scholarly journals The Gibbs paradox and the physical criteria for indistinguishability of identical particles

2016 ◽  
Vol 14 (06) ◽  
pp. 1640037 ◽  
Author(s):  
C. S. Unnikrishnan
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Interference ◽  
Classical Mechanics ◽  
Gibbs Paradox ◽  
Quantitative Criterion ◽  
Identical Particles ◽  
Von Neumann ◽  
Paradoxical Situation ◽  
Classical Picture

Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits quantum theory for enabling indistinguishability of identical particles to solve the problem. We argue that indistinguishability of identical particles is already a feature in classical mechanics and this is clearly brought out when the problem is treated in the language of information and associated entropy. We pinpoint the physical criteria for indistinguishability that is crucial for the treatment of the Gibbs’ problem and the consistency of its solution with conventional thermodynamics. Quantum mechanics provides a quantitative criterion, not possible in the classical picture, for the degree of indistinguishability in terms of visibility of quantum interference, or overlap of the states as pointed out by von Neumann, thereby endowing the entropy expression with mathematical continuity and physical reasonableness.

Subsequent and Subsidiary? Rethinking the Role of Applications in Establishing Quantum Mechanics

2015 ◽  
Vol 45 (5) ◽  
pp. 641-702 ◽  
Author(s):  
Jeremiah James ◽  
Christian Joas
Keyword(s):  
Molecular Structure ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Mechanics ◽  
Theory Construction ◽  
Science Pedagogy ◽  
Old Quantum Theory ◽  
The Common ◽  
Complete Quantum

As part of an attempt to establish a new understanding of the earliest applications of quantum mechanics and their importance to the overall development of quantum theory, this paper reexamines the role of research on molecular structure in the transition from the so-called old quantum theory to quantum mechanics and in the two years immediately following this shift (1926–1928). We argue on two bases against the common tendency to marginalize the contribution of these researches. First, because these applications addressed issues of longstanding interest to physicists, which they hoped, if not expected, a complete quantum theory to address, and for which they had already developed methods under the old quantum theory that would remain valid under the new mechanics. Second, because generating these applications was one of, if not the, principal means by which physicists clarified the unity, generality, and physical meaning of quantum mechanics, thereby reworking the theory into its now commonly recognized form, as well as developing an understanding of the kinds of predictions it generated and the ways in which these differed from those of the earlier classical mechanics. More broadly, we hope with this article to provide a new viewpoint on the importance of problem solving to scientific research and theory construction, one that might complement recent work on its role in science pedagogy.

scholarly journals Relativistic quantum mechanics

1932 ◽  
Vol 136 (829) ◽  
pp. 453-464 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Correspondence Principle ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Relativistic Quantum Mechanics ◽  
Classical Electrodynamics ◽  
Quantum Phenomena

The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approxi­mation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.

scholarly journals Measurement theory in classical mechanics

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
So Katagiri
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Measurement Theory ◽  
Classical Mechanics ◽  
The State ◽  
Measurement Device ◽  
Von Neumann ◽  
Relative Interpretation ◽  
Classical And Quantum Mechanics

Abstract We investigate measurement theory in classical mechanics in the formulation of classical mechanics by Koopman and von Neumann (KvN), which uses Hilbert space. We show a difference between classical and quantum mechanics in the “relative interpretation” of the state of the target of measurement and the state of the measurement device. We also derive the uncertainty relation in classical mechanics.

INTERPRETATION AND PREDICTABILITY OF QUANTUM MECHANICS AND QUANTUM COSMOLOGY

1988 ◽  
Vol 03 (07) ◽  
pp. 645-651 ◽  
Author(s):  
SUMIO WADA
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Physical Meaning ◽  
Quantum Cosmology ◽  
Degrees Of Freedom ◽  
Interpretation Of Quantum Mechanics ◽  
Probabilistic Interpretation ◽  
The Universe ◽  
Classical Picture

A non-probabilistic interpretation of quantum mechanics asserts that we get a prediction only when a wave function has a peak. Taking this interpretation seriously, we discuss how to find a peak in the wave function of the universe, by using some minisuperspace models with homogeneous degrees of freedom and also a model with cosmological perturbations. Then we show how to recover our classical picture of the universe from the quantum theory, and comment on the physical meaning of the backreaction equation.

