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The Physics of Quantum Mechanics

James Jeffrey Binney, David Skinner

Published Dec 5, 2008• 42 citations
OpenAlex

Abstract

The Physics of Quantum Mechanics aims to give students a good understanding of how quantum mechanics describes the material world. It shows that the theory follows naturally from the use of probability amplitudes to derive probabilities. It stresses that stationary states are unphysical mathematical abstractions that enable us to solve the theory's governing equation, the time-dependent Schroedinger equation. Every opportunity is taken to illustrate the emergence of the familiarclassical, dynamical world through the quantum interference of stationary states. The text stresses the continuity be

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The Physics of Quantum Mechanics aims to give students a good understanding of how quantum mechanics describes the material world. It shows that the theory follows naturally from the use of probability amplitudes to derive probabilities. It stresses that stationary states are unphysical mathematical abstractions that enable us to solve the theory's governing equation, the time-dependent Schroedinger equation. Every opportunity is taken to illustrate the emergence of the familiarclassical, dynamical world through the quantum interference of stationary states. The text stresses the continuity be
This PhD thesis is submitted as a \emph{compendium} of the articles \cite{NS, holographic, topological}. The following has been adapted from their abstracts. Quantum mechanics has been argued to be a coarse--graining of some underlying deterministic theory. Here we support this view by establishing mappings between non-relativistic quantum mechanics and thermodynamic theories, since the latter are the paradigm of an emergent theory. First, we map certain solutions of the Schroedinger equation to solutions of the irrotational Navier--Stokes equation for viscous fluid flow. Although this is formally a generalization of Madelung's hydrodynamical interpretation, the presence of a viscous term leads to a novel interpretation. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state--dependent) viscosity of this fluid is proportional to Planck's constant, while the volume density of entropy is proportional to Boltzmann's constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information--loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier--Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation). Then, we present a map of standard quantum mechanics onto classical thermodynamics of irreversible processes. In particular, the propagators of the quantum harmonic oscillator are mapped to the conditional probabilities that solve the Chapman-Kolmogorov equation for Markovian Gaussian processes. While no gravity is present in our construction, our map exhibits features that are reminiscent of the holographic principle of quantum gravity. Finally, the classical thermostatics of equilibrium processes is shown to possess a quantum mechanical dual theory with a finite dimensional Hilbert space of quantum states. Specifically, the kernel of a certain Hamiltonian operator becomes the Hilbert space of quasistatic quantum mechanics. The relation of thermostatics to topological field theory is also discussed in the context of the approach of emergence of quantum theory, where the concept of entropy plays a key role.
Applications of quantum mechanics have led to many successful predictions and explanations of puzzling phenomena, and we now apply quantum mechanics to gain, process, and communicate information in novel ways. We can understand quantum mechanics by understanding how we have applied it. We should not seek agreement on the nature of the world it represents, because this theory does not itself represent the physical world (though its applications do help us to represent it better). When applied to a quantum state, quantum mechanics yields probabiities for physical events: both state and probability are objective--not because they represent elements of phyiscal reality, but because each exerts norrmative authority over the beliefs of anyone who accepts quantum mechanics and applies it relative to a physical situation they may (but need not) occupy. These events may be described by statements that are meaningful in an appropriate environmental context, and quantum mechanics can help one to say when that is. Measurement creates an appropriate context, so here the Born rule indirectly yields probabilities of measurement outcomes. The quantum state of a system does not "collapse" on measurement: a new state must be assigned relative to a physical situation in which information about the outcome is accessible. Understood this way, there is no measurement problem, and violations of Bell inequalities does not demonstrate "spooky" non-local action. Quantum field theories have no physical ontology of their own: a quantum field is a mathematical object in a model whose application helps us to improve and extend our descriptions of the world in other terms. We cannot realise the scenario of Wigner's friend and its recent extensions: but the data that provide overwhelming evidence for quantum mechanics are objective in the same sense as the relative measurement outcomes described in those scenarios.
This introductory text emphasizes Feynman's development of path integrals and its application to wave theory for particles. Suitable for undergraduate and graduate students of physics, the well-written, clear, and rigorous text was written by two of the nation's leading authorities on quantum physics. A solid foundation in quantum mechanics and atomic physics is assumed. Early chapters provide background in the mathematical treatment and particular properties of ordinary wave motion that also apply to particle motion. The close relation of quantum theory to physical optics is stressed. Subsequent sections emphasize the physical consequences of a wave theory of material properties, and they offer extensive applications in atomic physics, nuclear physics, solid state physics, and diatomic molecules. Four helpful Appendixes supplement the text.
Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier–Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck’s constant, while the volume density of entropy is proportional to Boltzmann’s constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier–Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).
The quantum world is allegedly strange. But is it? What if there’s a simple mathematical explanation, and a simple solution? The success of quantum mechanics (QM) arises from the accuracy of its probability predictions, which are obtained by squaring amplitudes (the Born rule). Suppose for a moment that nature uses the negative of QM’s equations. When squared they would yield the same probabilities, confirmed by the same experiments and technological triumphs. If that were true, if nature uses the negative of QM’s equations, then the quantum world would become transparent, easy to understand. No more Schrödinger’s-cat. No quantum-eraser. No backwards-in-time cause-and-effect. No paradoxes nor enigmas. But, what’s a negative quantum equation? It could mean that particles follow zero-energy waves backwards, instead of forwards. That’s still an eccentric idea, a residual strangeness. Overall, it’s a bargain. We could swap one odd idea for another, because wave-particle duality is odd. We are accustomed to wave-particle duality. But we’ll show that experiments support the other arrangement: quantum particles follow zero-energy waves backwards. How could that possibly be true? We present substantial experimental evidence, new mathematics, and six dozen colorful illustrations. This is the Theory of Elementary Waves (TEW).