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

The Journal of Engineering and Exact Sciences

Calculation of the creation and annihilation operators of a free particle in the light cone coordinate formalism

Jorge Kysnney Santos Kamassury, Damião Pedro Meira Filho, Charles da Rocha Silva, João Bosco Soares Pampolha, Robhyson Denys Rodrigues da Silva, Asael Ribeiro Pinto

Published Oct 19, 2021• 1 citation
Publisher

Abstract

This work didactically presents the mathematical procedures required for the construction of the creation and annihilation operators for a free quantum particle considering the coordinates of the light cone. For that, the relationships between the aforementioned coordinates and the coordinates (ct, x, y, z) are listed, in addition to the use of the Klein-Gordon-Fock equation in the formalism of the light cone coordinates. Finally, the temporal evolution operator and the quantum operators of creation and annihilation of the integral type of motion are obtained.

DOI
10.18540/jcecvl7iss4pp13317-01-15e

Related papers

Scalar product according to Klein-Fock-Gordon equation in the context of light cone coordinates / Produto escalar associado a equação de Klein-Fock-Gordon no contexto de coordenadas de cone de luz

Damião Pedro Meira Filho, Eliesio Alves da Silva, Railene Martins de Araujo +3 more

The objective of the present work is to didactically analyze, in the context of relativistic quantum mechanics, the algebraic structuring of the dot product between two quantum wavefunctions relative to a spin-0 quantum particle, assuming the light cone coordinates. For that, the linear momentum tensor is used in four dimensions, considering the description of the operational and quantum character of energy and linear momentum in three dimensions, as well as defining a pseudo volumetric probability density from which the product is reached scalar, properly written in terms of light cone coordinates.

Brazilian Journal of Development•May 4, 2022•1 citation

A note on relativistic second quantization

T. S. Chang

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.

Mathematical Proceedings of the Cambridge Philosophical Society•Apr 1, 1947•1 citation

Introduction to the "second quantization" formalism for non-relativistic quantum mechanics: A possible substitution for Sections 6.7 and 6.8 of Feynman's "Statistical Mechanics"

Hal Tasaki

This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general $N$ body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)

arXiv:1812.10732•Dec 27, 2018•1 citation

Generalized Version of the Creation and Annihilation Operators for the Schrödinger Equation

Luis C. N. Santos, C. C. Barros

A generalized version of the creation and annihilation operators is constructed and the factorization of the Schrödinger equation is investigated. It is shown that the generalized version of factorization operators yield a factorization for the twelve different separable coordinates for the Schrödinger equation.

arXiv:1701.08581•Jan 30, 2017•0 citations

Elementary derivation of the quantum propagator for the harmonic oscillator

Jiushu Shao

Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.

American Journal of Physics•Sep 21, 2016•9 citations

- Creation and Annihilation Operators in Fock Space

Girish S. Setlur

In this chapter, we discuss the notion of creation and annihilation operators. These operators correspond to addition or removal of excitations of a system of particles or particles themselves. We show that these operators may be used to rewrite manybody Hamiltonians in a compact form where information about Bose or Fermi statistics is encoded in the Hamiltonian itself, which greatly reduces the effort required in studying its properties.

CRC Press eBooks•Dec 5, 2013•0 citations
This work didactically presents the mathematical procedures required for the construction of the creation and annihilation operators for a free quantum particle considering the coordinates of the light cone. For that, the relationships between the aforementioned coordinates and the coordinates (ct, x, y, z) are listed, in addition to the use of the Klein-Gordon-Fock equation in the formalism of the light cone coordinates. Finally, the temporal evolution operator and the quantum operators of creation and annihilation of the integral type of motion are obtained.
The objective of the present work is to didactically analyze, in the context of relativistic quantum mechanics, the algebraic structuring of the dot product between two quantum wavefunctions relative to a spin-0 quantum particle, assuming the light cone coordinates. For that, the linear momentum tensor is used in four dimensions, considering the description of the operational and quantum character of energy and linear momentum in three dimensions, as well as defining a pseudo volumetric probability density from which the product is reached scalar, properly written in terms of light cone coordinates.
The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.