Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Dirac Quantum Optics

F.J. Duarte

Published Oct 29, 2013• 0 citations
Publisher

Abstract

The Paul Adrien Maurice Dirac notation is one of the mathematical avenues that can be used to describe nature quantum mechanically. This mathematical notation was invented by Dirac in 1939 and is particularly well suited to describe quantum optics. In Principles of Quantum Mechanics, first published in 1930, Dirac discusses the essence of interference as a one-photon phenomenon. Albeit his discussion is qualitative, it is also profound. In 1965, R. P. Feynman discussed electron interference via a two-slit thought experiment using probability amplitudes and the Dirac notation as a tool. The Dirac notation offers a natural avenue to describe the propagation of particles from a source to a detection plane, via a pair of slits. This was done by Feynman in a thought experiment using electrons and two slits.

DOI
10.1201/b16055-4

Related papers

Introduction

F.J. Duarte

This chapter explores the opinion on this subject given by a few luminaries of quantum physics: Paul Adrien Maurice Dirac, R. P. Feynman, H. Haken, and W. E. Lamb. An advance directly related to the physics of entanglement is quantum teleportation. Quantum mechanics came to light via the work of Max Planck, published in 1901. In that contribution, Planck used concepts of thermodynamics to explain the energy distribution of light sources as a function of wavelength. The Principles of Quantum Mechanics is the landmark book written by one of the creators of quantum mechanics Dirac. The first edition of this masterpiece was published in 1930, the second edition in 1935, and the third edition in 1947. The use of quantum physics in macroscopic optics is not unique to Dirac. In 1965, Feynman used his path integrals to describe divergence and diffraction resulting from the passage through a Gaussian slit.

CRC Press eBooks•Oct 29, 2013•0 citations

Quantum Interference

F. J. Duarte

Quantum interference, which is a physical phenomenon, and probability amplitudes à la Dirac are miraculously like synonyms. Indeed, the way probability amplitudes are written down on paper is very much as a single photon, or an ensemble of indistinguishable photons, propagates from one plane to the next. And then, if at one of the planes an N -slit array alternative is offered then the probability amplitude notation à la Dirac cleverly describes the N -path alternative via a mathematical summation. Here we should emphasize that since the photon is a nonlocal quantum entity, all the microscopic slits in the N -slit array, or transmission diffraction grating, are illuminated simultaneously.

CRC Press eBooks•Apr 10, 2026•0 citations

The Dirac–Feynman Quantum Interferometric Principle

F.J. Duarte

The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.

CRC Press eBooks•Jan 15, 2024•0 citations

Quantum Principles and the Probability Amplitude

F.J. Duarte

Twelve mathematical statements that define the quantum mechanics of interference and entanglement are listed in the chapter. The importance of superposition probability amplitudes, via the Dirac–Feynman principle, in quantum mechanics is highlighted. Particular attention is given to the proper description of experimental configurations via probability amplitudes. The refinement of spatial accuracy, only limited by Nature’s availability of shorter and shorter wavelengths, in N -slit quantum interferometers is described. The transition from superposition probability amplitudes to quantum probabilities is discussed. In all this, the essential importance of the nonlocality of the photon is emphasized. The importance of the probability amplitude of quantum entanglement at the foundations of quantum physics is discussed relative to quantum interference. The intrinsic link between these two principles is highlighted via a series of transparent mathematical statements.

CRC Press eBooks•Jan 15, 2024•0 citations

Probability Amplitudes

F. J. Duarte

Quantum mechanics runs on probability amplitudes. Probability amplitudes must accurately describe the physics prior to calculating the quantum probability. Symbolical probability amplitudes are mathematically represented by complex wave functions. Many physicists in the field don’t even mention the words probability amplitude and prefer to refer to wave function, or state, instead. When they do that they miss out on the subtleties of the physics at hand. In this brief chapter we emphasize probability amplitudes à la Dirac since that is the way Feynman and Ward preferred to describe quantum mechanics. The Dirac notation is also ideally suited to describe quantum interference and quantum entanglement.

