P. M. Mathews

CXVII. A stochastic problem relating to counters

Summary The probability distribution function π(n, t) of the number of registered events n in a time interval t for a counter in the case when the registered and unregistered events are followed by different dead times, is derived by using the method of product densities formulated recently by one of us (R.).

Observables of a Dirac Particle

The most general forms, consistent with reasonable physical requirements, for the position and other observables of a Dirac particles are obtained.

Observables in the extreme relativistic representation of the dirac equation

A choice of operators is given to represent dynamical variables of a Dirac particle, in the extreme relativistic representation. (auth)

The Observables and Localized States of a Dirac Particle

possibilities consistent with reasonable physical requirements for defining the position operator (and hence the operators for the other dynamical variables too) of a Dirac particle. It was found there that in each of these four sets, the operator representatives for the dynamical variables were such that all the usual commutation relations were obeyed. This invariance of the commutation relations in passing from one set to another suggests that the different sets are related by unitary transformations. In the following we obtain the actual unitary transformations involved. We then establish the relationship between the Newton-Wigner (N.W.) 2l position operator and the position operator found by Foldy and Wouthuysen (F.W.) 3l and Pryce,4l the latter of which is the simplest of our four operators. This relationship enables us to deduce the localized eigenstates for these four operators. An examination of these in the light of the requirements laid down by Newton and Wigner2l shows that all the states (and hence the corresponding operators) are admissible in the case of a zero mass particle, but only the F.W. operator remains in the finite mass case. § 2. The unitary transformations In reference 1), we showed that there are four possible sets of choices for the operator representatives of the dynamical variables of a Dirac particle, and that in particular, in the C representation*) in which the Hamiltonian takes the form

Stochastic processes associated with a symmetric oscillatory poisson process

A symmetric oscillatory Poisson process is defined and its stochastic features studied. The process represented by the symbolic integral of this oscillatory Poisson process is then discussed in detail. The results obtained are applied to the well-known stochastic problem of multiple scattering of charged particles in their passage through matter.

On a class of stochastic integro-differential equations

SummaryThe solutions of the stochastic integro-differential equation involving, in addition to the random transitions of the stochastic variable, a deterministic change, are obtained under various approximations. It is shown that the solution in the case of a negative deterministic change is strikingly different from the solution when the change is positive. Examples from physics and astro-physics are cited to illustrate such stochastic processes.