scholarly journals Operator calculus in the electron theory of metals

1940 ◽  
Vol 176 (965) ◽  
pp. 214-228 ◽  
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Differential Operator ◽  
Multiplication Operator ◽  
Position Operator ◽  
Electron Theory ◽  
Operator Calculus ◽  
The Common ◽  
Finite Space ◽  
Proper Functions

An operator calculus is developed applicable to problems in the electron theory of metals. It differs from the common operator calculus of the quantum theory in the fact that the wave function is defined in a finite space (the atomic polyhedron) bounded by a finite surface. This leads to the introduction of surface operators. The position operator x cannot be developed with respect to the proper functions of the Hamiltonian. Instead an operator ξ is introduced, which is essentially the Fourier development of x. Thus there are three fundamental types of operators: the differential operator p, the multiplication operator ξ and the surface operators. It is shown that with the help of these a consistent calculus can be developed.

On Electromagnetic Spinors and Electron Theory

1989 ◽  
Vol 44 (4) ◽  
pp. 327-328
Author(s):  
Warren H. Inskeep
Keyword(s):  
Quantum Theory ◽  
Differential Operator ◽  
Finite Mass ◽  
Dirac Theory ◽  
Electron Theory ◽  
Proper Place ◽  
Mechanical Approach ◽  
Quantum Operators ◽  
Energy And Momentum ◽  
The Relationship

Abstract The relationship between the Dirac theory and electromagnetic spinors is extended to the case of finite mass. Certain products of the electromagnetic fields give rise to the Dirac differential operator upon the usual subsitutions for the energy and momentum. By placing mass in the proper place for the wave mechanical approach to quantum theory, the algebra of the fields, interpreted as quantum operators, may be deduced.

Operators and meaning of wave function

2016 ◽  
pp. 4039-4042
Author(s):  
Viliam Malcher
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
State Vector ◽  
The State ◽  
Physical Significance ◽  
Central Problem ◽  
Quantum Mechanical ◽  
Mechanical Description ◽  
Quantum Mechanical Description ◽  
Interpretation Of Quantum Theory

The interpretation problems of quantum theory are considered. In the formalism of quantum theory the possible states of a system are described by a state vector. The state vector, which will be represented as |ψ> in Dirac notation, is the most general form of the quantum mechanical description. The central problem of the interpretation of quantum theory is to explain the physical significance of the |ψ>. In this paper we have shown that one of the best way to make of interpretation of wave function is to take the wave function as an operator.

Causality Then and Now

1993 ◽  
Vol 10 (2) ◽  
pp. 165-177
Author(s):  
Karen Harding
Keyword(s):  
Quantum Theory ◽  
Common Sense ◽  
Mechanical Model ◽  
Physical World ◽  
Scientific Data ◽  
The World ◽  
The Common ◽  
The Universe ◽  
The Way

Ate appearances deceiving? Do objects behave the way they do becauseGod wills it? Ate objects impetmanent and do they only exist becausethey ate continuously created by God? According to a1 Ghazlli, theanswers to all of these questions ate yes. Objects that appear to bepermanent are not. Those relationships commonly tefemed to as causalare a result of God’s habits rather than because one event inevitably leadsto another. God creates everything in the universe continuously; if Heceased to create it, it would no longer exist.These ideas seem oddly naive and unscientific to people living in thetwentieth century. They seem at odds with the common conception of thephysical world. Common sense says that the universe is made of tealobjects that persist in time. Furthermore, the behavior of these objects isreasonable, logical, and predictable. The belief that the univetse is understandablevia logic and reason harkens back to Newton’s mechanical viewof the universe and has provided one of the basic underpinnings ofscience for centuries. Although most people believe that the world is accutatelydescribed by this sort of mechanical model, the appropriatenessof such a model has been called into question by recent scientificadvances, and in particular, by quantum theory. This theory implies thatthe physical world is actually very different from what a mechanicalmodel would predit.Quantum theory seeks to explain the nature of physical entities andthe way that they interact. It atose in the early part of the twentieth centuryin response to new scientific data that could not be incorporated successfullyinto the ptevailing mechanical view of the universe. Due largely ...

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 Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

scholarly journals Semi-classical approximations based on Bohmian mechanics

2020 ◽  
Vol 35 (14) ◽  
pp. 2050070 ◽  
Author(s):  
Ward Struyve
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Expectation Values ◽  
Usual Approach ◽  
Classical Approximation

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

scholarly journals The Strange (Hi)story of Particles and Waves

2016 ◽  
Vol 71 (3) ◽  
pp. 195-212
Author(s):  
H. Dieter Zeh
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Excited States ◽  
Configuration Space ◽  
Historical Context ◽  
Quantum States ◽  
Wave Functions ◽  
General Readership ◽  
Boson Fields

AbstractThis is an attempt of a non-technical but conceptually consistent presentation of quantum theory in a historical context. While the first part is written for a general readership, Section 5 may appear a bit provocative to some quantum physicists. I argue that the single-particle wave functions of quantum mechanics have to be correctly interpreted as field modes that are “occupied once” (i.e. first excited states of the corresponding quantum oscillators in the case of boson fields). Multiple excitations lead to apparent many-particle wave functions, while the quantum states proper are defined by wave function(al)s on the “configuration” space of fundamental fields, or on another, as yet elusive, fundamental local basis.

scholarly journals Space-time from Collapse of the Wave-function

2019 ◽  
Vol 74 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Hilbert Space ◽  
Wave Function ◽  
Quantum Theory ◽  
Minkowski Space ◽  
Time Scales ◽  
Quantum Interference ◽  
Quantum Dynamics ◽  
Space Time ◽  
Minkowski Space Time ◽  
Macroscopic Objects

AbstractWe propose that space-time results from collapse of the wave function of macroscopic objects, in quantum dynamics. We first argue that there ought to exist a formulation of quantum theory which does not refer to classical time. We then propose such a formulation by invoking an operator Minkowski space-time on the Hilbert space. We suggest relativistic spontaneous localisation as the mechanism for recovering classical space-time from the underlying theory. Quantum interference in time could be one possible signature for operator time, and in fact may have been already observed in the laboratory, on attosecond time scales. A possible prediction of our work seems to be that interference in time will not be seen for ‘time slit’ separations significantly larger than 100 attosecond, if the ideas of operator time and relativistic spontaneous localisation are correct.

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