Quantum Mechanical Probability Research Articles

Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics

The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one pair of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. We also discuss the relation of our results with the quantum comb formalism.

On probabilities in quantum mechanics

This is an attempt to clarify certain concepts related to a debate on the interpretation of quantum mechanics, a debate between Khrennikov on the one hand and Stacey and Schack on the other hand. Central to this debate is the notion of quantum probabilities. I first take up the probability concept in the QBist school, as seen from my point of view, and then give my own arguments for the Born formula for calculating quantum probabilities. In that connection, I also sketch some consequences of my approach toward the foundation and interpretation of quantum theory. I discuss my general views on QBism as a possible alternative interpretation of quantum mechanics before I give some final remarks.

Network analysis of students’ conceptual understanding of mathematical expressions for probability in upper-division quantum mechanics

One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well. Published by the American Physical Society 2024

Quantum versus classical quenches and the broadening of wave packets

The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.

Non-Markovian dynamics of time-fractional open quantum systems

The Time-Fractional Schrödinger Equation (TFSE) is suitable to describe the non-Markovian dynamics of a quantum system exposed to an environment, which is instructive for understanding and characterizing the time behavior of actual physical systems. Three popular TFSEs, namely Naber’s TFSE I, Naber’s TFSE II, and XGF’s TFSE, have been introduced in the form of ∂β∂tβ with β∈0,1. However, they suffer from the drawbacks. In their respective descriptions, the total probability for finding a particle in the entire space equals one only when β=1, implying that time-fractional quantum mechanics violates quantum mechanical probability conservation. By applying the three TFSEs to a basic single-qubit open system model, we discover that in describing the non-Markovian evolution of the system, the three TFSEs are limited by their own applicable ranges of β. This indicates that the three TFSEs cannot effectively describe the non-Markovian quantum dynamics. To address these issues, we introduce a well-performed TFSE by constructing a new analytic continuation of time without iβ combined with the conformable fractional derivative. In its description, for one thing, the total probability for finding the system in any single-qubit state equals one when β∈0,1. For another, the system evolves correctly in the non-Markovian manner at all values of β. Furthermore, we study the performances of the four TFSEs applying to a two-qubit open system model and show that our TFSE still possesses the above two advantages compared with the other three TFSEs.

An Interdisciplinary interpretation of literature through the lens of Quantum Theory

This paper explores the interdisciplinary intersection between Quantum Theory and literary analysis, particularly focusing on William Shakespeare’s "Hamlet." By examining the character of Hamlet through the principles of quantum mechanics, specifically Schrödinger's wave function and the concept of superposition, we aim to demonstrate how scientific metaphors can provide novel insights into literary texts. This approach not only deepens our understanding of characters and narratives but also bridges the gap between the sciences and the humanities, highlighting their interconnectedness and mutual relevance. The paper concludes by discussing the implications of this interdisciplinary methodology, its limitations, and future directions for research. By employing quantum theory and the Schrödinger equation as metaphors, we gain deeper insights into the intricate layers of Hamlet’s character and the overarching themes of the play. This interdisciplinary approach enriches our understanding of Shakespeare’s work, highlighting the profound connections between the uncertainties and probabilities in quantum mechanics and the existential dilemmas faced by Hamlet. This perspective not only offers a unique lens for literary analysis but also demonstrates the potential for cross-disciplinary interpretations to enhance our appreciation of classic literature.

Second response theory: a theoretical formalism for the propagation of quantum superpositions

The propagation of general electronic quantum states provides information of the interaction of molecular systems with external driving fields. These can also offer understandings regarding non-adiabatic quantum phenomena. Well established methods focus mainly on propagating a quantum system that is initially described exclusively by the ground state wavefunction. In this work, we expand a previously developed size-extensive formalism within coupled cluster theory, called second response theory, so it propagates quantum systems that are initially described by a general linear combination of different states, which can include the ground state, and show how with a special set of time-dependent cluster operators such propagations are performed. Our theory shows strong consistency with numerically exact results for the determination of quantum mechanical observables, probabilities, and coherences. We discuss unperturbed non-stationary states within second response theory and their ability to predict matrix elements that agree with those found in linear and quadratic response theories. This work also discusses an approximate regularized methodology to treat systems with potential instabilities in their ground-state cluster amplitudes, and compares such approximations with respect to reference results from standard unitary theory.