Quantum logic

2018 ◽  
Author(s):  
Peter Forrest
Keyword(s):  
Quantum Theory ◽  
Quantum Logic ◽  
Algebraic System ◽  
Classical Mechanics ◽  
Remarkable Case ◽  
Von Neumann ◽  
Sentential Calculus ◽  
Subsequent Discussion ◽  
Term Logic ◽  
Formal Properties

The topic of quantum logic was introduced by Birkhoff and von Neumann (1936), who described the formal properties of a certain algebraic system associated with quantum theory. To avoid begging questions, it is convenient to use the term ‘logic’ broadly enough to cover any algebraic system with formal characteristics similar to the standard sentential calculus. In that sense it is uncontroversial that there is a logic of experimental questions (for example, ‘Is the particle in region R?’ or ‘Do the particles have opposite spins?’) associated with any physical system. Having introduced this logic for quantum theory, we may ask how it differs from the standard sentential calculus, the logic for the experimental questions in classical mechanics. The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity. All this can be discussed without deciding whether quantum logic is a genuine logic, in the sense of a system of deduction. Putnam argued that quantum logic was indeed a genuine logic, because taking it as such solved various problems, notably that of reconciling the wave-like character of a beam of, say, electrons, as it passes through two slits, with the thesis that the electrons in the beam go through one or other of the two slits. If Putnam’s argument succeeds this would be a remarkable case of the empirical defeat of logical intuitions. Subsequent discussion, however, seems to have undermined his claim.

The basis of statistical quantum mechanics

1929 ◽  
Vol 25 (1) ◽  
pp. 62-66 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Dynamical System ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Degrees Of Freedom ◽  
Present Note ◽  
Classical Mechanics ◽  
The Other ◽  
Statistical Nature ◽  
Actual System ◽  
Quantum Treatment

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.

scholarly journals Irreversibility in quantum mechanics

2004 ◽  
Vol 2004 (1) ◽  
pp. 75-83 ◽  
Author(s):  
R. C. Bishop ◽  
A. Bohm ◽  
M. Gadella
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Boundary Conditions ◽  
Liouville Equation ◽  
Quantum Systems ◽  
Arrow Of Time ◽  
Time Asymmetry ◽  
Classical Physics ◽  
Von Neumann ◽  
Signal Features

Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time-asymmetric boundary conditions of time-symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann-Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time-asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.

scholarly journals The Statistical Origins of Quantum Mechanics

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
U. Klein
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Mechanics ◽  
Statistical Interpretation ◽  
Interpretation Of Quantum Mechanics ◽  
Local Conservation ◽  
Expectation Values ◽  
Strong Argument ◽  
Probability Current ◽  
The Mean

It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.

Popper and the Quantum Theory

1995 ◽  
Vol 39 ◽  
pp. 163-176
Author(s):  
Michael Redhead
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Scientific Discovery ◽  
Von Neumann ◽  
Interpretation Of Probability ◽  
Propensity Interpretation

Popper wrote extensively on the quantum theory. In Logic der Forschung (LSD) he devoted a whole chapter to the topic, while the whole of Volume 3 of the Postscript to the Logic of Scientific Discovery is devoted to the quantum theory. This volume entitled Quantum Theory and the Schism in Physics (QTSP) incorporated a famous earlier essay, ‘Quantum Mechanics without “the Observer”’ (QM). In addition Popper's development of the propensity interpretation of probability was much influenced by his views on the role of probability in quantum theory, and he also wrote an insightful critique of the 1936 paper of Birkhoff and von Neumann on nondistributive quantum logic (BNIQM).

scholarly journals ON KOOPMAN–VON NEUMANN WAVES

2002 ◽  
Vol 17 (09) ◽  
pp. 1301-1325 ◽  
Author(s):  
D. MAURO
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Wave Functions ◽  
Quantum Case ◽  
Von Neumann ◽  
Interference Experiment ◽  
Classical Wave

In this paper we study the classical Hilbert space introduced by Koopman and von Neumann in their operatorial formulation of classical mechanics. In particular we show that the states of this Hilbert space do not spread, differently from what happens in quantum mechanics. The role of the phases associated to these classical "wave functions" is analyzed in detail. In this framework we also perform the analog of the two-slit interference experiment and compare it with the quantum case.

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