A non-symmetric transition probability in quantum mechanics

This quotation is perhaps the most famous statement of the ensemble interpretation of quantum mechanics. The role of the ensemble in quantum mechanics ranges from innocuous to profound, and even controversial. The innocuous role of the ensemble stems from the fact that quantum mechanics does not predict the actual events, but only the probabilities of the various possible outcomes ( probability in quantum mechanics) of the various possible events. In order to compare the predictions of quantum mechanics with experiment, one must prepare a state and measure some dynamical variable, and repeat this preparation–measurement sequence many times. The relative frequencies of the various outcomes in this ensemble of results can then be compared with the theoretical probabilities predicted by quantum mechanics. Thus it is natural to say that quantum mechanics describes the statistics of an ensemble of similarly prepared systems. Here, as in classical statistical mechanics, one should not confuse the ensemble of systems with an assembly of systems into a composite. For example, if the system is a single particle, then the ensemble is a conceptual set of replicas of it, each in its own environment, whereas the assembly would be a many-particle system. The role of the ensemble is to enable statistical analysis; its members do not interact with or influence each other. The more significant role of the ensemble interpretation is exemplified by

Quantum mechanical calculations are compared to quasiclassical trajectory forward (QCT) calculations for the collinear, endoergic reaction H2(n1)+I→H+HI for two different potential energy surfaces, a rotated-Morse-curve (RMC) surface and the semiempirical valence-bond surface of Raff et al. Vibrationally state-selected reaction probabilities and rate constants and Arrhenius parameters are presented. Thermally averaged rate constants and their Arrhenius parameters are also given. For one of the potential energy surfaces, quasiclassical trajectory reverse histogram (QCTRH) calculations were also performed. The results show that classical mechanics and quantum mechanics are in significant qualitative agreement for state-selected properties. Specifically, for the n1=0 state of the Raff et al. surface the quantum mechanical reaction probabilities are very small (less than 0.005) and the QCT method predicts this state to be totally non-reactive. For all other states on both surfaces quantum mechanics and QCT and QCTRH results all agree that reaction probabilities attain much higher values (up to 0.85). For both surfaces quantum mechanical and QCT results predict that excited vibrational states make significant contributions to the thermal reaction rates, although the methods disagree as to which vibrational state is quantitatively most important. Quantitative agreement with quantum mechanical results is obtained only with the QCTRH method for thermally averaged rate constants (agreement within 2%) and with both QCT and QCTRH methods for the Arrhenius parameters (agreement within a few tenths kcal mol−1 for activation energy). However, to achieve such agreement the QCT method had to be suitably modified to correct unphysical discrepancies in the threshold energy region. We present tables of these and many other results as a function of temperature. These should be useful in assessing the validity of trajectory studies of various kinds of reaction attributes under conditions where they are used to interpret experiments.

In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space. Quanta 2017; 6: 53–56.

In quantum physics, the term 'contextual' can be used in more than one way. One usage, here called 'Bell contextual' since the idea goes back to Bell, is that if A, B and C are three quantum observables, with A compatible (i.e. commuting) with B and also with C, whereas B and C are incompatible, a measurement of A might yield a different result (indicating that quantum mechanics is contextual) depending upon whether A is measured along with B (the {A, B} context) or with C (the {A, C} context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell non-contextual: the outcome of a particular A measurement when A is measured along with B would have been exactly the same if A had, instead, been measured along with C. A different definition, here called 'globally (non)contextual' refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

After reviewing paradigmatic cases of “experimental metaphysics” basing inferences against local realism and determinism on experimental tests of Bells theorem (and successors), we concentrate on clarifying the meaning and status of “objective probability” in quantum mechanics. The terms “objective” and “subjective” are found ambiguous and inadequate, masking crucial differences turning on the question of what the numerical values of probability functions measure vs. the question of the nature of the “events” on which such functions are defined. This leads naturally to a 2×2 matrix of types of interpretations, which are then illustrated with salient examples. (Of independent interest are the splitting of “Copenhagen interpretation” into “objective” and “subjective” varieties in one of the dimensions and the splitting of Bohmian hidden variables from (other) modal interpretations along that same dimension.) It is then explained why Everett interpretations are difficult to categorize in these terms. Finally, we argue that Bohmian mechanics does not seriously threaten the experimental-metaphysical case for ultimate randomness and purely physical probabilities.

