Showing papers on "Quantum state published in 1971"

Classical S Matrix for Linear Reactive Collisions of H+Cl2

Abstract: Classical trajectories are computed for linear H+Cl2 collisions and used to construct the classical limit of the S matrix for reactive and nonreactive transitions between individual quantum states. An interesting feature of this system is that both ``direct'' and ``complex'' mechanisms participate in the collision dynamics. The two mechanisms contribute additively to individual S‐matrix elements, and within a ``random phase approximation'' for the complex contribution it is seen that they also contribute additively to the transition probability. The complex contribution to a transition probability is strictly classical, but interference structure may be prominent in the direct contribution. Comparison is made with quantum calculations for the same potential and the agreement is rather poor, even at a fairly coarse level. The reason for this disagreement is not completely clear, but it may be connected with the fact that complex formation is prominent.

Classical S Matrix for Rotational Excitation; Quenching of Quantum Effects in Molecular Collisions

Abstract: A previously developed theory in which exact solutions of the classical equations of motion for a complex collision system (i.e., numerically computed trajectories) can be used to generate the classical limit of the quantum mechanical S matrix (the “classical S matrix”) for the scattering process is applied to rigid rotor–atom collisions (rotational excitation). Comparison with essentially exact quantum results shows that transition probabilities (the square modulus of an S‐matrix element) between individual quantum states are given reasonably accurately by classical dynamics provided the interference terms are properly accounted for; a strictly classical approach (neglect of interference) gives poor agreement with the quantum values. For averaged collision properties, however, it is found that interference and tunneling effects are rapidly quenched. The linear atom–diatom system (vibrational excitation) and the rigid rotor–atom system are both investigated with regard to this question, namely, how much averaging is necessary to quench these quantum effects. Results indicate that even summation over a few quantum states is often sufficient to make a completely classical treatment appropriate.

Markov Approach to Density Fluctuations Due to Transport and Scattering. I. Mathematical Formalism

Abstract: A kinetic approach to fluctuations and correlations of stochastic processes depending on a continuous set of parameters y is presented. In particular, we consider particle densities np(y, t) which may refer to macroscopic densities in position space, or to microscopic quantities such as distributions in phase space, or occupancies of quantum states of which the labeling is continuous (as with Bloch states in solids). In a Markovian sense such processes are infinite dimensional. We describe the fluctuating particle densities in a Hilbert space: the analog of de Groot's a‐space for non‐spatial‐dependent variables. Mainly, we employ a Langevin description; i.e., we start from presumed phenomenological equations, amended with source densities ξp(y, t). A theorem is derived for the density‐density or two‐point covariance function (Λ theorem). In its general form, the theorem applies to the nonequilibrium steady state. It closely resembles the generalized g‐r theorem for finite‐dimensional processes. However, t...

A general theory of empirical state determination in quantum physics: Part I

Abstract: This paper develops a method for extracting from data the quantum theoretical state representation belonging to any reproducible empirical scheme for preparing a physical system, provided only that at least one observable has its possible values limited to a finite set. In Part I, we formulate a general systematic procedure, based on the concept of irreducible tensor operators, for the selection of sets of observables sufficiently large to permit the unambiguous determination of an unknown quantum state.

Construction of Non-Linear Local Quantum Processes.

Abstract: : A species of singular perturbation theory applicable to a general class of nonlinear local quantum fields is developed. The theory is applied in detail to the arbitrary scalar relativistic field in two space time dimensions with positive-energy self-interaction. (Author)

Observables and the Field in Quantum Mechanics

Abstract: Corresponding to any irreducible proposition system L in general quantum mechanics there is a division ring D with an anti‐automorphism * and a vector space (V, D) over D with a definite sesquilinear form φ such that L is isomorphic to the set of φ closed subspaces of (V, D). The main task remaining in connecting the general quantum mechanics to the conventional quantum theory in a complex Hilbert space is to give physical arguments which force D to be the complex field. In this paper it is shown that if L admits a certain type of observable (together with other structure which seems to be physically justified), then D contains the real field as a subfield. Steps are then indicated that can be taken to move from the reals to the complexes or quaternions.

From mendeléev's atom to the collapsing star*†

Abstract: A decade before Planck’s 1900 formulation of the quantum principle, Mendeleev out of his researches on chemistry and the periodic system of the elements recognized the general character that the new principle must have, dealing with no “deathlike inactivity”, establishing “individuality amid continuity”, and destined to “hasten the advent of true chemical mechanics”. Implicit in the Bohr-Rutherford 1911 model of the atom was the paradox of atomic collapse. No cheap way out offered itself. Only application of the quantum principle resolved the paradox. This development led (1927) to the ‘chemical mechanics’ envisaged in outline by Mendeleev. It may be symbolized today by an electron orbit with the shape of a double necklace, the ‘chemical orbit’ (Powers). A new crisis confronts physics today in the predicted phenomenon of gravitational collapse, both at the level of a star (‘black hole physics’) and at the level of the universe itself. Again no way out is evident except to call on the quantum principle. It leads to the conclusion that the dynamics of the universe goes on in superspace. In superspace alternative dynamical histories of the universe (cycles of expansion and recontraction) not only dynamically couple to each other in the era of collapse, as well as one can judge, but also ‘coexist’. If the vision of Clifford and Einstein in updated form is taken as guide and particles are regarded as quantum states of excitation of a dynamic geometry, then it is natural to believe that each period of collapse sees the universe ‘reprocessed’, with the previous spectrum of particle masses extinguished, and a new pattern of masses established. On this view particle masses are as far removed from any possibility of being calculated from first principles as the ‘initial conditions of dynamics’ themselves. For an early test of this framework of ideas nothing looks so promising as the prediction that a black hole, formed by whatever combination of baryons, photons, leptons, and other entities, is characterized by nothing but mass, charge, and angular momentum (‘transcendence of the law of conservation of baryons and leptons’).

Elementary derivation of the Boltzmann distribution law

Abstract: The author considers an elementary collision process in which one or more particles of a system undergo an upward transition through quantum states.

Myriotic Fields in Quantum Statistical Mechanics

Abstract: A Hilbert space formulation of quantum statistical mechanics is developed by using the notion of myriotic fields. The nonseparable nature of the Hilbert space is investigated, and it is shown that in the classical limit, ℏ → 0, the Hilbert space approaches, in the absence of point eigenvalues, that Hilbert space appropriate to classical statistical mechanics.

On the statistical description of quantum systems

Abstract: It is emphasized that the use of statistical methods is a consequence of the incomplete isolation of physical systems. The description of a partially specified quantum state by a density operator is shown to be equivalent to a description in terms of an ensemble of systems. Several definitions of the statistical operator are shown to be identical.

Photon Spin Formalism

Abstract: A new approach to quantum electrodynamics is considered, in which photon spin terms are incorporated phenomenologically into the equations of classical electrodynamics in such a way as to yield quantized equations in the Heisenberg picture, for which relativistic requirements are satisfied. Photon momentum and energy analyses in the new formalism are then compared to the corresponding analyses in conventional field‐theoretic formalisms, and questions regarding mathematical and physical consistency are discussed. Different notations are used to distinguish between physical and Hilbert space vectors, noting that the Hilbert space of the vector wavefunction has a 3‐dimensional subspace resembling physical space, mathematically, in a way that allows interesting results, which can be described by two formalisms when auxiliary notation is necessary to analyze relationships involving both kinds of vectors.