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

Institute of Physics eBooks

A theory of polarized photons

K. Heyde, John L. Wood

Published Dec 1, 2019• 1 citation
Publisher

Abstract

The basic logic of quantum systems is presented. The system chosen is the simplest possible example: plane-polarized photons and polarizing filters, a two-state system. The mathematics of linear spaces and its role as the logical language of quantum mechanics is introduced. The use of Dirac bras and kets is demonstrated.
DOI
10.1088/978-0-7503-2179-2ch1

Related papers

Polarization: photons and spin-1/2 particles

Michel Le Bellac

In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.

Cambridge University Press eBooks•Mar 23, 2006•0 citations

Proposal for quantum physics in secondary school

Marisa Michelini, R. Ragazzon, Lorenzo Gianni Santi +1 more

A possible strategy to introduce the basic formalism of the `Dirac formulation' of quantum mechanics without requiring an advanced mathematical or physical background is proposed, generalizing the description of a simple two-state system, namely the linear polarization of photons interacting with polaroids and birefringent crystals. The relation between physical observables and linear operators is discussed on both a conceptual and a formal level. A feasibility study has been performed in the final class of a Liceo.

Physics Education•Nov 1, 2000•99 citations

Simplest Quantum Devices: Polarizers and Beam Splitters

Muhammad Suhail Zubairy

Maxwell showed that light consists of electric and magnetic fields that oscillate in directions perpendicular to the direction of propagation. Associated with this picture of light as an electromagnetic wave is an important property—the polarization of light. The polarization of light is related to the direction of oscillation of the electric field in an electromagnetic wave. In this chapter, the basic principles of quantum mechanics are discussed by studying the polarization property of a single photon. First the properties of a polarizer are presented and Malus’ law for polarized light is derived. Next it is shown that the basic features of quantum mechanics can be understood via an analysis of a single photon passing through a polarizer. This simple system allows an introduction of Dirac’s ket and bra notations for a quantum state. Finally the transformation properties of the quantum beam splitter and the polarization beam splitters are discussed.

Oxford University Press eBooks•May 7, 2020•0 citations

Mathematical Formalism of Quantum Mechanics

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem

•Sep 7, 2023•0 citations

Linear Operators for Quantum Mechanics

Thomas F. Jordan, Robert H. Good

This compact treatment highlights the logic and simplicity of the mathematical structure of quantum mechanics. Suitable for advanced undergraduates and graduate students, it treats the language of quantum mechanics as expressed in the mathematics of linear operators.Originally oriented toward atomic physics, quantum mechanics became a basic language for solid-state, nuclear, and particle physics. Its grammar consists of the mathematics of linear operators, and with this text, students will find it easier to understand and use the language of physics. Topics include linear spaces and linear fun

American Journal of Physics•Nov 1, 1969•143 citations

Basic formalism

Bipin R. Desai

We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.

Cambridge University Press eBooks•Dec 3, 2009•0 citations
In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.
A possible strategy to introduce the basic formalism of the `Dirac formulation' of quantum mechanics without requiring an advanced mathematical or physical background is proposed, generalizing the description of a simple two-state system, namely the linear polarization of photons interacting with polaroids and birefringent crystals. The relation between physical observables and linear operators is discussed on both a conceptual and a formal level. A feasibility study has been performed in the final class of a Liceo.
Maxwell showed that light consists of electric and magnetic fields that oscillate in directions perpendicular to the direction of propagation. Associated with this picture of light as an electromagnetic wave is an important property—the polarization of light. The polarization of light is related to the direction of oscillation of the electric field in an electromagnetic wave. In this chapter, the basic principles of quantum mechanics are discussed by studying the polarization property of a single photon. First the properties of a polarizer are presented and Malus’ law for polarized light is derived. Next it is shown that the basic features of quantum mechanics can be understood via an analysis of a single photon passing through a polarizer. This simple system allows an introduction of Dirac’s ket and bra notations for a quantum state. Finally the transformation properties of the quantum beam splitter and the polarization beam splitters are discussed.
Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem
This compact treatment highlights the logic and simplicity of the mathematical structure of quantum mechanics. Suitable for advanced undergraduates and graduate students, it treats the language of quantum mechanics as expressed in the mathematics of linear operators.Originally oriented toward atomic physics, quantum mechanics became a basic language for solid-state, nuclear, and particle physics. Its grammar consists of the mathematics of linear operators, and with this text, students will find it easier to understand and use the language of physics. Topics include linear spaces and linear fun
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.