Probabilities in Quantum Mechanics

2021 ◽  
pp. 205-238
Author(s):  
Wayne C. Myrvold
Keyword(s):  
Quantum Mechanics ◽  
Dynamical Evolution ◽  
Degrees Of Belief ◽  
Basic Formalism ◽  
Precise Knowledge

This chapter examines the role played by probabilities on each of the major approaches to understanding quantum mechanics. It is argued that the sorts of considerations brought up in previous chapters, having to do with limitations on precise knowledge of physical states, and the result of applying dynamical evolution to agents’ degrees of belief about those states, have a part to play on each of those approaches. The chapter includes an introduction to the basic formalism of quantum mechanics.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

2021 ◽  
Vol 10 (9) ◽  
pp. 3241-3251
Author(s):  
H. Umair ◽  
H. Zainuddin ◽  
K.T. Chan ◽  
Sh.K. Said Husein
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Uncertainty Relation ◽  
Dynamical Behavior ◽  
Dynamical Evolution ◽  
Riemannian Metric ◽  
Hamiltonian Flow ◽  
Space Dynamics ◽  
Geometric Quantum Mechanics ◽  
Complex Projective

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

scholarly journals Quantum Bayesianism Assessed

The Monist ◽  
2019 ◽  
Vol 102 (4) ◽  
pp. 403-423
Author(s):  
John Earman
Keyword(s):  
Quantum Mechanics ◽  
Measurement Problem ◽  
Degrees Of Belief ◽  
Quantum Probabilities

Abstract The idea that the quantum probabilities are best construed as the personal/subjective degrees of belief of Bayesian agents is an old one. In recent years the idea has been vigorously pursued by a group of physicists who fly the banner of quantum Bayesianism (QBism). The present paper aims to identify the prospects and problems of implementing QBism, and it critically assesses the claim that QBism provides a resolution (or dissolution) of some of the long-standing foundations issues in quantum mechanics, including the measurement problem and puzzles of nonlocality.

scholarly journals QUANTUM GROUP SCHRÖDINGER FIELD THEORY

1993 ◽  
Vol 08 (23) ◽  
pp. 2213-2221 ◽  
Author(s):  
MARCELO R. UBRIACO
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Quantum Group ◽  
Arbitrary Number ◽  
Quantum Deformation ◽  
Step Size ◽  
Basic Formalism ◽  
Nonlinear Lattice

We show that a quantum deformation of quantum mechanics given in a previous work is equivalent to quantum mechanics on a nonlinear lattice with step size ∆x=(1−q)x. Then, based on this, we develop the basic formalism of quantum group Schrödinger field theory in one spatial quantum dimension, and explicitly exhibit the SU q(2) covariant algebras satisfied by the q-bosonic and q-fermionic Schrödinger fields. We generalize this result to an arbitrary number of fields.

scholarly journals Semiclassical limits, Lagrangian states and coboundary equations

2017 ◽  
Vol 17 (02) ◽  
pp. 1750014 ◽  
Author(s):  
Artur O. Lopes ◽  
Joana Mohr
Keyword(s):  
Quantum Mechanics ◽  
Quantum State ◽  
Discrete Time ◽  
Semiclassical Limit ◽  
Dynamical Evolution ◽  
Transformation Formula ◽  
Continuous Transformation ◽  
State Formula ◽  
Continuous Potential ◽  
Semiclassical Limits

Assume that [Formula: see text] is a continuous transformation [Formula: see text]. We consider here the cases where [Formula: see text] is either the transformation [Formula: see text] or [Formula: see text] is a smooth diffeomorphism of the circle [Formula: see text]. Consider a fixed continuous potential [Formula: see text], [Formula: see text] and [Formula: see text] (a quantum state). The transformation [Formula: see text] acting on [Formula: see text], [Formula: see text], defined by [Formula: see text] describes a discrete time dynamical evolution of the quantum state [Formula: see text]. Given [Formula: see text] we define the Lagrangian state [Formula: see text] In this case [Formula: see text]. Under suitable conditions on [Formula: see text] the micro-support of [Formula: see text], when [Formula: see text], is [Formula: see text]. One of the meanings of the semiclassical limit in Quantum Mechanics is to take [Formula: see text] and [Formula: see text]. We address the question of finding [Formula: see text] such that [Formula: see text] satisfies the property: [Formula: see text], we have that [Formula: see text] has micro-support on the graph of [Formula: see text] (which is the micro-support of [Formula: see text]). In other words: which [Formula: see text] is such that [Formula: see text] leaves the micro-support of [Formula: see text] invariant? This is related to a coboundary equation for [Formula: see text], twist conditions and the boundary of the fat attractor.

