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.     

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 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.     

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.     

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 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.     

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.     

Conservation laws in asymptotically separable quantum mechanics

A notion of chronological disorder is introduced to explain the physical reason why many familiar conservation laws should not be regarded as sacrosanct and valid for all times. Their disappearance as absolute laws in an asymptotically separable theory is a desirable feature, rather than a difficulty, of the theory.     

Notes on the classical and the nonrelativistic limits in quantum mechanics

A method is presented which can be used to discuss both the classical and also the nonrelativistic limit of quantum mechanics. A one-to-one correspondence may be established between the asymptotic convergence of the resolvent and that of the timedependent solution. In so far as the question of dynamics is concerned we investigate the relation between families of nonrelativistic Hamiltonians and the corresponding Dirac-Hamiltonians when c± or when c±0. The nonrelativistic free theory formally shows the same pattern when ±0 (the classical limit) or when ±. The investigation finally shows how the asymptotic convergence of the relativistic theory can take place under some fairly general conditions of the radiation field.     

Axiomatic foundations of nonrelativistic quantum mechanics: A realistic approach

A realistic axiomatic formulation of nonrelativistic quantum mechanics for a single microsystem with spin is presented, from which the most important theorems of the theory can be deduced. In comparison with previous formulations, the formal aspect has been improved by the use of certain mathematical theories, such as the theory of rigged spaces, and group theory. The standard formalism is naturally obtained from the latter, starting from a central primitive concept: the Galilei group.     

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.     

Incompressible quantum liquids and new conservation laws

In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least q-fold degenerate when the filling factor is p/q, where and are coprime integers. Interestingly, the simplest Hamiltonian respecting this type of symmetry encapsulates two prominent examples of novel states of matter, namely, the fractional quantum Hall liquid and the quantum dimer liquid. We discuss the relevance of this class of Hamiltonian to the search for featureless Mott insulators.