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AIP conference proceedings

Classical and Quantum Random Processes

Leon W. Cohen

Published Jan 1, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.2361208

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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.
In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.