In this work, exact quantum mechanical (QM) and classical (CL) transition probabilities for the Cl+XY (X,Y=H,D,T) collinear system are compared. The calculations were performed using a semiempirical LEPS surface. The main features discussed are tunneling and threshold behavior for both the ground state and the first excited state of the hydrogen molecule. In the second part, kinetic isotope effects are presented and discussed. Those were calculated in three different ways, using (i) QM transition probabilities, (ii) CL transition probabilities, and (iii) transition state theory (TST-1D). Tunneling coefficients kQM/kCL and kQM/kTST were calculated and discussed. Finally, a brief comparison with experimental data was performed.

We show that the newly found complete and orthonormal representation , which is constructed in terms of guiding centers and canonical momenta of an electron moving in a uniform magnetic field, also helps us to establish a systematic phase operator theory. The unitary phase operator, interpreted as a quantity canonically conjugated to the electron angular momentum, manifestly exhibits in the representation its phase behavior. As a byproduct, the phase distribution function, in the correct sense of probability in quantum mechanics, can also be defined by the representation.

Non-commutative measure theory serves as a mathematical basis to compare the classic 1 and the quantum mechanical probability scheme.

In this work, we discuss a formal way of dealing with the properties of contextual systems. Our approach is to assume that properties describing the same physical quantity, but belonging to different measurement contexts, are indistinguishable in a strong sense. To construct the formal theoretical structure, we develop a description using quasi-set theory, which is a set-theoretical framework built to describe collections of elements that violate Leibnitz's principle of identity of indiscernibles. This framework allows us to consider a new ontology in order to study the properties of quantum systems. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

This book introduces the reader to statistical reasoning and its use in physics. It is based on a course taught to non-science majors at Cornell, and differs from other treatments by its wide-ranging use of quantitative methods, which are built up in a constructive way and assume only that the reader can add, subtract, multiply and divide with confidence. The author begins with a self-contained introduction to the everyday uses of probability, including the quantitative assessment of statistical information. Following a chapter on useful mathematical concepts, he develops the basic ideas of mechanical motion, the molecular theory of gases, entropy as a measure of molecular agitation, limitations on the conversion of heat to work, the physics of the direction of time, chaos, and the role of probability in quantum mechanics. To aid self-instruction, there are solved problems at the end of each chapter.

Algorithmic probability has shown some promise in dealing with the\nprobability problem in the Everett interpretation, since it provides an\nobjective, single-case probability measure. Many find the Everettian cosmology\nto be overly extravagant, however, and algorithmic probability has also\nprovided improved models of subjective probability and Bayesian reasoning. I\nattempt here to generalize algorithmic Everettianism to more Bayesian and\nsubjectivist interpretations. I present a general framework for applying\ngenerative probability, of which algorithmic probability can be considered a\nspecial case. I apply this framework to two commonly vexing thought experiments\nthat have immediate application to quantum probability: the Sleeping Beauty and\nReplicator experiments.\n

Quantum logic originated with the Birkhoff-von Neumann paper of 1936.2 The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.

Two problems will be considered: the question of hidden parameters and the problem of Kolmogorovity of quantum probabilities. Both of them will be analyzed from the point of view of two distinct understandings of quantum mechanical probabilities. Our analysis will be focused, as a particular example, on the Aspect-type EPR experiment. It will be shown that the quantum mechanical probabilities appearing in this experiment can be consistently understood as conditional probabilities without any paradoxical consequences. Therefore, nothing implies in the Aspect experiment that quantum theory is incompatible with a deterministic universe.

The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra–ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities.

Numerically exact quantum-mechanical transition probabilities for the collinear collision between an atom and a diatomic oscillator interacting via a strongly attractive Morse potential are compared with those obtained in various simple perturbation approximations. It is concluded that many of the improved perturbation approximations which work well for purely repulsive interactions are also quantitatively accurate for Morse interactions having deeply attractive wells. A simple distorted wave correction factor, derived, for example, via second-order perturbation theory, yields quite accurate transition probabilities, even for very small reduced collision energies (Ecoll/D=0.05) where attractive forces dominate in determining the dynamics of the collision. Furthermore, when the perturbation limit is no longer applicable, the so-called exponential approximation provides a viable method for calculating both direct and multiquantum transition probabilities.