scholarly journals Remarks on the order for quantum observables

2007 ◽  
Vol 57 (6) ◽  
Author(s):  
S. Pulmannová ◽  
E. Vinceková
Keyword(s):  
Classical Case ◽  
Effect Algebras ◽  
Quantum Case ◽  
Quantum Observables ◽  
Bounded Sets ◽  
Generalized Effect Algebras

AbstractRelations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.

XII.—On the Theory of Unimolecular Gas Reactions: A Quantum Harmonic Oscillator Model

1954 ◽  
Vol 64 (2) ◽  
pp. 161-174 ◽  
Author(s):  
N. B. Slater
Keyword(s):  
High Pressure ◽  
Classical Mechanics ◽  
Classical Case ◽  
Constant Factor ◽  
Quantum Case ◽  
Fundamental Vibration ◽  
Fixed Temperature ◽  
Dissociation Probability ◽  
Pressure Scale ◽  
Continuum Of States

SynopsisThe writer's theory of unimolecular dissociation rates, based on the treatment of the molecule as a harmonically vibrating system, is put in a form which covers quantum as well as classical mechanics. The classical rate formulæ are as before, and are also the high-temperature limits of the new quantum formulæ. The high-pressure first-order rate k∞ is found first from the Gaussian distribution of co-ordinates and momenta of harmonic systems, and is justified for the quantum-mechanical case by Bartlett and Moyal's phase-space distributions. This leads to a re-formulation of k∞ as a molecular dissociation probability averaged over a continuum of states, and to a general rate for any pressure of the gas.The high-pressure rate k∞ is of the form ve-F/kT, where v and F depend, in the quantum case, on the temperature T; but v is always between the highest and lowest fundamental vibration frequencies of the molecule. Concerning the decline of the general rate k with pressure at fixed temperature, k/k∞ is to a certain approximation the same function of as was tabulated earlier for the classical case, apart from a constant factor changing the pressure scale in the quantum case.

scholarly journals The Gibbs Paradox

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 552 ◽  
Author(s):  
Simon Saunders
Keyword(s):  
Statistical Mechanics ◽  
Cartesian Product ◽  
Classical Case ◽  
Particle State ◽  
Gibbs Paradox ◽  
Quantum Case ◽  
State Spaces ◽  
Open Questions ◽  
Thermodynamics And Statistical Mechanics ◽  
Sequence Position

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

scholarly journals Weakly ordered partial commutative group of self-adjoint linear operators densely defined on Hilbert space

2011 ◽  
Vol 50 (1) ◽  
pp. 63-78
Author(s):  
Jiří Janda
Keyword(s):  
Hilbert Space ◽  
Algebraic Structure ◽  
Complex Hilbert Space ◽  
Commutative Group ◽  
Pt Symmetry ◽  
Linear Operators ◽  
Effect Algebras ◽  
Infinite Dimensional ◽  
Densely Defined ◽  
Generalized Effect Algebras

ABSTRACT We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space ℋ. In [Paseka, J.- -Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72] there is introduced the notion of a weakly ordered partial commutative group and showed that linear operators on H with restricted addition possess this structure. In our work, we are investigating the set of self-adjoint linear operators on H showing that with more restricted addition it also has the structure of a weakly ordered partial commutative group.

Classical and quantum tensor product expanders

2009 ◽  
Vol 9 (3&4) ◽  
pp. 336-360
Author(s):  
M.B. Hastings ◽  
A.W. Harrow
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
Classical Case ◽  
Single System ◽  
Quantum Case ◽  
Kraus Operator ◽  
Expectation Value ◽  
Kraus Operators

We introduce the concept of quantum tensor product expanders. These generalize the concept of quantum expanders, which are quantum maps that are efficient randomizers and use only a small number of Kraus operators. Quantum tensor product expanders act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.

Topological Properties of Operator Generalized Effect Algebras

2012 ◽  
Vol 69 (3) ◽  
pp. 311-320 ◽  
Author(s):  
S. Pulmannová ◽  
Z. Riečanová ◽  
M. Zajac
Keyword(s):  
Effect Algebras ◽  
Topological Properties ◽  
Generalized Effect Algebras
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