Elsevier

Chemical Physics

Volume 538, 1 October 2020, 110886
Chemical Physics

The hydrogen atom in the momentum representation; a critique of the variables comprising the momentum representation

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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 momentum space.

Introduction

In this article we aim to examine the other half of wave mechanics. The establishment of a wave function, which contains all information about a system consistent with the uncertainty principle, is one goal of quantum theory. This wave function is generally obtained on solving the Schroedinger equation in terms of position variables in space of n dimensions; n is the number of degrees of freedom available to the system. This space is termed position space or the position representation. There is, however, an alternative representation, which is most readily understood in terms of the commutation relations between the position variables (qn) and the corresponding momentum variables (pn). The fact that these variables are conjugate to each other implies an equivalence between the wave function of the system in terms of the position variables and that in terms of the momentum variables. Wave functions in the latter space, the momentum space or momentum representation, also contain all information about a system consistent with the uncertainty principle, but this momentum space has been, by and large, neglected. The reasons for this practice are varied, but one glaring reason is that a proper definition of momentum space is lacking. Many scientists state that the connection between momentum space and position space is expressed in terms of a Fourier transform. Except in the case of Cartesian coordinates, this statement is incorrect; this misunderstanding has led many scientists in the wrong direction, and has led to many inconsistencies. Our intention here is to explore some inconsistencies and to develop a definition of momentum space that is quantum-mechanically consistent, so as to enable an establishment of procedures to solve the Schroedinger equation in momentum space. We then apply these ideas directly to a solution of the hydrogen atom in momentum space. Some advantages and disadvantages of exploring this half of wave mechanics are also examined.

The foundations of quantum mechanics were established nearly a century ago. The amplitude functions, which are algebraic solutions of the time-independent Schroedinger equation, are expressed explicitly in terms of either spatial coordinates or momentum variables. These solutions in position space were originally obtained for several simple but important systems such as the harmonic oscillator (see Appendix A) and the hydrogen atom (see below and Appendix B); these results laid the foundation for the extension of wave mechanics and provided a basis of our current knowledge and treatment of atomic and molecular structure. Much of that work involved obtaining valid functions that are solutions to the differential equation and the matrix formulations of the quantum theory. The fact that the momentum representation is totally equivalent to the position representation should presage considerable attention to the momentum functions, for the possibility that they could provide some advantage over functions in position space for various problems. This promise has not, however, been realised. We contend that part of the reason for this imbalance lies in the difficulties in establishing the proper momentum variables as analogues of the position coordinates. These difficulties stem from various misconceptions about the nature of the relations between the two representations, and the transformations used to generate them. For the particular case of the hydrogen atom, we investigate these difficulties in this article.

We first trace the development of the choice and use of variables used in determining the momentum analogues of the variables in position space. We explore the difficulties encountered and the misunderstandings that caused these problems. We then present a proper definition of the momentum representation, and illustrate its application to the foundational problem of the hydrogen, or one-electron, atom.

Section snippets

Conjugate variables in multiple dimensions

In the particular case of a system involving only one spatial dimension, the displacement q and momentum p/h are directly conjugate variables. For a system in three spatial dimensions other than for Cartesian coordinates, a problem of the conjugacy of variables arises. Because the commutator between displacement q and momentum p in one spatial dimension in assumed Cartesian coordinates, i.e. [p, q] = pq − qp, evaluates not to zero but to ih, products such as p2 and q−1p q p are inequivalent.

Derivation of Podolsky and Pauling for the one-electron atom

In 1929, Podolsky and Pauling [2] derived what they claimed to be a momentum distribution for hydrogen-like atoms. They abandoned Podolsky's promising approach of adopting a quantum-mechanically correct Hamiltonian in favor of the utilization of a method resembling a Fourier transform. They chose a momentum coordinate system (pr, θp, ϕp) in which pr represents the magnitude of the total momentum as pr = (px2 + py2 + pz2)½; θp and ϕp denote angles that specify the orientation of the momentum

Deficiencies of the derivation by Podolsky and Pauling

The above approach presents several important difficulties. We note first that variable pr represents the magnitude of the total momentum, and is not chosen conjugate to any variable in position space; no valid commutation relation is given for pr. Uncertainty relations with a corresponding coordinate in position space are not presented. Although pr has units of momentum, and represents a measurable quantity, it represents no proper quantum–mechanical variable that can serve as a foundation of

Definition of momentum space

In Cartesian space, variables p, q range from [-∞,∞]. For a single particle in one dimension, the Hamiltonian is typically written asHp,q=p2/2m+Vqin which μ is the mass of the particle and V(q) is the potential-energy function. To convert this classical Hamiltonian function to a quantum–mechanical operator, we express the momentum as an operator,p→-thd/dq

We seek solutions of the resulting differential equation for the eigenvalues represented by E:Hψ(q)=Eψq

Ψ(q)are the eigen functions, expressed

The function for the hydrogen atom in momentum space

We proceed to develop an amplitude function for the hydrogen atom in the momentum representation based on DeWitt's formulation of momentum variables in spherical polar coordinates [8]. We consider three momentum variables (pr, θp, ϕp), which must be chosen as conjugates to the corresponding position variables (r, θ, ϕ) [11]. As the hydrogen atom is separable in the position representation, we transform separately the radial equation for R(r) and the angular equations for Θ(θ) and Φ(ϕ). In this

Discussion

We present some graphical properties of the quantum-mechanically correct amplitude functions in momentum space. Fig. 1 presents the squares of the amplitude functions of pr according to Eq. (33). In contrast to the radial eigenfunctions in space coordinates (r, θ, ϕ), these functions become narrower as n increases. Whereas the radial functions (not shown) of Podolsky and Pauling [2] have n − 1 nodes along the momentum axis between pr = 0 and ∞ at which the squared radial momentum equals zero,

Conclusion

A great mystery of this rather complicated history of the momentum-space formulation of the hydrogen atom is that Podolsky, in his 1928 paper [1], was on the verge of obtaining the correct solution. He derived a correct expression for the Hamiltonian (see Eq. (2)), and, with the proper definition of the momentum variables, could easily have solved the eigenvalue problem with simple differential or integral equations. Instead, discarding this promising approach, the very next year he

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (28)

  • B. Lohmann et al.

    Phys. Lett. A

    (1981)
  • K.T. Leung et al.

    Chem. Phys.

    (1983)
  • M. Defranceschi

    Chem. Phys.

    (1987)
  • D. Judge

    Phys. Lett.

    (1963)
  • W.H. Louisell

    Phys. Lett.

    (1963)
  • B. Podolsky

    Phys. Rev.

    (1928)
  • B. Podolsky et al.

    Phys. Rev.

    (1929)
  • B.H. Bransden et al.

    Physics of Atoms and Molecules

    (1983)
  • E.A. Hylleraas

    Z. Phys.

    (1932)
  • V. Fock

    Z. Phys.

    (1935)
  • A. Messiah

    Quantum Mechanics

    (1961)
  • P. A. M. Dirac, Principles of Quantum Mechanics, (Oxford University Press, Oxford UK, 1958, (fourth edition), p....
  • E. Merzbacher

    Quantum Mechanics

    (1961)
  • B.S. DeWitt

    Phys. Rev.

    (1952)
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