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arXiv:2605.25857 · Physical Review A

Photon position eigenstates in configuration space

Artemio González-López, Luis Martínez Alonso

quant-phmath-phphysics.opticsSubmitted May 25, 2026• Announced May 26, 2026• 0 citations• Revised Aug 24, 2026 (v2)
v1 · May 25, 2026v2 · Aug 24, 2026
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Abstract

The expressions of the eigenfunctions of the Hawton photon position operator in configuration space are derived for several classes of wave function, including the Riemann--Silberstein and Landau--Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals $K(κ)$ and $E(κ)$ with modulus $κ$ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value $\mathbf q$ of the position eigenvalue, but also on a plane containing $\mathbf q$ and that they decay as inverse powers of the distance from $\mathbf q$.

Journal reference
Phys.Rev. A 114 (2026) 022204
DOI
10.1103/ll57-rj9l
Comments
17 pages, 2 figures. Minor changes from previous version to match the published paper

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The expressions of the eigenfunctions of the Hawton photon position operator in configuration space are derived for several classes of wave function, including the Riemann--Silberstein and Landau--Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals K(κ)K(κ)K(κ) and E(κ)E(κ)E(κ) with modulus κκκ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value q\mathbf qq of the position eigenvalue, but also on a plane containing q\mathbf qq and that they decay as inverse powers of the distance from q\mathbf qq.
It is shown that the photon position operator X⃗^\hat{\vec{X}}X^ with commuting components introduced by Margaret Hawton can be written in the momentum representation as X⃗^=iD⃗^\hat{\vec{X}}=i \hat{\vec{D}}X^=iD^, where D⃗^\hat{\vec{D}}D^ is a flat connection in the tangent bundle T(R3∖{(0,0,k3)∈R3:k3≥0})T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})T(R3∖{(0,0,k3​)∈R3:k3​≥0}) over R3∖{(0,0,k3)∈R3:k3≥0}\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}R3∖{(0,0,k3​)∈R3:k3​≥0} equipped with the Cartesian structure. Moreover, D⃗^\hat{\vec{D}}D^ is such that the tangent 222-planes orthogonal to the momentum are parallelly propagated with respect to D⃗^\hat{\vec{D}}D^ and, also, D⃗^\hat{\vec{D}}D^ is an anti-Hermitian operator with respect to the scalar product ⟨Ψ∣H^−2s∣Φ⟩\langle \mathbf{\Psi} | \hat{H}^{-2s} |\mathbf{\Phi} \rangle⟨Ψ∣H^−2s∣Φ⟩. The eigenfunctions ΨX⃗(x⃗)\mathbf{\Psi}_{\vec{X}} (\vec{x})ΨX​(x) of the position operator X⃗^\hat{\vec{X}}X^ are found.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
It is a commonplace that eigenfunctions in momentum space are the Fourier transform of eigenfunctions in position space. However, if we choose that space for which the relevant variables are chosen canonically conjugate to the position space variables, then the transform is not Fourier, but must involve the Jacobian of transformation between Cartesian coordinates and the curvilinear position space coordinates. (For Cartesian space the transform is Fourier.) In this work we find the correct form of the Hamiltonian in momentum space for the Yukawa potential. It is shown that the potential function in momentum space becomes an operator. The first term is shown to be the Coulomb operator in momentum space, while the remaining terms just mix other hydrogen atom momentum space functions. The resulting eigenvalues are in agreement with the comparable position space results. The analytic behavior of the eigenfunctions is examined. It is shown that all the singularities are poles (the function is meromorphic) and that all the poles lie along the imaginary axis no farther from the origin than p0 (the inverse of the Bohr radius). This result should be of considerable value in simplifying calculations with these functions.
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev. A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of quantum mechanics, this wave function gives the probability to count a photon at any position in space.
Some properties of the operator ${\stackrel{^}{n}}_{V,t}$ representing the number of photons localized in a finite volume VVV at time ttt are investigated. To some extent these properties reflect the well-known difficulty of localizing photons in space-time. However, it is shown that, when the linear dimensions of VVV are large compared with the wave-length of any occupied mode of the field, the ${\stackrel{^}{n}}_{V,t}$ operator acquires some simple properties. The commutator of ${\stackrel{^}{n}}_{V,t}$ and the detection operator $\stackrel{^}{\mathrm{A}}(\mathrm{x}, t)$ is expressible in an interesting form. The commutator and relations derived from it become particularly simple for certain space-time regions which we label conjoint and disjoint. An orthogonal set of eigenstates of ${\stackrel{^}{n}}_{V,t}$ is found, together with the corresponding eigenvalues, and it is shown that an arbitrary state is expressible in terms of these eigenstates. Some NthN\mathrm{th}Nth-order correlations of the ${\stackrel{^}{n}}_{V,t}$ operators are evaluated, and the results are used to calculate the probability distribution of eigenvalues of ${\stackrel{^}{n}}_{V,t}$ for an arbitrary state of the field.
A method for solution of the Schrödinger equation in momentum space is described. If the Schrodinger equation in momentum space is represented by Oχ=0, where O is an integral operator and χ is the momentum function, the method involves two steps: (i) the representation of χ as a rational function of P and (ii) expansion of Oχ in a power series in some appropriately chosen variable t (‖t‖≤1) which is a function of P. Equating to zero the coefficients of the first N powers of t (which determines the eigenvalues) makes Oχ differ from zero by terms of order tN. As N increases the eigenvalues approach the correct limiting values if the method converges. Within its circle of convergence a power series converges for both real and complex values of the variable and hence the Schrödinger equation, an integral equation, is forced to hold for both real and complex values of the variable. This is in contrast to variational methods which involve only real values of the independent variable. To test the method the motion of a particle in the field of a Yukawa potential is studied. Only the lowest energy state is considered. Two extreme cases, involving linear combinations of rational basis sets, are studied. Both basis sets are complete but the power series method converges only for one basis set. For the other set the power series method provides an eigenvalue which oscillates around the correct value without approaching a limit but a variation method using the same basis set converges rapidly to the correct eigenvalue. The variation method succeeds because the function is defined only on the real axis while the power series method fails because, in the complex plane, the basis set can only represent a meromorphic function (one whose only finite singular points are poles) and the momentum eigenfunction cannot be meromorphic. Thus the choice of a wrong basis set in a variation method can completely conceal the true analytic nature of an eigenfunction. It is shown that momentum eigenfunctions are not meromorphic because they possess branch points. This is true not only for the Yukawa potential but for several one-electron models including the Hartree–Fock equations.