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Physical Review A

Heisenberg's uncertainty relations for a hydrogen atom confined by an impenetrable spherical cavity

R. Reyes-García, Salvador A. Cruz, R. Cabrera‐Trujillo

Published Aug 15, 2024• 14 citations
Publisher

Abstract

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 ${r}_{o}$. 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 $n$ 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 ${r}_{o}<1$ a.u.

DOI
10.1103/physreva.110.022814

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This paper presents a theoretical investigation on effects of lateral position uncertainty of captivity electrons within spherical electron shells of Bohr hydrogen-like atoms. A captivity electron, which is spatially confined in Bohr orbits, introduces a lateral position uncertainty that can be determined by considering the area of the electron shell. After deriving uncertainty relation for position and kinetic energy, author theoretically demonstrates that, due to the lateral position uncertainties of electrons in spherical shells, Heisenberg uncertainty principle suggests uncertainty bounds in measurement of kinetic energy states of captivity electrons that orbits non-relativistic hydrogen-like Bohr atom. Afterward, these analyses are extended for relativistic hydrogen-like Bohr atom case.
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.
Abstract This study investigates the impact of spatial confinement and the Hellmann potential on the Shannon entropy of a hydrogenic atom. A hydrogenic atom screened by the Hellmann potential and confined within an impenetrable spherical potential is analyzed. The Schrödinger equation is solved numerically using the finite difference method to determine energy eigenvalues and wavefunctions. These wavefunctions are examined in both position and momentum spaces to calculate the Shannon entropy in position space , the Shannon entropy in momentum space , and the total Shannon entropy . How the confinement radius and the screening parameter affect these entropies is investigated, and the results are compared to those of unconstrained systems. The findings are contrasted with previously reported results and discussed in relation to the Beckner–Bialynicki–Birula–Mycielski (BBM) inequality, offering insights into the behavior of quantum states under confinement. Significant variations in Shannon entropy and energy levels are observed with changes in confinement and screening parameters, providing a deeper understanding of quantum mechanics in confined systems. The study reveals a critical screening parameter at which the behavior of entropy transitions, shedding light on the interplay between spatial confinement, screening effects, and quantum uncertainties.
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.
We make use of the Dirac equation to provide a relativistic calculation of the binding energy of a hydrogen-like atom confined within a penetrable spherical barrier. We take the potential to be Coulombic within the barrier and constant outside the barrier. The derived energy values are compared with the energy values obtained from a non-relativistic calculation based on the Schrodinger equation, where significant differences are found for small values of the barrier radius. In addition, we show that without the introduction of the principle quantum number nnn, all energy states of the confined atom, determined by a single quantum number kkk, transfer into the known energy states of the free atom as the radius of the barrier becomes large.
This paper presents a detailed analysis of the radial uncertainty product for quantum systems with spherically symmetric potentials. Using the principles of quantum mechanics, the study derives the radial uncertainty relation analogous to the Cartesian form and investigates its implications for three key spherically symmetric potentials: the Hydrogen atom, the infinite spherical potential well, and the spherical harmonic oscillator, all within the non-relativistic regime. Employing the Schrodinger equation in spherical coordinates, the paper rigorously evaluates the normalized radial wave functions, expectation values, and uncertainties associated with both position and momentum. Analytical derivations and numerical computations highlight the dependence of the uncertainty product on quantum numbers and system-specific parameters.