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The Journal of Chemical Physics

The Yukawa potential in momentum space: Analytic behavior of the eigenfunctions

John R. Lombardi

Published Jul 15, 1986• 5 citations
Publisher

Abstract

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.

DOI
10.1063/1.451250

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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.
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.
Abstract The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.
The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.
The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.
We consider Fock's fundamental theory of the hydrogen atom in momentum space which allows a realization of the previously predicted rotation group of a three-dimensional (3D) sphere in four-dimensional (4D) space. We then modify Fock's theory and abandon the momentum space description. To transform and simplify the theory, we use invariant tensor methods of electrostatics in 3D and 4D spaces. We find a coordinate 4D space where the Schrodinger equation becomes the 4D Laplace equation. The transition from harmonic 4D polynomials to original 3D physical space is algebraic and involves derivatives with respect to a coordinate that is interpreted as time. We obtain a differential equation for eigenfunctions in the momentum space and find its solutions. A concise calculation of the quadratic Stark effect is given. The Schwinger resolvent is derived by the method of harmonic polynomials. Vector ladder operators are also considered.
The coordinate-space wave function ψ(x)ψ(x)ψ(x) of quasi-one-dimensional atoms is defined in the x≥0x\geq 0x≥0 region only. This poses a typical problem to write a physically acceptable momentum-space wave function ϕ(p)ϕ(p)ϕ(p) from the Fourier transform of ψ(x)ψ(x)ψ(x). We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function ϕ(p)ϕ(p)ϕ(p) as a function of ppp and confirm that ϕi(p)ϕ_i(p)ϕi​(p) (the imaginary part of ϕ(p)ϕ(p)ϕ(p)) represents the correct momentum-space wave function. We make use of the results for ψ(x)ψ(x)ψ(x) and ϕi(p)ϕ_i(p)ϕi​(p) to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.