A succinct history of the Confined Atom problem is presented. The hydrogen atom confined to the centre of an impenetrable sphere counts amongst the exactly soluble problems of physics, alongside much more noted exact solutions such as Black Body Radiation and the free Hydrogen atom in absence of any radiation field. It shares with them the disadvantage of being an idealisation, while at the same time encapsulating in a simple way particular aspects of physical reality. The problem was first formulated by Sommerfeld and Welker [1]—henceforth cited as SW—in connection with the behaviour of atoms at very high pressures, and the solution was published on the occasion of Pauli's 60th birthday celebration. At the time, it seemed that there was not much other connection with physical reality beyond a few simple aspects connected to the properties of atoms in solids, for which more appropriate models were soon developed. Thus, confined atoms attracted little attention until the advent of the metallofullerene, which provided the first example of a confined atom with properties quite closely related to those originally considered by SW. Since then, the problem has received much more attention, and many more new features of quantum confinement, quantum compression, the quantum Faraday cage, electronic reorganisation, cavity resonances, etc have been described, which are relevant to real systems. Also, a number of other situations have been uncovered experimentally to which quantum confinement is relevant. Thus, studies of the confined atom are now more numerous, and have been extended both in terms of the models used and the systems to which they can be applied. Connections to thermodynamics are explored through the properties of a confined two‐level atom adapted from Einstein's celebrated model, and issues of dynamical screening of electromagnetic radiation by the confining shell are discussed in connection with the Faraday cage produced by a confining conducting shell. The conclusions are shown to be relevant to a proposed 'quantum computer'. The description of the actual geometry of C60, as opposed to a purely spherical approximation, leads to some qualification of the computed results.
A succinct history of the Confined Atom problem is presented. The hydrogen atom confined to the centre of an impenetrable sphere counts amongst the exactly soluble problems of physics, alongside much more noted exact solutions such as Black Body Radiation and the free Hydrogen atom in absence of any radiation field. It shares with them the disadvantage of being an idealisation, while at the same time encapsulating in a simple way particular aspects of physical reality. The problem was first formulated by Sommerfeld and Welker [1]—henceforth cited as SW—in connection with the behaviour of atoms at very high pressures, and the solution was published on the occasion of Pauli's 60th birthday celebration. At the time, it seemed that there was not much other connection with physical reality beyond a few simple aspects connected to the properties of atoms in solids, for which more appropriate models were soon developed. Thus, confined atoms attracted little attention until the advent of the metallofullerene, which provided the first example of a confined atom with properties quite closely related to those originally considered by SW. Since then, the problem has received much more attention, and many more new features of quantum confinement, quantum compression, the quantum Faraday cage, electronic reorganisation, cavity resonances, etc have been described, which are relevant to real systems. Also, a number of other situations have been uncovered experimentally to which quantum confinement is relevant. Thus, studies of the confined atom are now more numerous, and have been extended both in terms of the models used and the systems to which they can be applied. Connections to thermodynamics are explored through the properties of a confined two‐level atom adapted from Einstein's celebrated model, and issues of dynamical screening of electromagnetic radiation by the confining shell are discussed in connection with the Faraday cage produced by a confining conducting shell. The conclusions are shown to be relevant to a proposed 'quantum computer'. The description of the actual geometry of C60, as opposed to a purely spherical approximation, leads to some qualification of the computed results.
Quantum Confinement is in some sense a new subject. International meetings dedicated to Quantum Confinement have occurred only recently in Mexico City (the first in 2010 and the second, in September 2011). However, at least in principle, the subject has existed since a very long time. Surprisingly perhaps, it lay dormant for many years, for want of suitable experimental examples. However, when one looks carefully at its origin, it turns out to have a long and distinguished history. In fact, the problem of quantum confinement raises a number of very interesting issues concerning boundary conditions in elementary quantum mechanics and how they should be applied to real problems. Some of these issues were missed in the earliest papers, but are implicit in the structure of quantum mechanics, and lead to the notion of Confinement Resonances, the existence of which was predicted theoretically more than ten years ago. Although, for several reasons, these resonances remained elusive for a very long time, they have now been observed experimentally, which puts the whole subject in much better shape and, together with the advent of metallofullerenes, has contributed to its revival.
