The three dimensional rotation group, SO(3), is a symmetry group of the normal hydrogen atom. Each reducible representation of this group can be associated with a degenerate energy level. If this atom is placed in an external magnetic field, the interaction between the orbital magnetic moment with this field will lead to a symmetry breaking where the symmetry group of the atom is a new group distinct from the SO(3) group. This phenomenon describes the normal Zeeman effect, where a degenerate energy level splits into several new energy levels. It is explicitly shown that each of the new energy levels can be associated with an irreducible representation of the new symmetry group.
The three dimensional rotation group, SO(3), is a symmetry group of the normal hydrogen atom. Each reducible representation of this group can be associated with a degenerate energy level. If this atom is placed in an external magnetic field, the interaction between the orbital magnetic moment with this field will lead to a symmetry breaking where the symmetry group of the atom is a new group distinct from the SO(3) group. This phenomenon describes the normal Zeeman effect, where a degenerate energy level splits into several new energy levels. It is explicitly shown that each of the new energy levels can be associated with an irreducible representation of the new symmetry group.
As is known, the hydrogen atom energy levels depend on the principal quantum number n only and are independent of the azimuthal quantum number l. One can say (by using the common but not very suitable term) that there is degeneracy (i.e., the multiplicity of the level) with respect to the azimuthal quantum number. On the other hand, there is a general rule according to which the multiplicity of the Schro¨dinger equation eigenvalues relates to the invariance of the equation under a definite group of transformations. So, for example, the invariance under a common rotation group (spherical symmetry) leads to the energy level independence on the magnetic quantum number m. Therefore, it should be expected that the energy level independence of the azimuthal quantum number is explained by the existence of a certain transformation group that is more general than the three-dimensional rotation group. Up to now, this group of transformations for the Schro¨dinger equation has not been found. In the present work, we shall show that this group is equivalent to the rotation group in the four-dimensional Euclidean space.
This study aimed to numerically investigate the impact of dipole-dipole interactions on the energy levels of a hydrogen atom, specifically the energy shift caused by the interaction of the atom's dipole moment with external electric fields. This research involved applying perturbation theory and numerical techniques to quantify the changes in energy levels due to dipole orientations and magnitudes, contributing to a deeper understanding of atomic behavior and its implications in various scientific disciplines. The results presented showed that the wavefunction of the hydrogen atom exhibits degeneracy, especially n=2, n=3 states. This implies that n=2, n=3, have different states but the same energy as a result of the subshells with angular momentum l, which contains 2l+1 distinct state corresponding to positive value of m (magnetic angular momentum quantum number) m=-l±1. The total degeneracy of the energy levels n is caused by n2. We find out that as the orientation of the electron charges (θ,ϕ) change the dipole moment also changes and thus has a perturbing effect on the energy levels of the hydrogen atom leading to a shift in its energy. This is confirmed, as the energy shift increases slightly from the ground state energy function Φ100 to the second exicted state Φ300, while the degeneracy state of Φ310 experiences infinitesimal energy shift above. This is because the probability of the electron been found in the lower energy level is greater than at higher energy level and so it is mostly affected there by energy shift. Also, the dipole moment could increase liberational energy of the atom and thus lead to more excitement even though its impact is quite negative. The outcome of this study could have implications in precision measurements, quantum computing, and the understanding of complex atomic systems.
The hydrogen atom exhibits an exact SO(4) symmetry that has long been recognized but seldom explained. Conventional quantum mechanics reproduces the spectrum through algebraic methods, yet leaves unanswered why spin-½, orbital degeneracies, and the Pauli exclusion principle exist. In this work, we show that these features arise naturally if the electron is modeled as a four-dimensional rotor constrained to the three-sphere, S 3 . The two independent spin modes of the rotor generate the SU(2) + × SU(2) - structure, whose balanced irreducible representations project into three-dimensional orbitals with the familiar quantum numbers n, l, and m. The radial equation reduces to the associated Laguerre form, yielding exact hydrogenic wavefunctions and energies. Orbital filling, exclusion, and hybridization emerge as geometric necessities rather than axioms. The model also predicts small but definite deviations from ideal quantum mechanics, including corrections from finite rotor inertia, spin–spin couplings, and symmetry-breaking effects. These deviations are accessible to precision spectroscopy and condensed matter experiments, providing falsifiable tests. By grounding quantum mechanics in four-dimensional geometry, the rotor framework preserves all known successes while offering new explanatory clarity and predictive power.
Abstract In this chapter we discuss the intrinsic angular momentum called spin. Starting from the magnetic dipole moment related to orbital angular momentum, we focus on the hydrogen atom and consider the effect of an external magnetic field, resulting in Larmor precession and Zeeman splitting. We then turn to the discovery of spin in the Stern-Gerlach experiment. After providing a mathematical description of spin-1/2 systems in terms of two-component spinors and Pauli matrices, we illustrate projective measurements for the example of spin. The relationship between the Lie groups SU(2) and SO(3) is discussed in the context of the transformation of spinors under rotations. We analyse how the presence of spin alters the description of the hydrogen atom. Discussing the addition of angular momenta and the Pauli exclusion principle leads us to the structure of electronic orbitals. Finally, we discuss the occupation of electronic energy levels in heavier nuclei
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.