Principal Quantum Number Research Articles

Sayed`s Theory in Quantum Mechanics: An Innovative Approach for Wave Function Equations Measurability, Efficacy and Universality

Allah (God) is ubiquitous and the only Creator of Everything. Quantum mechanics is the world of the microscopic scale of physics. The universe is a macroscopic manifestation of the quantum approach. In this revolutionary article, correctness and rectification of Schrodinger wave equations that has huge limitations and shortcomings, were proudly achieved. The Sayed quantum wave function (SQWF) was derived and defined (Ψs) for free electron and in Laplace coordinates as a function in wavelength and light speed. The SQWF was calculated and quantified for de Broglie moving electron; 0.4033x10-20 Second. Using SQWF equation, the speed of light was accurately calculated; ~3x108 m/s. respectively. An outstanding Sayed`s quantum wavelength (SQW) was proposed and elucidated. The Sayed time dependent wave function equations (STDWFE) were derived and given to be a generic alternative to deal with all atoms rather than the one electron atoms of Schrodinger. The crucial parameters; wavelength, Rydberg constant RH, atomic number, principle quantum number, and mc2 were introduced in the STDWFE. It is a quantized formula. The (Ψs) was calculated for H, He, B and found to be , ,The Sayed quantum constantRH mc2, was quantified to be 8.93x10-7 J/m.

Electron-impact ionization for Ne[formula omitted] and Ne[formula omitted]

Ionization cross sections are studied for energy levels of the ground configurations of the Ne3+ and Ne4+ ions. The distorted wave (DW) approximation is used to analyze experimental data. The scaled DW cross sections are used to explain measurements for the Ne3+ ion. Study includes analysis of contributions from the direct and indirect ionization processes. Convergences of excitation-autoionization (EA) channels are estimated by analyzing excitations up to shells with the principal quantum number n⩽20. This study shows that the EA process provides from ∼10% to 20% for the energy levels of the ground configurations of the Ne3+ and Ne4+ ions.

Hanle Effect for Lifetime Determinations in the Soft X-Ray Regime.

By exciting a series of 1s^{2} ^{1}S_{0}→1snp^{1}P_{1} transitions in heliumlike nitrogen ions with linearly polarized monochromatic soft x rays at the Elettra facility, we found a change in the angular distribution of the fluorescence sensitive to the principal quantum number n. In particular it is observed that the ratio of emission in directions parallel and perpendicular to the polarization of incident radiation increases with higher n. We find this n dependence to be a manifestation of the Hanle effect, which served as a practical tool for lifetime determinations of optical transitions since its discovery in 1924. In contrast to traditional Hanle effect experiments, in which one varies the magnetic field and considers a particular excited state, we demonstrate a "soft x-ray Hanle effect" which arises in a static magnetic field but for a series of excited states. By comparing experimental data with theoretical predictions, we were able to determine lifetimes ranging from hundreds of femtoseconds to tens of picoseconds of the 1snp^{1}P_{1} levels, which find excellent agreement with atomic-structure calculations. We argue that dedicated soft x-ray measurements could yield lifetime data that are beyond current experimental reach and cannot yet be predicted with sufficient accuracy.

Phase Prediction via Crystal Structure Similarity in the Periodic Number Representation.

The periodic number (PN) representation of the chemical systems, introduced by Dmitri Mendeleev, uncovers the fundamental principle of chemical similarity in a straightforward way. In this framework, the rows correspond to the principal quantum numbers of the elements' electronic configurations when considered isolated atoms. This systematic arrangement allows for a deeper understanding of the relationships and patterns among the elements. In this study, we propose a novel strategy for structure type (prototype) prediction by utilizing the PN concept to identify possible modifications and phase stability of unexplored chemical systems. Our PN-based crystal structure prediction (PNcsp) program, which evaluates similarity through PN neighboring in the phase map, provides the most probable prototypes for unknown and unreported modifications of given phases in binary and higher order chemical systems. We applied PNcsp to 59 distinct chemical systems whose equimolar phases are indicated in the respective phase diagrams but lack accurate experimental structure determination. Our methodology identified 93 prototypes for these 59 equiatomic phases, of which 47 exhibit mechanical and dynamic stability. Notably, this approach discovered 19 entirely novel, fully stable polymorphic phases, thereby expanding the known landscape of potential materials. Furthermore, we demonstrated that this method is also effective for nonequimolar and higher order systems.

High-Resolution Continuous-Wave Laser Spectroscopy of Long-Lived Rydberg States in NO.