The exact solution to the problem of a hydrogen atom confined in a spherical well (CHA) is discussed; the standard results for the unconfined hydrogen atom (UHA) are recovered as the sphere size becomes large compared to the Bohr radius. The solutions are characterized by a set of three quantum numbers N (= 1, 2, 3,…), L (= 0, 1, 2,…), and M (= − L, − L + 1,…, L − 1, L), and the energy eigenvalues, in contrast to the situation in the UHA, depend on both N and L. All members of a given family n = N + L, however, evolve asymptotically toward the same energy level in the large-sphere limit, recovering the typical n2 degeneracy of the UHA. Besides numerically exact solutions for arbitrary sphere sizes, rigorous analytical approximations are provided for the physically relevant strong- and weak-confinement regimes. A conjecture concerning the ordering of the energy levels is rigorously confirmed. The validity of the virial theorem, Kato's cusp condition, and the role played by the density as an alternative basic variable for the case of the CHA are discussed.
Spatial confinement of quantum systems has been used to model atoms under high pressure, quantum dots, and Bose–Einstein condensates. Spherical boxes are commonly employed, yet they are energy-minimizing, and other box geometries will be more relevant for some applications. We study generic single-particle quantum systems comprising a central physical potential confined within a three-dimensional infinite potential well corresponding to an impenetrable box having prescribed geometric properties; we refer to this as geometric confinement. There is a marked distortion of the wavefunction and energy spectrum, relative to the standard unconfined system, when a confining box is made small enough. We employ asymptotic analysis in combination with numerical simulations to better understand this behaviour, determining lengthscales on which each of the physical potential and confining box respectively dominate the structure of quantum states, with Padé approximants employed to connect asymptotic scalings between these two regimes. This combination of methods permits us to determine, in a fairly general manner, how and to what extent box size and shape modify a given single-particle quantum system held under geometric confinement. We provide a variety of examples of single-particle central potentials and confining geometries to illustrate our methods and motivate the theoretical findings.
Since the pioneering works of Michels et. al., followed by Sommerfeld and Welker, over eighty years ago, there is much interest in modelling confinement of the ground state hydrogen atom by an impenetrable sphere of confining radius, Rc. The literature “truncated probability density” model assumes that, upon confinement, the nodeless free hydrogen atom ground state probability density is replaced by the innermost node of an excited state probability density, fitted to Rc by a non-quantum parameter, λ. Since the literature model is an excited state, the confined electron energy, E1c, is inversely proportional to the excited state Bohr radius, Rλ (or aoλ2), which becomes larger than Rc as λ increases. However, due to the infinite potential barrier at radii greater than Rc, such large values of Rλ are not allowed. In contrast, for the Particle in an Infinite Box (PIB), Harmonic Oscillator (HO) and the proposed compressed probability density ground state models, the particle energy is inversely proportional to the confined length (Lc or Rc), the maximum length allowed. Additionally, the Dirac/Sommerfeld fine structure formula calculation of E1c is in agreement with the proposed model, but does not support the literature truncated probability density model, indicating the literature model is incorrect.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].
An idealised model to treat the effect of spherically confining the electron in a hydrogen-like atom is studied, where the potential is infinite in all space except for a spherical shell. The exact solution of the Schrödinger equation is obtained in terms of two independent solutions of the Kummer equations. It is found that, in some cases, it is necessary to use the standard Kummer M function and a non-standard second solution. In other cases we may use the Kummer U function and in a limiting case the two standard solutions of Bessel's equation. The effect of an imposed dipole field on the shell is treated using the first-order perturbation equation from which the polarisability can be calculated. In addition, the exact wavefunction is used to calculate the Shannon entropies of both position and momentum and it is shown that these measures give insight into the form of the wavefunction.