Electron spin In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r , θ, ϕ. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. In a quantum-mechanical treatment of an alkali metal atom, the lone valence electron may be considered as moving in the combined field of the nucleus and the core electrons. In contrast to the hydrogen-like atom, the energy levels of this valence electron are found to depend on both the principal and the azimuthal quantum numbers. The experimental spectral line pattern corresponding to transitions between these energy levels, although more complex than the pattern for the hydrogen-like atom, is readily explained. However, in a highly resolved spectrum, an additional complexity is observed; most of the spectral lines are actually composed of two lines with nearly identical wave numbers. In an alkaline-earth metal atom, which has two valence electrons, many of the lines in a highly resolved spectrum are split into three closely spaced lines. The spectral lines for the hydrogen atom, as discussed in Section 6.5, are again observed to be composed of several very closely spaced lines, with equation (6.83) giving the average wave number of each grouping.
Electron spin In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r , θ, ϕ. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. In a quantum-mechanical treatment of an alkali metal atom, the lone valence electron may be considered as moving in the combined field of the nucleus and the core electrons. In contrast to the hydrogen-like atom, the energy levels of this valence electron are found to depend on both the principal and the azimuthal quantum numbers. The experimental spectral line pattern corresponding to transitions between these energy levels, although more complex than the pattern for the hydrogen-like atom, is readily explained. However, in a highly resolved spectrum, an additional complexity is observed; most of the spectral lines are actually composed of two lines with nearly identical wave numbers. In an alkaline-earth metal atom, which has two valence electrons, many of the lines in a highly resolved spectrum are split into three closely spaced lines. The spectral lines for the hydrogen atom, as discussed in Section 6.5, are again observed to be composed of several very closely spaced lines, with equation (6.83) giving the average wave number of each grouping.
Wave mechanics failed badly in accounting for the multiplicity of atomic energy levels. This was most conspicuous in the case of the alkali metals, lithium, sodium, potassium, and so on. It was known that an atom of any of these elements can be treated as a more-or-less inert core, consisting of the nucleus and Z − 1 inner electrons, together with a single outer electron whose transitions between energy levels are responsible for spectral lines. Since the electrostatic field felt by the outer electron is not a Coulomb field, its energy levels in the absence of external fields depend on the orbital angular momentum quantum number ℓ as well as a radial quantum number n , but because of the spherical symmetry of the atom, not on the angular momentum z -component ħ m . (See Eq. (2.1.30).) For each n , ℓ , and m there should be just one energy level. But observations of atomic spectra showed that in fact all but the s states were doubled. For instance, even a spectroscope of low resolution shows that the D line of sodium, which is produced in a 3 p → 3 s transition, is a doublet, with wavelengths 5896 and 5890 Angstroms. Pauli was led to propose that there is a fourth quantum number for electrons in such atoms, in addition to n , ℓ , and m , with the fourth quantum number taking just two values in all but s states. But the physical significance of this fourth quantum number was obscure.
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.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
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.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.