Chemical Physics 538 (2020) 110886 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys The hydrogen atom in the momentum representation; a critique of the T variables comprising the momentum representation ⁎ J.R. Lombardia, , J.F. Ogilvieb,c,d a Department of Chemistry, The City College of the City University of New York, New York, NY 10031, USA b Centre for Experimental and Constructive Mathematics, Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada c Escuela de Quimica, Universidad de Costa Rica, San Pedro de Montes de Oca, San Jose 11501-2060, Costa Rica d Institute of Quantum Physics, Irkutsk National Research Technical University, 664074 Irkutsk, Russia ABSTRACT We examine critically the solutions for the hydrogen atom in momentum space. We demonstrate that the approach by Podolsky and Pauling (Physical Review 1928, 34, 109) to such a transformation was inconsistent with Podolsky's preceding analysis (Physical Review 1928, 32, 812) and yields functions that fail to utilize quantum-mechanically acceptable variables in momentum space. This practice arose from the commonplace belief that functions in momentum space are Fourier transforms of those in position space. We show that proper quantum–mechanical functions are obtainable through presentation of a clear definition of momentum space based on DeWitt's transformation (Physical Review 1952, 85, 653). This method allows us to obtain proper wave functions for the hydrogen atom in mo- mentum space. 1. Introduction quantum-mechanically consistent, so as to enable an establishment of procedures to solve the Schroedinger equation in momentum space. We In this article we aim to examine the other half of wave mechanics. then apply these ideas directly to a solution of the hydrogen atom in The establishment of a wave function, which contains all information momentum space. Some advantages and disadvantages of exploring this about a system consistent with the uncertainty principle, is one goal of half of wave mechanics are also examined. quantum theory. This wave function is generally obtained on solving The foundations of quantum mechanics were established nearly a the Schroedinger equation in terms of position variables in space of n century ago. The amplitude functions, which are algebraic solutions of dimensions; n is the number of degrees of freedom available to the the time-independent Schroedinger equation, are expressed explicitly in system. This space is termed position space or the position re- terms of either spatial coordinates or momentum variables. These so- presentation. There is, however, an alternative representation, which is lutions in position space were originally obtained for several simple but most readily understood in terms of the commutation relations between important systems such as the harmonic oscillator (see Appendix A) and the position variables (qn) and the corresponding momentum variables the hydrogen atom (see below and Appendix B); these results laid the (pn). The fact that these variables are conjugate to each other implies an foundation for the extension of wave mechanics and provided a basis of equivalence between the wave function of the system in terms of the our current knowledge and treatment of atomic and molecular struc- position variables and that in terms of the momentum variables. Wave ture. Much of that work involved obtaining valid functions that are functions in the latter space, the momentum space or momentum re- solutions to the differential equation and the matrix formulations ofthe presentation, also contain all information about a system consistent quantum theory. The fact that the momentum representation is totally with the uncertainty principle, but this momentum space has been, by equivalent to the position representation should presage considerable and large, neglected. The reasons for this practice are varied, but one attention to the momentum functions, for the possibility that they could glaring reason is that a proper definition of momentum space is lacking. provide some advantage over functions in position space for various Many scientists state that the connection between momentum space problems. This promise has not, however, been realised. We contend and position space is expressed in terms of a Fourier transform. Except that part of the reason for this imbalance lies in the difficulties in es- in the case of Cartesian coordinates, this statement is incorrect; this tablishing the proper momentum variables as analogues of the position misunderstanding has led many scientists in the wrong direction, and coordinates. These difficulties stem from various misconceptions about has led to many inconsistencies. Our intention here is to explore some the nature of the