The intrinsic and dynamic kinetic energies and the potential energies of electron states in the hydrogen atom were determined using the operator formalism in Schrödinger’s nonrelativistic equation. Intrinsic energies were determined using the momentum operator, while for ℓ ≠ 0, the additional dynamic energies of the spinning fields were determined using the angular momentum operator. All 10 states up to the principal quantum number n = 3 and all 4 m states of n = 7, l = 3 were analyzed. The two forms of kinetic energy can only be explained with an electron field representation. All total kinetic and potential energies conformed with the well-known 1/n2 rule. Angular momentum analysis of the 2P1/2 state provided a field spinning rate; in addition, the dynamic kinetic energy of the spinning field determined by both operator analysis and explicit calculation based on the spinning rate gave the same energy results.
The intrinsic and dynamic kinetic energies and the potential energies of electron states in the hydrogen atom were determined using the operator formalism in Schrödinger’s nonrelativistic equation. Intrinsic energies were determined using the momentum operator, while for ℓ ≠ 0, the additional dynamic energies of the spinning fields were determined using the angular momentum operator. All 10 states up to the principal quantum number n = 3 and all 4 m states of n = 7, l = 3 were analyzed. The two forms of kinetic energy can only be explained with an electron field representation. All total kinetic and potential energies conformed with the well-known 1/n2 rule. Angular momentum analysis of the 2P1/2 state provided a field spinning rate; in addition, the dynamic kinetic energy of the spinning field determined by both operator analysis and explicit calculation based on the spinning rate gave the same energy results.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the later cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrodinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
A rigorous derivation of the Schrödinger theory from the Pauli (or Dirac) theory implies that the Schrödinger equation describes an electron in an eigenstate of spin. Furthermore, the ground-state kinetic energy is completely determined by the electron spin density. This can be explained by interpreting the spin as an orbital angular momentum, which is necessarily accompanied by a kinetic energy. Thus, the spin is a zero-point angular momentum associated with the zero-point energy of the electron. Since the dispersion in electron momentum is determined by the zero-point energy, the Heisenberg uncertainty relations for an electron can be interpreted as a property of the electronspin motion. The kinetic interpretation of spin and the statistical interpretation of quantum mechanics can be jointly sustained by regarding the electron as a point particle. It follows that stationary electron states are sources of fluctuating electric fields. There is reason to believe that these fluctuating fields are responsible for the Van der Waals force and can be identified with electromagnetic vacuum field fluctuations. The kinetic interpretation of spin then implies that the Van der Waals forces are spin dependent. These ideas are not only consistent with the conventional mathematical formalism of quantum mechanics, they provide a more complete and coherent interpretation of many details in the formalism than does the alternative Copenhagen interpretation. They present some difficulties, however, which, if the kinetic interpretation of spin is correct, probably require some modification of quantum electrodynamics to be resolved.
The solution of the Dirac equation with the Coulomb potential is used to analyze bound electron states in a hydrogen atom. The analysis is based on the fact that such states are characterized by a set of quantum numbers which describe definite values of the complete set of physical quantities that can be determined simultaneously. This set includes the energy, square of the total angular momentum, one of its component and a spinor invariant. The latter, spinor invariant gives two-valued quantum number which determines the sign of its eigenvalue. In addition to the known Dirac and Johnson–Lippman invariant, there exists a new one. Operators of these three spinor invariants do not commute between themselves which results in the degeneracy of the energy levels with respect to the two-valued quantum number. Three different systems of the eigenbispinor corresponding to the three spinor invariants are obtained and the generalized solution with free parameters is calculated. Variation of the free parameters transforms one particular solution into any other. It is shown that the electron probability densities and spin polarizations in an electron cloud depend essentially on the invariant set, demonstrating physical difference of the states corresponding to different spinor invariants.
The purpose of this study is to find the expected value of the intrinsic kinetic energy of deuterium atoms in radial and total wave functions at quantum numbersn≤3 . The method used in this study is non-experimental research in the field of theoretical-mathematical physics concerning hydrogen atoms. The results of the analysis of the intrinsic kinetic energy expectation value of deuterium atoms calculated through the radial wave functionR_(n,l) (r) show that the energy value decreases significantly as the orbital quantum numberl increases for each fixed value ofn . Meanwhile, the calculation of the intrinsic kinetic energy expectation value of deuterium atoms using the total wave function ψ_(n,l,m) (r,θ,φ) shows that the intrinsic kinetic energy expectation value obtained in the1s state is〈K〉=ℏ^2/(2m〖a_0〗^2 ) , in the2s state, the〈K〉=ℏ^2/(8m〖a_0〗^2 ) value, the 2p state value〈K〉=ℏ^2/(24m〖a_0〗^2 ) , the 3s state value〈K〉=ℏ^2/(18m〖a_0〗^2 ) , the 3p state value 〈K〉=〖5ℏ〗^2/(162m〖a_0〗^2 ) , and the 3d state value 〈K〉=ℏ^2/(90m〖a_0〗^2 ) have the same results as the calculations using the radial wave function. This indicates that the expectation value originates from the radial component, while the contributions from the polar angle(θ) and azimuth(φ) show small variations in the total kinetic energy expectation. Thus, the intrinsic energy expectation of the electron can be obtained through the calculation of the radial wave functionR_(n,l) (r) or the total wave functionψ_(n,l,m) (r,θ,φ).