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Relativistic Description of Confined Hydrogen-Like Atoms

Jacob Noon, Robert T. Deck

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

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 $n$, all energy states of the confined atom, determined by a single quantum number $k$, transfer into the known energy states of the free atom as the radius of the barrier becomes large.

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A Fully Relativistic Treatment of Confined Hydrogen-Like Atoms

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The Dirac equation is used 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. Binding energies are derived for the ground state of hydrogen for various barrier heights and confining radii. In addition, it is shown that without the introduction of the principle quantum number $n$, all energy states of the confined relativistic hydrogen atom, determined by a single quantum number $k$, transfer into the known energy states of the free relativistic hydrogen atom as the radius of confinement becomes large.

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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.

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In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

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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.
The Dirac equation is used 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. Binding energies are derived for the ground state of hydrogen for various barrier heights and confining radii. In addition, it is shown that without the introduction of the principle quantum number nnn, all energy states of the confined relativistic hydrogen atom, determined by a single quantum number kkk, transfer into the known energy states of the free relativistic hydrogen atom as the radius of confinement becomes large.
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.
In 1928, the brilliance of Paul Dirac granted him the insight to develop a particle equation that was fully consistent with both the principles of quantum mechanics and the theory of special relativity. In this bachelor thesis we have studied the fundamentals of relativistic quantum mechanics including the derivation of the Dirac equation in a similar fashion to the way originally done by Paul Dirac. Additionally we have studied the analytical solution for hydrogen-like atoms. Furthermore we have numerically analysed the energy spectrum, radial characteristics and correctional terms using programs written in Python and MATLAB. Finally we have investigated the implications and applications ofthe Dirac Hamiltonian and its use to effectively describe the attributes of gold atoms.We find that relativistic contribution scales approximately by Z2 for large Z. In addition, we conclude that the Dirac Hamiltonian is of vital importance for an accurate description of chemical and physical properties of large atoms. Furthermore, we have completed calculations of the radial solution for the Coulomb potential and also for a free particle.
The Bohr radius was derived using the non-relativistic de Broglie approximation, not the relativistic de Broglie equation. In this work, we will derive a relativistic Bohr radius from both the relativistic de Broglie approximation and relativistic kinetic energy. The Bohr model of the hydrogen atom has since been replaced by modern quantum mechanics and the Schrodinger equation. It is well known that the Schrodinger equation confirms the first Bohr radius as the most probable radius, but the Schrodinger equation itself is non-relativistic. The Dirac equation can likely be used, but there seem to be very few publications on the relativistically corrected Bohr radius. This is something we will discuss.
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.
In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.