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CRC Press eBooks

Interferometry via Dirac's Notation

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

Various interferometers are described quantum mechanically using Dirac’s bra ket notation. Interferometers included in this description are the Fabry–Perot, the Hanbury Brown–Twiss, the HOM, the Mach–Zehnder, the Michelson, the N -slit, and the Sagnac interferometers.
DOI
10.1201/9781003398707-7

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The genesis of quantum optics can be found in the Dirac discussion on interference as disclosed in The Principles of Quantum Mechanics . This is a masterful and prophetic discussion that begins by considering a roughly monochromatic light source. The discussion continues by considering a beam of light consisting of a large number of photons. Besides Dirac, useful references on interferometry include Steel, Meaburn and Born and Wolf. The Hanbury Brown–Twiss effect originates in interferometric measurements performed by an “intensity interferometer” used for astronomical observations. A diagram of the stellar intensity interferometer used to determine the diameter of stars is depicted. Two-beam interferometers are optical devices that divide and then recombine a light beam. It is on recombination of the beams that interference occurs. The most well-known two-beam interferometers are the Sagnac interferometer, the Mach–Zehnder interferometer, and the Michelson interferometer.
According to Dirac's bra-ket notation, in an inner-product space, the inner product ⟨x ∣ y⟩\langle x\,|\,y\rangle⟨x∣y⟩ of vectors x,yx,yx,y can be viewed as an application of the bra ⟨x∣\langle x|⟨x∣ to the ket ∣y⟩|y\rangle∣y⟩. Here ⟨x∣\langle x|⟨x∣ is the linear functional ∣y⟩↦⟨x ∣ y⟩|y\rangle \mapsto \langle x\,|\,y\rangle∣y⟩↦⟨x∣y⟩ and ∣y⟩|y\rangle∣y⟩ is the vector yyy. But often -- though not always -- there are advantages in seeing ∣y⟩|y\rangle∣y⟩ as the function a↦a⋅ya \mapsto a\cdot ya↦a⋅y where aaa ranges over the scalars. For example, the outer product ∣y⟩⟨x∣|y\rangle\langle x|∣y⟩⟨x∣ becomes simply the composition ∣y⟩∘⟨x∣|y\rangle \circ \langle x|∣y⟩∘⟨x∣. It would be most convenient to view kets sometimes as vectors and sometimes as functions, depending on the context. This turns out to be possible. While the bra-ket notation arose in quantum mechanics, this note presupposes no familiarity with quantum mechanics.
Abstract Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also includes complete orthonormal sets of eigenfunctions, the translation operator, current, spin, equation of continuity, gauge transformation, determinant & permanent multiparticle energy eigenfunctions for noninteracting particles and the Pauli exclusion principle. Attention is then focused on Dirac bra-ket notation and complete sets of commuting observables. In this connection, representations and transformation among representations are discussed in detail for the Schrödinger system state vector and the eigenstates, as well as bra-ket matrix elements of operators. Finally, Schwinger’s interpretation of ket-bra matrix operator structures (Schwinger “Measurement Symbols”) in terms of annihilation and creation of systems in eigenstates is introduced.
Dirac introduced a very useful notation of state vectors Ψ i in terms of "bra" vectors. This notation allows to make the formal expressions of quantum physics more transparent and easier to manipulate. The bra-ket notation extends to action of operators on state vectors. The projection operator and the completeness relation are very important quantities in quantum physics. They are very useful in calculations and are often employed to represent an arbitrary operator and a wave function (state) in terms of orthonormal states. To illustrate the procedure of representations, this chapter considers an arbitrary operator  and shows how to represent the operator in terms of projection operators of known orthonormal states. The example in the chapter illustrates the concept of orthonormal states. It presents the application of the completeness relation. Using the completeness relation, that matrix elements of the product operator in the same orthonormal basis can be found from the multiplication of the matrix elements.