Stochastic Quantum Probability and Subatomic Particle Experiment Design

This article is the third of a trilogy of articles on the nature of probability in quantum mechanics. The first article [1] began by noting that superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into the position probability density of an atomic particle, which happens nowhere else in probability theory. It went on to also note that there is an unexplained coincidence in quantum mechanics in that the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables and went on to examine whether there could be an archetypical variable, in the Platonic sense of true form, behind quantum probability that would reconcile quantum probability with classic probability. This examination found such a variable, which can encompass both local and nonlocal quantum events. The second article [2] provided evidence that quantum probability has such a stochastic nature. This evidence was based on the number of electrons that need to be sent through a two-slit interferometer to gain a clear pattern of self-interference, which when compared with the number that would be expected to be sufficient in order for the position probability distribution of the self-interference wavefunction to take clear shape suggests that there is more variability present than that described by the formulation of quantum mechanics, which implies the presence of an underlying and as yet unrecognized physical process. This final article completes the trilogy by considering how a key aspect of experimental design would be affected by the increased variability that would be present if quantum probability is itself stochastic in the manner suggested in the previous two articles.

Description of temperature effects on proton radioactivity

Role of thermal effects of hot parent nuclei in the characteristics of proton emission process has been studied for the first time by analyzing the surface energy coefficient γ entering in the calculation of proximity potential for 15 experimentally detected proton emitters (67≤Z≤83) in the isomeric states. In the present work, the proton-nucleus interaction potentials are obtained by using one of the latest versions of proximity potential formalism proposed by Zhang et al. in 2013. In addition, the quantum mechanical tunneling probability and consequently the proton radioactivity half-lives are calculated within the framework of the semiclassical WKB method. We present a new temperature-dependent (TD) form of the coefficient γ based on thermal properties of hot proton emitters in the framework of the finite-temperature generalized liquid-drop model. By integrating temperature dependence into the surface energy coefficient of proximity potential Zhang 2013, a reasonable description of the experimental half-lives of proton radioactivity is achieved. We extend the modified form of the Zhang 2013 model to predict the proton radioactivity half-lives of 6 proton emitters in the isomeric states, whose proton radioactivity is energetically allowed or observed, but has not been quantified yet. The obtained results reveal that the predictions by our model are in good agreement with the other theoretical methods, namely UDLP proposed by Qi et al. (2012) [19] and NGNL proposed by Chen et al. (2019) [49]. In this work, we also attempt to introduce an empirical formula for estimating the half-lives of one-proton emission process from the excited states by taking into account both the thermal effects of all the available 15 hot proton emitters and the contribution of centrifugal potential. Our results indicate that the calculated proton radioactivity half-lives by the current formula are in good agreement with experimental data. This means that the correlation between the half-lives of proton decay processes and the nuclear temperature T of proton emitters can be suitable to deal with the proton radioactivity one-proton transition from isomeric states.

The Bell Inequality, Inviolable by Data Used Consistently with Its Derivation, Is Satisfied by Quantum Correlations Whose Probabilities Satisfy the Wigner Inequality

It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1’s that are writable on paper. This surprising fact is not immediately obvious from Bell’s inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.

Quantum work statistics in regular and classical-chaotic dynamical billiard systems.

In the thermodynamics of nanoscopic systems the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we have chosen two two-dimensional billiard systems. Both systems are studied in the classical and the quantum mechanical settings. The classical conditional probability density p(E,L|E_{0},L_{0}) as well as the quantum mechanical transition probability P(n,l|n_{0},l_{0}) are calculated, which build the basis for the statistical analysis. We calculate the work distribution for one particle. The results in the quantum case in particular are of special interest since a suitable definition of mechanical work in small quantum systems is already controversial. Furthermore, we analyze the probability of both zero work and zero angular momentum difference. Using connections to an exactly solvable system analytical formulas are given for both systems. In the quantum case we get numerical results with some interesting relations to the classical case.

