The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. In general, this is a multi-dimensional problem, and not one of the simpler one-dimensional problems with which we have been concerned in the previous chapters. Once we begin to treat multiple dimensions, then state degeneracy begins to arise more frequently, and the most common problem treated is that of the hydrogen atom vs. the helium atom. These extra dimensions provide more degrees of freedom and more complexity. In the hydrogen atom, there is only a single electron, and the problem is solvable. On the other hand, adding a second electron for the helium atom makes the problem extremely difficult to solve, as it has become a three-body problem in which the interactions are difficult to handle.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.
It is shown that the approximation of a strong Coulomb interaction between electrons results in a new model of the atom with a spatial quantization of electrons accompanied by their quantization in energy. This model implies that electrons rotate in circular orbits centered outside the atomic nucleus and only orbit axes pass through it. The Coulomb interaction between electrons leads to a spherically symmetric distribution of their orbits on the surfaces of equipotential spheres of a spherically symmetric electrostatic field of the nucleus. The distribution is similar to “inscribing” electron orbits into faces of regular nucleus-centered polyhedra so each polyhedron corresponds to a certain electron state (s, p, d, f), and a certain set of polyhedra corresponds to a certain period of the Mendeleev Table. It is shown that a spherically symmetric distribution of electron orbits gives rise to the formation of electron pairs in which electron orbits with a common axis are located symmetrically with respect to the nucleus and the orbital magnetic moments of the electrons are oppositely directed. The physical meaning of the electron spin concept becomes clear. The spin turns out to be related to the orbital magnetic moment of an electron and reflects the fact that two electrons of a pair rotate in opposite directions relative to their common axis. So the spin is one of characteristics of the electron state in the atom associated with electron rotation in the orbit centered outside the nucleus. The atomic model gives an insight into the periodicity of changes in the atomic properties with increasing nuclear charge and the reasons for an electron double energy quantization associated with different states and periods. The model shows that the atomic structure and properties can be explained by using concepts of classical mechanics and classical electrodynamics which regard the electron as a particle.
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.
This issue comprises two dozen research papers which are all in one sense or another devoted to models in which the interaction is singular and sharply localized; a typical example is a quantum particle interacting with a family of δ-type potentials. Such an idealization usually makes analysis of their properties considerably easier, sometimes allowing us to reduce it to a simple algebraic problem—this is why one speaks about solvable models. The subject can be traced back to the early days of quantum mechanics; however, the progress in this field was slow and uneven until the 1960s, mostly because singular interactions are often difficult to deal with mathematically and intuitive arguments do not work. After overcoming the initial difficulties the `classical' theory of point interactions was developed, and finally summarized in 1988 in a monograph by Albeverio, Gesztesy, Høegh-Krohn, and Holden, which you will find quoted in numerous places within this issue. A reliable way to judge theories is to observe the progress they make within one or two decades. In this case there is no doubt that the field has witnessed a continuous development and covered areas which nobody had thought of when the subject first emerged. The reader may see it in the second edition of the aforementioned book which was published by AMS Chelsea only recently and contained a brief survey of these new achievements. It is no coincidence that this topical issue appears at the same time; it has been conceived as its counterpart and a forum at which fresh results in the field can demonstrated. Let us briefly survey the contents of the issue. While the papers included have in common the basic subject, they represent a broad spectrum philosophically as well as technically, and any attempt to classify them is somewhat futile. Nevertheless, we will divide them into a few groups. The first comprises contributions directly related to the usual point-interaction ideology. M Correggi and one of the editors study a toy model of a decay under the influence of a time-periodic δ potential. E Demiralp describes the spectrum of a spherical harmonic oscillator amended with a concentric family of δ-shell interactions. Another of the editors presents an isoperimetric problem for point interactions arranged at vertices of a polygon. W Huddell and R Hughes show how singular perturbations of a one-dimensional Dirac operator can be approximated by regular potentials, and J Brasche constructs a family of Hamiltonians in which the singular interaction has a more complicated support, namely a Brownian path. Finally, B Pavlov and I Antoniou apply the singular perturbation technique to another classical Hamiltonian, that of a generalized Friedrichs model; no matter that the unperturbed observable is called momentum in their paper. The three papers in the following group are distinguished by the fact that they consider systems which are fully or partially periodic. F Bentosela and M Tater analyse scattering on a crystalline `slab' modelled by point interactions distributed periodically on a finite number of parallel plates. E de Prunelé studies evolution of wavepackets in crystal models of different geometries, and M Avdonin et al discuss a simple model of a spin-dependent scattering on a one-dimensional array of quantum dots. The next group of papers is devoted to a topic which was untouched at the time of the aforementioned first edition, namely quantum graphs, which became a subject of interest after numerous applications of such systems to semiconductor, carbon and other nanostructures. Most contributions here deal with the `usual' model in which the Hamiltonian is a Schrödinger operator supported by the graph. P Kuchment describes spectral properties of such graphs, in particular periodic ones and those with decorations. S Albeverio and K Pankrashkin present a modification of Krein's formula which is suitable for constructing Hamiltonians of quantum graphs using boundary conditions at vertices directly. Two papers are devoted to inverse problems in this context: M Harmer studies inverse scattering for the matrix Schrödinger operator on the halfline with applications to star graphs, while P Kurasov and M Nowaczyk give a mathematically rigorous version of the known Gutkin–Smilansky result on the inverse spectral problem. The paper by O Post contributes to the question of how graphs can be approximated by more realistic `fat' graphs, and describes a class leading to disconnected quantum graphs. Finally, S Kondej and one of the editors study scattering in the context of `leaky' graphs which takes quantum tunnelling into account. While most results in this field describe one-particle Hamiltonians, more complicated systems have also been studied. In this issue we have three examples. C Cacciapuito, R Carlone, and R Figari discuss decoherence in a simple model of two particles, one heavy and one light, interacting through a δ potential; they give a rigorous meaning to a formula derived by Joos and Zeh. A related model by R Figari and A Teta is used to describe ionization. M Hallnäs, E Langmann, and C Paufler treat a true N -body situation, namely a model of one-dimensional gas of distinguishable particles interacting through generalized point interactions; they write the Bethe ansatz and present the solution of a particular case. The last group is a collection of contributions which in one sense or another are outside quantum mechanics, either modifying its postulates or applying it to a different physical situation. The latter applies to the paper of D Noja and A Posilicano in which they study nonlinear wave equations with point perturbations and show the existence of a solution to the Cauchy problem. F Coutinho et al discuss one-dimensional point interactions with energy-dependent coupling constant, S Albeverio and S Kuzhel examine a class of point interactions which are not symmetric but P-symmetric, where P is the parity operator, and M Znojil and V Jakubský consider a `double-well' PT-symmetric model with two δ interactions with an imaginary coupling. The last two papers present mathematical constructions. Yu Shondin demonstrates a way to define an interaction more singular than the usual δ potentials obtained by means of self-adjoint extensions, and V Koshmanenko presents a construction of strongly singular perturbations leading to rather unusual `Hamiltonians'.
Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.