Regularized potentials in nonrelativistic quantum mechanics

Using scaling technique, we describe all self-adjoint extensions of Schrödinger operator with a singular potential in one dimension as a limit of Schrödinger operators with regular local potentials.I am indebted to Dr. H. Englisch for stimulating discussions. I am also grateful to the Department of Mathematics at the Karl-Marx-University Leipzig, where this work was made possible.     

Axioms for nonrelativistic quantum mechanics

On the basis of the axioms assumed it is proved that the logic of propositions concerning any quantum-mechanical system may be endowed with the structure of an orthomodular atomistic complete lattice satisfying the covering postulate, and hence, as a consequence of these axioms, the Piron-MacLaren representation theorem for the logic is obtained.     

Regularized potentials in nonrelativistic quantum mechanics

Using scaling technique we describe various self-adjoint extensions of the three-dimensiona Schrödinger operator with singular potential as a limit of Schrödinger operators with regularized potentials.I am indebted to Dr. H. Englisch for stimulating discussions. I am also grateful to the Department of Mathematics of the Karl Marx University Leipzig where this work was written.     

Complex mass in nonrelativistic quantum mechanics

The use of complex mass in nonrelativistic quantum mechanics and its relation to the influence of vacuum field fluctuation in electron motion are discussed.     

Coupling constant thresholds in nonrelativistic quantum mechanics

We extend the analysis of absorbtion of eigenvalues for the two body case to situations where absorbtion occurs at a two cluster threshold in anN-body system. The result depends on a Birman-Schwinger kernel for such anN-body system, an object which we apply in other ways. In particular, we control the number of discrete eigenvalues in the 0 limit.Research partially supported by U.S.N.S.F. under Grant MCS-78-01885.     

Differential conservation laws in nonrelativistic quantum mechanics

Differential conservation laws for quantum mechanical operator fields are studied from a) some general point of view and b) to find the particle, momentum and energy conservation law, if the Hamiltonian isnot local as usually assumed. Concerning (b): If the interaction in a many body system is nonlocal, the particle current operator in the continuity equation has to be redefined. Physical properties of the interaction current are discussed. Similar nonlocal effects must be taken care of in the stress and energy current operator. Concerning (a): Besides the mentioned conservation laws there are arbitrary many other ones. In fact, for each arbitrary field a class of corresponding currents exists and vice versa, which together are related by a differential conservation law. Some physical aspects for defining current operators are given and discussed.     

Finite total cross-sections in nonrelativistic quantum mechanics

We present a simple geometric method for estimating total cross-sections in two-body and more generally two cluster scattering. We discuss a variety of aspects of total cross-sections including large coupling constant behavior.On leave from Dept. of Physics, University of Bielefeld, Federal Republic of Germany; address after April 1, 1980: Dept. of Mathematics, University of Bochum, D-4630 Bochum 1, Federal Republic of GermanyResearch partially aupported by USNSF under Grant MCS-78-01885     

The operator formalism of quantum mechanics from the viewpoint of short disturbances in nonrelativistic classical motion

The effect of short disturbances on nonrelativistic motion is formulated in terms of operators. Analogies with quantum mechanics are developed and some disparities noted. For the one-dimensional particle we obtain analogues of the de Broglie wave commonly associated with particle motion, Heisenberg's commutation relation, Schrödinger's equation, and the statistical interpretation. Whether these results have any bearing on quantum mechanics itself is left an open question.     

Light velocity in nonrelativistic quantum mechanics on a circle

A single-valued Schrödinger wave function on a circle is discussed in the framework of the path integral. It is clarified how the light velocity comes into play and how terms corresponding to superluminal transmission of signals are suppressed.     

The problem of decay in nonrelativistic quantum mechanics

In this paper the time-dependent Schrödinger equation with an arbitrary potential with finite effective radius is solved. Outside the effective radius of the potential a universal solution is obtained which does not depend on the initial state and which describes the evolution of a resonance state. In the single-resonance approximation various space-time domains are studied in which an exponential decay law can be observed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 38–43, December, 1972.     

Nonspreading quasi-classical wave packets in nonrelativistic quantum mechanics

A general method of constructing nonspreading quasi-classical wave packets in nonrelativistic quantum mechanics is outlined. The nonspreading wave packet for a free particle is constructed in explicit form. It is shown that, in this case, the specified accuracy of the quasi-classical solution is not lost over time.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 94–99, June, 1994.     

Superposition in quantum and classical mechanics

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.Dedicated to the Memory of Charles H. Randall.     

The use of projective limits in classical statistical mechanics and Euclidean quantum field theory

The Gibbs states of an infinite classical statistical system correspond to the states of reservoir at infinity. It is shown that its configuration space can be thought of as a generalized projective limit of configuration spaces of remote reservoirs. This notion of projective limit is defined and it is noted that it can also be used e.g. for proofs of the existence of Gibbs states in the thermodynamic limit and their decomposition into pure phase. A similar approach to (nonperturbative) Euclidean quantum field theory is suggested and connections with the concept of renormalizability are found.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.The first-named author (R. K.) would like to express his thanks to Prof. V. Votruba for general support and constant encouragement.     

Continual measurements in the spacetime formation of nonrelativistic quantum mechanics

We study the nature and effects of some continual measurements in nonrelativistic quantum mechanics, a concept introduced by R. P. Feynman in his path integral formulation of quantum mechanics. We prove the existence in various senses of mathematically rigorous objects formally equivalent to the path integrals associated with such measurements, by means of both a limiting definition and more general techniques; and we derive some quantitative results concerning the effects on the wave-function and the numerical value of certain associated probabilities.     

Spin & statistics in nonrelativistic quantum mechanics,II

Recently a sufficient and necessary condition for Pauli's spin‐statistics connection in nonrelativistic quantum mechanics has been established [1]. The two‐dimensional part of this result is extended to n‐particle systems and reformulated and further simplified in a more geometric language.     

The underlying Brownian motion of nonrelativistic quantum mechanics

Nonrelativistic quantum mechanics can be derived from real Markov diffusion processes by extending the concept of probability measure to the complex domain. This appears as the only natural way of introducing formally classical probabilistic concepts into quantum mechanics. To every quantum state there is a corresponding complex Fokker-Planck equation. The particle drift is conditioned by an auxiliary equation which is obtained through stochastic energy conservation; the logarithmic transform of this equation is the Schrödinger equation. To every quantum mechanical operator there is a stochastic process; the replacement of operators by processes leads to all the well-known results of quantum mechanics, using stochastic calculus instead of formal quantum rules. Comparison is made with the classical stochastic approaches and the Feynman path integral formulation.     

Born's postulate and reconstruction of theψ-function in nonrelativistic quantum mechanics

A continuous family of self-adjoint operators is constructed such that their measurement data are insufficient to reproduce uniquely via Born's postulate the underlying quantum state. Moreover, no pair of operators has a common invariant subspace. This rejects a conjecture given by Moroz. On the other hand, strengthening results obtained by Kreinovitch, it is shown that already one special potential and the related localization measurement data at different moments of time can guarantee the uniqueness of reconstruction.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.