CRC Press eBooks•Apr 10, 2026•0 citations

Quantum Interference without Quantum Mechanics

Arend Niehaus

A recently proposed model of the Dirac electron, which has been shown to describe several observed properties of the particle correctly, is in the present paper shown to be also able to explain quantum interference by classical probabilities. According to this model, the electron is not point-like, but rather an “entity with structure”, formed by a fast periodic motion of a “light-like object”, whose momentum (p) causes the angular momentum responsible for the spin, and whose energy (E = pc) is equal to the energy of the electron, mc2. A qualitative description of the model is given, together with the quantitative formulae that allow to discuss interference. Applied to the experimental situation of the “two-slit” experiment, the formulae yield the same time dependence of the detection probability as the quantum mechanical treatment, and hence the same interference pattern. In contrast to quantum mechanics, the pattern is due to “particle interference” rather than to “wave interference”. No wave-particle paradox arises. The merits of the model are summarized, and its physical content discussed.

arXiv:1804.00496•Journal of Modern Physics•Jan 1, 2019•3 citations
The Paul Adrien Maurice Dirac notation is one of the mathematical avenues that can be used to describe nature quantum mechanically. This mathematical notation was invented by Dirac in 1939 and is particularly well suited to describe quantum optics. In Principles of Quantum Mechanics, first published in 1930, Dirac discusses the essence of interference as a one-photon phenomenon. Albeit his discussion is qualitative, it is also profound. In 1965, R. P. Feynman discussed electron interference via a two-slit thought experiment using probability amplitudes and the Dirac notation as a tool. The Dirac notation offers a natural avenue to describe the propagation of particles from a source to a detection plane, via a pair of slits. This was done by Feynman in a thought experiment using electrons and two slits.
This chapter explores the opinion on this subject given by a few luminaries of quantum physics: Paul Adrien Maurice Dirac, R. P. Feynman, H. Haken, and W. E. Lamb. An advance directly related to the physics of entanglement is quantum teleportation. Quantum mechanics came to light via the work of Max Planck, published in 1901. In that contribution, Planck used concepts of thermodynamics to explain the energy distribution of light sources as a function of wavelength. The Principles of Quantum Mechanics is the landmark book written by one of the creators of quantum mechanics Dirac. The first edition of this masterpiece was published in 1930, the second edition in 1935, and the third edition in 1947. The use of quantum physics in macroscopic optics is not unique to Dirac. In 1965, Feynman used his path integrals to describe divergence and diffraction resulting from the passage through a Gaussian slit.
Quantum interference, which is a physical phenomenon, and probability amplitudes à la Dirac are miraculously like synonyms. Indeed, the way probability amplitudes are written down on paper is very much as a single photon, or an ensemble of indistinguishable photons, propagates from one plane to the next. And then, if at one of the planes an N -slit array alternative is offered then the probability amplitude notation à la Dirac cleverly describes the N -path alternative via a mathematical summation. Here we should emphasize that since the photon is a nonlocal quantum entity, all the microscopic slits in the N -slit array, or transmission diffraction grating, are illuminated simultaneously.
The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.
Twelve mathematical statements that define the quantum mechanics of interference and entanglement are listed in the chapter. The importance of superposition probability amplitudes, via the Dirac–Feynman principle, in quantum mechanics is highlighted. Particular attention is given to the proper description of experimental configurations via probability amplitudes. The refinement of spatial accuracy, only limited by Nature’s availability of shorter and shorter wavelengths, in N -slit quantum interferometers is described. The transition from superposition probability amplitudes to quantum probabilities is discussed. In all this, the essential importance of the nonlocality of the photon is emphasized. The importance of the probability amplitude of quantum entanglement at the foundations of quantum physics is discussed relative to quantum interference. The intrinsic link between these two principles is highlighted via a series of transparent mathematical statements.
Quantum mechanics runs on probability amplitudes. Probability amplitudes must accurately describe the physics prior to calculating the quantum probability. Symbolical probability amplitudes are mathematically represented by complex wave functions. Many physicists in the field don’t even mention the words probability amplitude and prefer to refer to wave function, or state, instead. When they do that they miss out on the subtleties of the physics at hand. In this brief chapter we emphasize probability amplitudes à la Dirac since that is the way Feynman and Ward preferred to describe quantum mechanics. The Dirac notation is also ideally suited to describe quantum interference and quantum entanglement.
A recently proposed model of the Dirac electron, which has been shown to describe several observed properties of the particle correctly, is in the present paper shown to be also able to explain quantum interference by classical probabilities. According to this model, the electron is not point-like, but rather an “entity with structure”, formed by a fast periodic motion of a “light-like object”, whose momentum (p) causes the angular momentum responsible for the spin, and whose energy (E = pc) is equal to the energy of the electron, mc2. A qualitative description of the model is given, together with the quantitative formulae that allow to discuss interference. Applied to the experimental situation of the “two-slit” experiment, the formulae yield the same time dependence of the detection probability as the quantum mechanical treatment, and hence the same interference pattern. In contrast to quantum mechanics, the pattern is due to “particle interference” rather than to “wave interference”. No wave-particle paradox arises. The merits of the model are summarized, and its physical content discussed.