scholarly journals Time Observables in a Timeless Universe

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 354
Author(s):  
Tommaso Favalli ◽  
Augusto Smerzi
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Hermitian Operator ◽  
Dynamical Evolution ◽  
Emergent Property ◽  
Time Operator ◽  
Static Universe ◽  
Unequally Spaced ◽  
Energy Ratios ◽  
Time In Quantum Mechanics

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially undistinguishables.

scholarly journals Schwinger’s picture of quantum mechanics II: Algebras and observables

2019 ◽  
Vol 16 (09) ◽  
pp. 1950136 ◽  
Author(s):  
F. M. Ciaglia ◽  
A. Ibort ◽  
G. Marmo
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Dynamical Evolution ◽  
Transition Functions ◽  
Classical Transition ◽  
Instrumental Role

The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.

Statistical results on discovered near-Earth asteroids

1999 ◽  
Vol 173 ◽  
pp. 81-86
Author(s):  
S. Berinde
Keyword(s):  
Dynamical Evolution ◽  
Minor Planet ◽  
Orbital Elements ◽  
The Earth ◽  
Near Earth Asteroids ◽  
Statistical Results

AbstractThe first part of this paper gives a recent overview (until July 1st, 1998) of the Near-Earth Asteroids (NEAs) database stored at Minor Planet Center. Some statistical interpretations point out strong observational biases in the population of discovered NEAs, due to the preferential discoveries, depending on the objects’ distances and sizes. It is known that many newly discovered NEAs have no accurately determinated orbits because of the lack of observations. Consequently, it is hard to speak about future encounters and collisions with the Earth in terms of mutual distances between bodies. Because the dynamical evolution of asteroids’ orbits is less sensitive to the improvement of their orbital elements, we introduced a new subclass of NEAs named Earth-encounter asteroids in order to describe more reliably the potentially dangerous bodies as impactors with the Earth. So, we pay attention at those asteroids having an encounter between their orbits and that of the Earth within 100 years, trying to classify these encounters.

From the Oort cloud to Halley-type comets

1999 ◽  
Vol 173 ◽  
pp. 327-338 ◽  
Author(s):  
J.A. Fernández ◽  
T. Gallardo
Keyword(s):  
Total Population ◽  
Complex Function ◽  
Dynamical Evolution ◽  
Oort Cloud ◽  
Capture Process ◽  
Influx Rate ◽  
Short Period ◽  
Dynamical Properties ◽  
Orbital Periods ◽  
Probability Of Capture

AbstractThe Oort cloud probably is the source of Halley-type (HT) comets and perhaps of some Jupiter-family (JF) comets. The process of capture of Oort cloud comets into HT comets by planetary perturbations and its efficiency are very important problems in comet ary dynamics. A small fraction of comets coming from the Oort cloud − of about 10−2− are found to become HT comets (orbital periods < 200 yr). The steady-state population of HT comets is a complex function of the influx rate of new comets, the probability of capture and their physical lifetimes. From the discovery rate of active HT comets, their total population can be estimated to be of a few hundreds for perihelion distancesq <2 AU. Randomly-oriented LP comets captured into short-period orbits (orbital periods < 20 yr) show dynamical properties that do not match the observed properties of JF comets, in particular the distribution of their orbital inclinations, so Oort cloud comets can be ruled out as a suitable source for most JF comets. The scope of this presentation is to review the capture process of new comets into HT and short-period orbits, including the possibility that some of them may become sungrazers during their dynamical evolution.

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