Time in Quantum Mechanics

2011 ◽  
Author(s):  
Jan Hilgevoord ◽  
David Atkinson
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Classical Mechanics ◽  
Special Problem ◽  
Uncertainty Relations ◽  
Operator Formalism ◽  
Heisenberg Uncertainty ◽  
Time In Quantum Mechanics ◽  
Time Operators

Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.

scholarly journals Relaxed uncertainty relations and information processing

2009 ◽  
Vol 9 (9&10) ◽  
pp. 801-832 ◽  
Author(s):  
G. Ver Steeg ◽  
S. Wehner
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Random Access ◽  
The Other ◽  
Uncertainty Relations ◽  
Expectation Values ◽  
Chsh Inequality ◽  
Local Correlations ◽  
Non Local

We consider a range of "theories'' that violate the uncertainty relation for anti-commuting observables derived. We first show that Tsirelson's bound for the CHSH inequality can be derived from this uncertainty relation, and that relaxing this relation allows for non-local correlations that are stronger than what can be obtained in quantum mechanics. We continue to construct a hierarchy of related non-signaling theories, and show that on one hand they admit superstrong random access encodings and exponential savings for a particular communication problem, while on the other hand it becomes much harder in these theories to learn a state. We show that the existence of these effects stems from the absence of certain constraints on the expectation values of commuting measurements from our non-signaling theories that are present in quantum theory.

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 An analysis on Quantum mechanical Stability of Regular Polygons on a Point Base Using Heisenberg Uncertainty Principle

2021 ◽  
Vol 9 (10) ◽  
pp. 74-79
Author(s):  
Anurag Chapagain
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Stability Problem ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Heisenberg Uncertainty Principle ◽  
Heisenberg Uncertainty ◽  
Degree Of Stability ◽  
The Stability ◽  
Heisenberg's Uncertainty Principle

Abstract: It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg’s Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1- dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg’s uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world. Keywords: Quantum mechanics, Heisenberg Uncertainty principle, degree of stability, polygon, the highest degree of stability

scholarly journals On deformations of classical mechanics due to Planck-scale physics

2020 ◽  
Vol 29 (10) ◽  
pp. 2050070
Author(s):  
Olga I. Chashchina ◽  
Abhijit Sen ◽  
Zurab K. Silagadze
Keyword(s):  
Quantum System ◽  
Evolution Equations ◽  
Hidden Variables ◽  
Classical Mechanics ◽  
Planck Scale ◽  
Leibniz Rule ◽  
Uncertainty Relations ◽  
Interesting Possibility ◽  
Heisenberg Uncertainty ◽  
Space Formulation

Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

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.

scholarly journals BLACK HOLE UNCERTAINTIES

1993 ◽  
Vol 08 (20) ◽  
pp. 1925-1941
Author(s):  
ULF H. DANIELSSON
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Quantum Theory ◽  
Uncertainty Relation ◽  
Wave Functions ◽  
Uncertainty Relations ◽  
Two Dimensional ◽  
Black Hole Mass ◽  
Dilaton Black Holes

In this work the quantum theory of two-dimensional dilaton black holes is studied using the Wheeler-De Witt equation. The solutions correspond to wave functions of the black hole. It is found that for an observer inside the horizon, there are uncertainty relations for the black hole mass and a parameter in the metric determining the Hawking flux. Only for a particular value of this parameter can both be known with arbitrary accuracy. In the generic case there is instead a relation that is very similar to the so-called string uncertainty relation.

scholarly journals Quantum Mechanics Needs No Interpretation

2005 ◽  
Vol 70 (5) ◽  
pp. 621-637 ◽  
Author(s):  
Lubomír Skála ◽  
Vojtěch Kapsa
Keyword(s):  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Mathematical Structure ◽  
Uncertainty Relations ◽  
Mean Values ◽  
Vector Potentials ◽  
Probability Densities ◽  
Klein Gordon ◽  
Definition Of

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.

scholarly journals Uncertainty Relations in Hydrodynamics

2020 ◽  
Author(s):  
Gyell Gonçalves de Matos ◽  
Takeshi Kodama ◽  
Tomoi Koide
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Quantum Particle ◽  
Navier Stokes ◽  
Pitaevskii Equation ◽  
Uncertainty Relations ◽  
Derived Relations ◽  
Stochastic Variational Method ◽  
Virtual Trajectory ◽  
Fourier Equation

Uncertainty relations in hydrodynamics are numerically studied. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys.\ Lett.\ A\textbf{382}, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schr\"{o}dinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid.

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