation with a stochastic process defined by the set of Bohmian momentum time histories from an ensemble of particles. We showed that the alternative stochastic mechanics with a purely random process characterized by an n‐th order joint probability density in the form of products of delta functions is equivalent to quantum mechanics. In this paper, we show that, for a particular non‐purely random process, the alternative stochastic mechanics is not equivalent to quantum mechanics. Hence, the alternative stochastic mechanics is generally not equivalent to quantum mechanics.

Ying Oon Tan

Previously [2004, AIP Conf. Proc., 750, 361], we considered a stochastic mechanics in the form of a stochastic differential equation mdxdt = μ(t) where μ(t) is a stochastic process defined by the set of Bohmian momentum time histories from an ensemble of particles. In this paper, we consider another stochastic differential equation d2xdt2 = 1m[−dV(x)dx + ζ(t)] where the stochastic force ζ(t) is a stochastic process defined by the set of quantum force time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process characterized by an n‐th order joint probability density in the form of products of delta functions, then the stochastic mechanics with the stochastic force ζ(t) is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non‐purely random process, we show that the stochastic mechanics with the stochastic force ζ(t) is not equivalent to quantum mechanics. Therefore, this stochastic mechanics is also generally not equivalent to quantum mechanics.

Harish Parthasarathy