We study the bound-state solutions of vanishing angular momentum in a quaternionic spherical square-well potential of finite depth. As in standard quantum mechanics, such solutions occur for discrete values of energy. At first glance, it seems that the continuity conditions impose a very restrictive constraint on the energy eigenvalues and, consequently, no bound states were expected for energy values below the pure quaternionic potential. Nevertheless, a careful analysis shows that pure quaternionic potentials do not remove bound states. It is also interesting to compare these new solutions with the bound state solutions of the trial-complex potential. The study presented in this paper represents a preliminary step towards a full understanding of the role that quaternionic potentials could play in quantum mechanics. Of particular interest for the authors is the analysis of confined wave packets and tunnelling times in this new formulation of quantum theory.
In looking for qualitative differences between quaternionic and complex formulations of quantum physical theories, we provide a detailed discussion of the behavior of a wave packet in the presence of a quaternionic time-independent potential step. In this paper, we restrict our attention to diffusion phenomena. For the group velocity of the wave packet moving in the potential region and for the reflection and transmission times, the study shows a striking difference between the complex and quaternionic formulations which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation.
Abstract. In looking for qualitative differences between quaternionic and complex formulations of quantum physical theories, we provide a detailed discussion of the behavior of a wave packet in presence of a quaternionic time-independent potential step. In this paper, we restrict our attention to diffusion phenomena. For the group velocity of the wave packet moving in the potential region and for the reflection and transmission times, the study shows a striking difference between the complex and quaternionic formulations which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation. PACS. 03.65.Fd – 03.65.Nk I.
We solve the stationary Schrödinger equation for a particle confined to a 3D spherical wedge -- the region
{(r,θ,φ):0≤r≤R,0≤θ≤π,0≤φ≤Φ} with Dirichlet BCs on all surfaces. This exactly solvable constrained-domain model exhibits spectral reorganisation under symmetry-breaking BCs and provides an operator-domain viewpoint on angular momentum quantisation. We obtain three main results. First, the stationary states are standing waves in the azimuthal coordinate and consequently are \emph{not} eigenstates of
L^z; we prove
⟨Lz⟩=0 with
ΔLz=ℏnφπ/Φ=0, demonstrating that angular momentum projection becomes an observable with genuine quantum uncertainty rather than a good quantum number. Second, the effective azimuthal quantum number
μ=nφπ/Φ is generically non-integer, and square-integrability of the polar wavefunctions at both poles requires the angular eigenvalue parameter
ν to satisfy
ν−μ∈Z≥0. This regularity constraint yields a hierarchy: sectoral solutions (
ν=μ, satisfying the first-order highest-weight condition) exist for any real
μ>0, while tesseral and zonal solutions require integer steps, appearing only when
μ itself is integer. Third, application to a Coulomb potential shows that the familiar integer angular momentum spectrum of hydrogen arises from the periodic identification
φ∼φ+2π that defines the full-sphere Hilbert space domain; modified boundary conditions yield a reorganised spectrum with non-integer effective angular momentum. The model clarifies the distinct roles of single-valuedness (selecting integer
m via azimuthal topology) and polar regularity (selecting integer
ℓ≥∣m∣ via analytic constraints) in the standard quantisation of orbital angular momentum.
The introduction of complex numbers marked a significant leap in mathematics, introducing the imaginary unit i to represent the square root of -1. This innovative concept proved invaluable in solving equations involving square roots of negative numbers. The extension to quaternions involved introducing additional imaginary units denoted as j and k . A quaternion in an interesting concept that extends complex numbers to 4-D.The manuscript is about using quaternions to calculate wave packets in one-dimension, and anti-hermitian operators to obtain the results in quaternionic form including expectation values of position, momentum and energy. The results are compared to the exisiting results on complex wave packets in one-dimension.