A generic approach to the quantum mechanical transition probability

In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.

Wave packet dynamics in an harmonic potential disturbed by disorder: Entropy, uncertainty, and vibrational revivals.

We investigate the quantum and classical wave packet dynamics in an harmonic oscillator that is perturbed by a disorder potential. This perturbation causes the dispersion of a Gaussian wave packet, which is reflected in the coordinate-space and the momentum-space Shannon entropies, the latter being a measure for the amount of information available on a system. Regarding the sum of the two quantities, one arrives at an entropy that is related to the coordinate-momentum uncertainty. Whereas in the harmonic case, this entropy is strictly periodic and can be evaluated analytically, this behavior is lost if disorder is added. There, at selected times, the quantum mechanical probability density resembles that of a classical oscillator distribution function, and the entropy assumes larger values. However, at later times and dependent on the degree of disorder and the chosen initial conditions, quantum mechanical revivals occur. Then, the observed effects are reversed, and the entropy may decrease close to its initial value. This effect cannot be found classically.

Logical entropy and negative probabilities in quantum mechanics

The concept of Logical Entropy, SL = 1-Σni=1 pi2, where the pi are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of SL as a measure of the distinctions of a partition on a given set. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. The quadratic form of the logical entropy lends itself to a generalization of the probabilities that include negative values, an idea that goes back to Feynman and Wigner. Here, we analyze and reinterpret negative probabilities in the light of the concept of logical entropy. Several intriguing quantum-like properties of the logical entropy are derived and discussed in finite dimensional spaces. For infinite-dimensional spaces (continuum), we show that, under the sole hypothesis that the logical entropy and the total probability are preserved in time, one obtains an evolution equation for the probability density that is basically identical to the quantum evolution of the Wigner function in phase space, at least when one considers only the momentum variable. This result suggest that the logical entropy plays a profound role in establishing the peculiar rules of quantum physics.

An Application of Quantum Logic to Experimental Behavioral Science

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

A New Understanding of the Origin of Wave-Particle Duality in Quantum Mechanics

The wave-particle duality of quantum mechanics has always been an unsolved problem in physics. This article attempts to use one of the properties to explain the other, so as to eliminate the confusion of quantum mechanics probability waves. Specifically, this article finds that the discreteness of energy is the inherent property of all waves, so this article explains the particle nature of light from the perspective of light wave, thereby eliminating the confusion of light’s wave-particle duality; in addition, this article found that microscopic matter particles are only suitable for discussing the number of scattered particles or energy flow density, not their position and momentum, when they are forced to discuss their position and momentum, it will inevitably lead to confusion about probability waves, when only discussing the number of scattered particles or energy flow density, its volatility can be explained from the perspective of particle nature, thereby eliminating the confusion of microscopic matter particle probability waves. When the attribute of light as a wave is established, the light needs to overcome the Hamiltonian of the medium in different constraint systems (gravitational fields) during the propagation process, which will produce a universal redshift phenomenon, this can provide a new understanding of cosmic redshift; when the property of light as a wave is established, it means that the speed of light is a constant speed relative to the medium, which can provide a new understanding of the principle of constant speed of light.

Conspiracy in ontological models: λ sufficiency and measurement contextuality

The conspiracy of ontic states responding to measurements contextually to comply with noncontextual quantum mechanical probabilities is analyzed for general ontological models. A general physical picture of ontological space structure and how ontic states are disturbed by measurement contexts is presented. A common assumption in ontological models called lambda-sufficiency is analyzed and argued to be inconsistent with measurement contextuality for ontological models with certain symmetry in ontic space with respect to quantum states and measurements.

