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Cambridge University Press eBooks

The helium atom

P. van der Straten, Harold J. Metcalf

Published Dec 31, 2015• 0 citations
Publisher

Abstract

Introduction Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n , ℓ , and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δ ℓ , as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly. Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter. Symmetry The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented. The exchange operator In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P 12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.

DOI
10.1017/cbo9781316106242.015

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Introduction Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n , ℓ , and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δ ℓ , as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly. Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter. Symmetry The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented. The exchange operator In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P 12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.
This chapter addresses the quantum mechanics of identical particles, emphasizing the role of wavefunction symmetry in determining statistical behavior. Symmetric and antisymmetric wavefunctions are defined, and the implications of particle indistinguishability are examined, particularly regarding how it influences measurable outcomes and the concept of energy exchange degeneracy. The distinguishability of identical particles is discussed and shown to affect specific calculations. For systems where particle interactions can be neglected in the zero-order approximation, the total energy eigenfunction is expressed as a product of single-particle eigenfunctions. The Pauli exclusion principle is derived as a consequence of antisymmetrization. The relationship between symmetry type (bosonic or fermionic) and quantum statistics is elaborated. As an illustration, the ground state and the first excited sss-state of the helium atom are analyzed under a simplified approximation. Metadata/Keywords: identical particles, symmetric wavefunction, antisymmetric wavefunction, Pauli exclusion principle, quantum statistics, energy exchange degeneracy, indistinguishability, helium atom, symmetry and statistics
The original intention of the author’s preoccupation with the quantum-mechanical behaviour of simple atoms and molecules such as Hydrogen and Helium was, on the one hand, the elegant simplicity of Niels Bohr’s atom model for Hydrogen, describing the metastable states of the excited electrons by planar concentric electron orbits, and, on the other hand, the hardly intelligible wave mechanical approach of Heisenberg, Schrödinger and others, describing mainly atoms with multiple electrons by three-dimensional orbitals which were characterized by probabilities of presence. Thereby the question arose whether it would be possible to find alternative atom models with well-defined electron trajectories. Therein, Louis de Broglie’s thesis of the wavy nature of electron motion implicating standing waves would have to be implemented. Nevertheless, as reviewed in the introduction, the orthodox three-dimensional concept influenced the own thinking in such a way that three-dimensional constellations for the electronic excited states were conceived. The break-through was achieved for the electronic ground state in the form of the spin-orbit coupling where the spin acts as a perpetuum mobile, inducing the orbital angular momentum h/2π. Furthermore, the insight was gained that a circularly rotating electron intrinsically corresponds to a harmonic oscillator, thus fulfilling the condition of a standing wave. Based on this concept, a double planar model was established for the H2-molecule which could be empirically verified by X-ray data from literature. However, for the two electrons containing Helium a 2D-array seemed impossible since the Pauli-principle seemed to be violated. After a long stepwise succession of 3D-attempts which turned out to be impossible—not least since eccentric forces are not possible in such a system—the here presented 2D-version for Helium was found, composed by two imaginary orthogonal electron orbits. It will enable in a subsequent publication the quantum mechanical interpretation of the thermal-radiative behaviour of Helium which was reported in the author’s publication nine years ago.
We begin our study of atomic physics by relying on the phenomenological notions summarized in chapter 1 and by fully exploiting the quantum mechanical formalism outlined in chapter 2. A hierarchy of approximations will be introduced in the first instance to describe the hydrogen atom, which allows one to focus on the main features of its energy spectrum and quantum properties, including the key ‘atomic orbital’ concept. Next, these results will be extended to the more general case of hydrogenic atoms. The physical picture will be made more rigorous by introducing the magnetic interactions, included those linked to the intrinsic spin degree of freedom, leading to the full description of the fine and hyperfine structure of one-electron atoms. The effects of a static external magnetic or electric field will be eventually discussed by calculating the corresponding energy shifts and splittings of the energy levels.
Introduction Series of atomic spectral lines had been studied by emission from excited vapors even before Balmer described the regularity of the visible lines of hydrogen in 1884. However, direct excitation had been restricted to the few lowest-lying levels, particularly those accessible by electric dipole transitions from the ground state using classical light sources. The advent of tunable lasers in the 1970s changed such spectroscopic limitations enormonsly, allowing excitation to a very much larger number of states. Atoms in highly excited states, typically with principal quantum numbers n ≥ 10, are called Rydberg atoms. They are oversize, fragile states of significant intrinsic interest. They are very rare in nature because of their size and fragility, but they are found in interstellar space (H I regions). One of the best sources for comprehensive descriptions of Rydberg atoms and references to the literature describing their exploration is Ref. [120]. It describes their characteristic properties and behavior both classically and quantum mechanically, and especially their strongly enhanced interactions with external fields. The first thing to note is that Rydberg atoms are usually characterized by a single, highly excited electron while all the other electrons remain in their original states. Thus their properties are very nearly classical, and the atoms are much closer to the ideal hydrogen-like model of Chap. 7, along with appropriate quantum defects (see Chap. 10). The spectroscopy of such states is especially interesting because the active electron is much less perturbed by the other electrons in the atom than when it is in lower states. Moreover, there are some special circumstances in which the energy of a Rydberg state is degenerate with an atomic state having two electrons excited to lower levels, and such configurations yield further new information. Second, Rydberg atoms do not usually decay back to the ground state by emission of characteristic radiation, so their study is mediated by other techniques that reveal new and interesting properties. Such properties arise because Rydberg atoms have a huge dipole moment and very low binding energy resulting from the large average distance between the excited electron and the nucleus-plus-core of the rest of the atom with net charge e . Third, transitions between Rydberg states are quite different from those connecting to the ground state for at least three reasons.
The quantum mechanics of two-electron systems is reviewed, starting with the ground state of the helium atom and heliumlike ions with central charge Z. The case of Z≥2 is rather straightforward. In contrast, for negative hydrogen ion with Z=1, the stability of H− cannot be achieved using a product of individual electron wavefunctions and requires explicit account of the anticorrelation among the two electrons. The wavefunction proposed by Chandrasekhar is revisited, where the permutation symmetry is first broken and then restored by a counterterm. More difficult problems can be studied using the same strategy such as the stability of hydrogenlike ions for any value of the proton-to-electron mass ratio M/m, the energy of the lowest spin-triplet state of helium and heliumlike ions, and the stability of the doubly excited hydrogen ion with unnatural parity. The positronium molecule, which was predicted years ago and discovered recently, can also be shown to be stable against spontaneous dissociation. Emphasis is placed on symmetry breaking, which can either spoil or improve the stability.