relations between the two representations, and the inconsistencies and to develop a definition of momentum space that is transformations used to generate them. For the particular case of the ⁎ Corresponding author. E-mail address: [email protected] (J.R. Lombardi). https://doi.org/10.1016/j.chemphys.2020.110886 Received 14 May 2020; Accepted 6 June 2020 Available online 11 July 2020 0301-0104/ © 2020 Published by Elsevier B.V. J.R. Lombardi and J.F. Ogilvie Chemical Physics 538 (2020) 110886 v hydrogen atom, we investigate these difficulties in this article. Gegenbauer polynomials Cr (pr) in the radial component. In the de- We first trace the development of the choice and use of variables tailed description that follows, we apply a consistent notation, even used in determining the momentum analogues of the variables in po- though somewhat different from that of the original authors. sition space. We explore the difficulties encountered and the mis- In three spatial dimensions, the Fourier transform of amplitude understandings that caused these problems. We then present a proper function n,, l m (,,)x y z in Cartesian coordinates to a corresponding am- definition of the momentum representation, and illustrate its applica- plitude function in momentum variable p with components (px, py, pz) is tion to the foundational problem of the hydrogen, or one-electron, 2i (xpx+ yp y + zp z + atom. 1 h (,,)p p p = n,, l m (,,)x y z e dxdydz n,, l m x y z h3/2 2. Conjugate variables in multiple dimensions (3) Podolsky and Pauling [2] applied standard relations between Car- In the particular case of a system involving only one spatial di- tesian and spherical polar coordinates, mension, the displacement q and momentum p/h are directly conjugate x= r sin()cos( ),y= r sin()sin( ),z= r cos() variables. For a system in three spatial dimensions other than for (4) Cartesian coordinates, a problem of the conjugacy of variables arises. and assumed the analogous relations, despite Podolsky’s previous re- Because the commutator between displacement q and momentum p in cognition of the necessity of conjugate relations between the coordinate one spatial dimension in assumed Cartesian coordinates, i.e. [p, 2 and momentum variables [1]. Instead they chose for the components of q] = pq − qp, evaluates not to zero but to ih, products such as p and linear momentum: −1 q p q p are inequivalent. For variables (q1, q2,…, qn) in position space px= psin( r p )cos(p ),p y= psin( r p )sin(p ),p z= pcos( r p ) in any set of n dimensions, the volume element is dV = g(q1, q2,…, qn) (5) dq dq … dq , in which g(q , q , …q ) is the Jacobian of the trans- 1 2 n 1 2 n According to this equation for the transformation between co- formation between variables in the two sets. To each variable (q1, q2, ordinate space with ψn,l,m(r, θ, ϕ) = Rn,l(r) l,m(θ) φm(ϕ) and momentum …, q ) in position space there is a corresponding variable (p , p , …, p ) n 1 2 n space, in momentum space such that each momentum variable is chosen conjugate to a corresponding position space variable: [pk, qk] = ih. n,, l m (,,)pr p p In 1928, Podolsky [1] derived the quantum-mechanically correct 2i (sin( )sin(p )cos(p )+ cos( )cos(p ))rpr 1 2 form of the Hamiltonian for conservative systems. In that work, he = (,,)r e h r2 3/2 n,, l m began with the Schroedinger equation in Cartesian coordinates, and h 0 0 0 converted it to spherical polar coordinates. He then utilized the trans- sin( )d d dr (6) formation that, for each momentum variable pk (=pr, θp, φp), there exists an expression pk = −i ( / qk). He showed that, with such a Podolsky and Pauling proceeded to generate an expression for am- transformation, an equation is obtained that differs from the correct plitude functions in their presumed momentum space of an atom of Hamiltonian, and that hence cannot truly represent the momentum atomic number Z with one electron of this form, equation. To overcome this impediment, he introduced an arbitrary n- n,, l m (,,)pr p p dimensional coordinate transformation such that the element of length 2(2l+ 1)( l m )!( n l 1)! is = 2eim p n n ()!()!l+ m n + l 2 ds= g dur du s rs 5 r=1s = 1 l 2 +l (1) ( 4inpr ) l ! n ( Zp0 )2 P ( l , m , cos p )C n l in which grs= g sr are elements of the Jacobean of transformation g. He rs 2 2 2 2 l+2 introduced also the notation g = (minor of grs)/g. Letting V be the Z p+ n p 1 1 1,l + 1, 0 r potential energy, he then obtained the quantum-mechanically correct 2 2 2 2 2 2 2 2 Z p0 + n pr (Z p0 + n pr ) form of the Hamiltonian as (7) n n 1 in which appear associated Legendre functions P (l, |m|, cos(θp)) of the H = g1/4 p g 1/2 grs p g 1/4 + V 2µ r s first kind and Gegenbauer [or ultraspherical functions C(n − l − 1, r=1s = 1 (2) l + 1, …).