The 4-Dimensional Dirac Equation in Relativistic Field Theory

An important milestone in quantum physics has been reachedby the publication of the Relativistic Quantum MechanicalDirac Equation in 1928. However, the Dirac equationrepresents a 1-Dimensional quantum mechanical equationwhich is unable to describe the 4-Dimensional PhysicalReality.In this article the 4-Dimensional Relativistic QuantumMechanical Dirac Equation expressed in the vectorprobability functions  and the complex conjugated vectorprobability function  *will be published. To realize this, theclassical boundaries of physics has to be changed. It isnecessary to go back in time 300 years ago. More than 200years ago before the Dirac Equation had been published.A Return to the Inception of Physics. The time of IsaacNewton who published in 1687 in the “PhilosophiaeNaturalis Principia Mathematica” a Universal FundamentalPrinciple in Physics which was in Harmony with Science andReligion. The Universal Path, the Leitmotiv, the UniversalConcept in Physics. Newton found the concept of “UniversalEquilibrium” which he formulated in his famous thirdequation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity.The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason these material waves in Schrödinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function.In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger's wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger's wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values.

Pendulum beams: optical modes that simulate the quantum pendulum

The wave equation of electromagnetism, the Helmholtz equation, has the same form as the Schrödinger equation, and so optical waves can be used to study quantum mechanical problems. The electromagnetic wave solutions for non-diffracting beams lead to the two-dimensional Helmholtz equation. When expressed in elliptical coordinates the solution of the angular part is the same as the Schrödinger equation for the simple pendulum. The resulting optical eigenmodes, Mathieu modes, have an optical Fourier transform with a spatial intensity distribution that is proportional to the quantum mechanical probability for the pendulum. Comparison of Fourier intensities of eigenmodes are in excellent agreement with calculated quantum mechanical probabilities of pendulum stationary states. We further investigate wave-packet superpositions of a few modes and show that they mimic the libration and the nonlinear rotation of the classical pendulum, including revivals due to the quantized nature of superpositions. The ability to ‘dial a wavefunction’ with the optical modes allows the exploration of important aspects of quantum wave-mechanics and the pendulum that may not be possible with other physical systems.

A Return to the Inception of Physics

A Return to the Beginning (Inception|) of Physics. The time of Isaac Newton who published in 1687 in the “Philosophiae Naturalis Principia Mathematica” a Universal Fundamental Principle in Physics which was in Harmony with Science and Religion. The Universal Path, the Leitmotiv, the Universal Concept in Physics. Newton found the concept of “Universal Equilibrium” which he formulated in his famous third equation Action = - Reaction. This article presents a New Kind of Physics based on this Universal Fundamental Concept in Physics which results in a New Approach in Quantum Physics and General Relativity. The physical concept of quantum mechanical probability waves has been created during the famous 1927 5th Solvay Conference. During that period there were several circumstances which came together and made it possible to create an unique idea of material waves being complex (partly real and partly imaginary) and describing the probability of the appearance of a physical object (elementary particle). The idea of complex probability waves was new in the beginning of the 20th century. Since then the New Concept has been protected carefully within the Copenhagen Interpretation. When Schrödinger published his famous material wave equation in 1926, he found spherical and elliptical solutions for the presence of the electron within the atom. The first idea of the material waves in Schrödinger’s wave equation was the concept of confined Electromagnetic Waves. But according to Maxwell this was impossible. According to Maxwell’s equations Electromagnetic Waves can only propagate along straight lines and it is impossible that Light (Electromagnetic Waves) could confine with the surface of a sphere or an ellipse. For that reason these material waves in Schrödinger’s wave equation could only be of a different origin than Electromagnetic Waves. Niels Bohr introduced the concept of “Probability Waves” as the origin of the material waves in Schrödinger’s wave equation. And defined the New Concept that the electron was still a particle but the physical presence of the electron in the Atom was equally divided by a spherical probability function. In the New Theory it will be demonstrated that because of a mistake in the Maxwell Equations, in 1927 Confined Electromagnetic waves could not be considered to be the material waves expressed in Schrödinger’s wave equation. The New Theory presents a new equation describing electromagnetic field configurations which are also solutions of the Schrodinger’s wave equation and the relativistic quantum mechanical Dirac Equation and carry mass, electric charge and magnetic spin at discrete values