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American Journal of Physics

Exact solution of the compressed hydrogen atom

Jorge Mario Ferreyra, César R. Proetto

Published Oct 23, 2013• 12 citations
Publisher

Abstract

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.

DOI
10.1119/1.4820244

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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.
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.
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.
The incompatibility between two observables in quantum theory is described by Heisenberg's uncertainty principle. In this work, we study spatial confinement effects on Heisenberg's uncertainty principle for a hydrogen atom located at the center of an impenetrable spherical cavity with radius ro{r}_{o}ro​. Both the radial and vector representation of the uncertainty principle are considered. For this, we solve the Schr\"odinger equation numerically within a finite-differences approach. We find that for small cavity sizes the values of $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}\mathrm{\ensuremath{\Delta}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{r}$ (radial) bunch according to the number of nodes and that for Rydberg states, i.e., large excitation, they become more coherent, satisfying exactly Heisenberg's uncertainty principle, in contrast to the vector description. However, for the vector case, we find that $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{r}}$ degenerates for small cavity sizes and bunches according to the principal quantum number nnn for large cavities. We find that the behavior of $\mathrm{\ensuremath{\Delta}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ is responsible for the breaking of the energy degeneracy for confined quantum systems. This occurs when the confinement radius is of the order of the orbital size, as determined by the electron average distance $\ensuremath{\langle}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}\ensuremath{\rangle}$. In addition, we estimate the critical cavity size for which relativistic effects become relevant and verify that the relativistic corrections to the energy, obtained from first-order perturbation theory, become important when the total energy of the atom surpasses 93.845 Hartree, corresponding to 10% of the speed of light, which is fulfilled for cavity sizes ro<1{r}_{o}<1ro​<1 a.u.
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The fuzzy onion model formed by connecting a set of concentric fuzzy spheres of increasing radius is motivated by studies of quantum space but can also be used to study standard physics. The main feature of the model is that functions in three-dimensional space—like scalar fields or wave functions—are expressed in terms of Hermitian matrices of a certain structure. Relevant equations are then matrix equations, and some problems, such as searching for the energy spectrum for fixed quantum numbers ( l , m ) , can be expressed as an eigenvalue problem. We show how this simple approach can reproduce the results of other studies analyzing the hydrogen atom in a spherical cavity. We also test the effect of the short-distance quantum structure of the space on these solutions—not looking for the phenomenological consequences, as the scale of quantum space is many orders below the order of the Bohr radius, but to understand the effect of quantum space in general. We observe a set of solutions without a classical counterpart which have been suggested also in a former theoretical study.
A straightforward way to study the electronic structure and certain physical properties of atoms and molecules under high pressures is through the confinement (caging) model, in which the atom is placed at the center of a spherical box with impenetrable walls. The hydrogen atom confined inside penetrable and impenetrable spherical cavities is perhaps the most studied compressed system, and exact solutions exist for both types of confinement. The physical properties of the hydrogen atom confined within impenetrable-wall cavities (AHCI) have been widely analyzed; however, the same system embedded in penetrable spherical cavities (AHCP) has received less attention. A more physical model suggests using a spherical box with penetrable walls. Therefore, in this work, we compute via the variational method, the Shannon and Fisher information entropy in position space,SrS_{r}Sr​andIrI_{r}Ir​, for the2p2 \mathbf{p}2pand 3d states of a hydrogen atom subjected to spatial confinement by penetrable and impenetrable spherical cavities. Additionally, we calculate the tunneling probability for the penetrable confinement and seek to relate it to the overlaps of the radial probability densities as a function of the cavity radius and the height of the confining barrier. To model the penetrable confinement, we employ a logistic potential which, unlike earlier reports that used a discontinuous step potential, is a continuous function. The wavefunction is expanded in terms of Slater functions that satisfy appropriate boundary conditions; in some cases, we find notable differences with respect to the impenetrable potential.