Classical-quantum versus exact quantum results for a particle in a box (original) (raw)

Quantization Conditions, 1900–1927

The Oxford Handbook of the History of Quantum Interpretations

We trace the evolution of quantization conditions from Max Planck’s introduction of a new fundamental constant (h) in his treatment of blackbody radiation in 1900 to Werner Heisenberg’s interpretation of the commutation relations of modern quantum mechanics in terms of his uncertainty principle in 1927.

On the Classical-Quantum Relation of Constants of Motion

Frontiers in Physics, 2018

Groenewold-Van Hove theorem suggest that is not always possible to transform classical observables into quantum observables (a process known as quantization) in a way that, for all Hamiltonians, the constants of motion are preserved. The latter is a strong shortcoming for the ultimate goal of quantization, as one would expect that the notion of "constants of motion" is independent of the chosen physical scheme. It has been recently developed an approach to quantization that instead of mapping every classical observable into a quantum observable, it focuses on mapping the constants of motion themselves. In this article we will discuss the relations between classical and quantum theory under the light of this new form of quantization. In particular, we will examine the mapping of a class of operators that generalizes angular momentum where quantization satisfies the usual desirable properties.

PROBLEMS OF QUANTUM MECHANICS [QM.2]

This paper reopens the debate on the failure of quantum mechanics (QM) to provide any understanding of micro-reality. A critique is offered of the commonly accepted 'Copenhagen Interpretation' of a theory that is only a mathematical approach to the level of reality characterized by atoms and electrons. This critique is based on the oldest approach to thinking about nature for over 2500 years, known as Natural Philosophy. Quantum mechanics was developed over the first quarter of the 20th Century, when scientists were enthralled by a new philosophy known as Positivism, whose foundations were based on the assumption that material objects exist only when measured by humans – this central assumption conflates epistemology (knowledge) with ontology (existence). The present critique rejects this human-centered view of reality by assuming material reality has existed long before (and will persist long after) human beings (" Realism "). The defensive view that the micro-world is too different to understand using regular thinking (and only a mathematical approach is possible) is also rejected totally. At least 12 earlier QM interpretations are critically analyzed, indicating the broad interest in " what does QM mean? " The standard theory of quantum mechanics is thus constructed on only how the micro-world appears to macro measurements-as such, it cannot offer any view of how the foundations of the world are acting when humans are not observing it - this has generated almost 100 years of confusion and contradiction at the very heart of physics. Significantly, we live in a world that is not being measured by scientists but is interacting with itself and with us. QM has failed to provide reasonable explanations: only recipes (meaningless equations), not insights. Physics has returned to the pre-Newtonian world of Ptolemaic phenomenology: only verifiable numbers without real understanding. The focus needs to be on an explicit linkage between the micro-world, when left to itself, and our mental models of this sphere of material reality, via the mechanism of measurement. This now limits the role of measurement to confirming our mental models of reality but never confusing these with a direct image of 'the thing in itself'. This implies a deep divide between reality and appearances, as Kant suggested. This paper includes an original analysis of several major assumptions that have been implicit in Classical Mechanics (CM) that were acceptable in the macroscopic domain of reality, demonstrated by its proven successes. Unfortunately, only a few of these assumptions were challenged by the developers of QM. We now show that these other assumptions are still generating confusions in the interpretation of QM and blocking further progress in the understanding of the microscopic domain. Several of these flawed assumptions were introduced by Newton to support the use of continuum mathematics as a model of nature. This paper proposes that it is the attempt to preserve continuum mathematics (especially calculus), which drives much of the mystery and confusion behind all attempts at understanding quantum mechanics. The introduction of discrete mathematics is proposed to help analyze the discrete interactions between the quintessential quantum objects: the electrons and their novel properties. A related paper demonstrates that it is possible to create a point-particle theory of electrons that explains all their peculiar (and 'paradoxical') behavior using only physical hypotheses and discrete mathematics without introducing the continuum mathematical ideas of fields or waves. Another (related) paper proves that all the known results for the hydrogen atom can also be exactly calculated from this new perspective with the discrete mathematics.

A Continuous Transition Between Quantum and Classical Mechanics. II

Foundations of Physics, 2002

In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative formulation of classical mechanics which provides a continuous transition to quantum mechanics via environment-induced decoherence.

Theory of Quanta and Topics of Advanced Quantum Mechanics

Physics Today, 1994

ing on this practical but previously little-studied aspect of quantum measurement, the authors show a simple way to understand the limits that quantum mechanics places on classical measurements of force, energy, displacement and velocity. As in Braginsky's previous book, Systems with Small Dissipation (U. of Chicago P., Chicago, 1985), the ideas, all subtle, fundamental and useful, are the original work of the authors. The material is accessible to graduate students and can be interpreted for undergraduates. Quantum Measurement will provide easy-to-understand examples for the quantum mechanics texts of the future, and it will influence the direction of research in quantum measurements.

Quantization in classical mechanics and reality of Bohm's psi-field

Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of Hamilton systems in the form of the Schrodinger equation. It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with the Bohm "quantum" potential. Within the framework of Bohmian quantum mechanics supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Schrodinger equation is equivalent semantically and syntactically to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters. The conditions for the correctness of the Bohm-Chetaev interpretation of quantum mechanics are discussed. Comment: 16 pages, significant impr...