High-resolution continuous-wave (cw) laser spectroscopy of nitric oxide (NO) molecules has been performed to study and characterize the energy-level structure of and effects of electric fields on the high Rydberg states. The experiments were carried out with molecules flowing through a room temperature gas cell. Rydberg-state photoexcitation was implemented using the resonance enhanced three-color three-photon excitation scheme. Excited molecules were detected by high-sensitivity optogalvanic methods. Detailed measurements were made of Rydberg states with principal quantum numbers n = 22 and 32 in the series converging to the lowest rotational and vibrational state of the NO+ cation. The experimental data were compared with the results of numerical calculations which provided insight into the orbital angular momentum character of the intermediate H 2Σ+ state, improved determinations of the nf and ng quantum defects, a bound on the magnitude of the nh quantum defect, and information on the decay rates of the nf and ng Rydberg states. These measurements represent a step-change in laser spectroscopic studies of high Rydberg states in small atmospheric molecules. They open opportunities for more detailed studies of slow decay processes of Rydberg NO molecules confined in electrostatic traps, the synthesis of ultralong range Rydberg bimolecules, and the development of optical methods for trace gas detection.

Photo-induced loss of H2 from H2S, CH4, H2O and SiH4, when and why is it possible

We have explored the possibility of forming H⋅ and H2 from H2S, CH4, H2O and SiH4 in a series of ab initio calculations. The key finding is that H2S can give rise to direct photolytic ultrafast formation of H2 This finding is in agreement with the available experimental data and is a result of a directed pathway that leads toward H2 formation at easily accessible wavelengths. It arises at the (conical) intersection between the two lowest-lying electronic states. This intersection is a result of the valence orbital interaction of the two hydrogen atoms. This interaction is becoming more favorable as the S-H bonds simultaneously becomes longer. The H-S σ-σ∗ bonding interaction cannot compete with the bond formation that takes place between the hydrogens as two bonds stretch simultaneously. For oxygen this interaction it predominant. The reason is that sulfur has a higher principal quantum number and the interaction with the 1s of hydrogen therefore is less favorable. The excited states that are involved in the H2 formation from H2S could be populated by, for example, a two-photon 400 nm excitation. The other hydrides have a more entangled potential energy landscape close to the Franck-Condon region in the direction of the reaction coordinate that leads to H2 loss and as a result the direct loss of a hydrogen atom is favored. A two-photon excitation could be induced by femtosecond laser pulses that at the same time is proposed as a future platform for sensing the H2S molecules with the backward transient absorption scheme. The implication would be that both sensing and photoinduced H2 formation could be in place at the same time which justifies the use of expensive femtosecond light sources getting “two for the price of one”.

Balmer Decrement Anomalies in Galaxies at z ∼ 6 Found by JWST Observations: Density-bounded Nebulae or Excited H i Clouds?

We investigate the physical origins of the Balmer decrement anomalies in GS-NDG-9422 and RXCJ2248-ID galaxies at z ∼ 6 whose Hα/Hβ values are significantly smaller than 2.7, the latter of which also shows anomalous Hγ/Hβ and Hδ/Hβ values beyond the errors. Because the anomalous Balmer decrements are not reproduced under the Case B recombination, we explore the nebulae with optical depths smaller and larger than the Case B recombination by physical modeling. We find two cases quantitatively explaining the anomalies: (1) density-bounded nebulae that are opaque only up to around Lyγ–Ly8 transitions and (2) ionization-bounded nebulae partly/fully surrounded by optically thick excited H i clouds. The case of (1) produces more Hβ photons via Lyγ absorption in the nebulae, requiring fine tuning in optical depth values, while this case helps ionizing photon escape for cosmic reionization. The case of (2) needs the optically thick excited Hi clouds with N 2 ≃ 1012−1013 cm−2, where N 2 is the column density of the hydrogen atom with the principal quantum number of n = 2. Interestingly, the high N 2 values qualitatively agree with the recent claims for GS-NDG-9422 with the strong nebular continuum requiring a number of 2s-state electrons and for RXCJ2248-ID with the dense ionized regions likely coexisting with the optically thick clouds. While the physical origin of the optically thick excited H i clouds is unclear, these results may suggest gas clouds with excessive collisional excitation caused by an amount of accretion and supernovae in the high-z galaxies.

Rydberg excitons in cuprous oxide: A two-particle system with classical chaos.

When an electron in a semiconductor gets excited to the conduction band, the missing electron can be viewed as a positively charged particle, the hole. Due to the Coulomb interaction, electrons and holes can form a hydrogen-like bound state called the exciton. For cuprous oxide, a Rydberg series up to high principle quantum numbers has been observed by Kazimierczuk et al. [Nature 514, 343 (2014)] with the extension of excitons up to the μm-range. In this region, the correspondence principle should hold and quantum mechanics turn into classical dynamics. Due to the complex valence band structure of Cu2O, classical dynamics deviates from a purely hydrogen-like behavior. The uppermost valence band in cuprous oxide splits into various bands resulting in yellow and green exciton series. Since the system exhibits no spherical symmetry, the angular momentum is not conserved. Thus, the classical dynamics becomes non-integrable, resulting in the possibility of chaotic motion. Here, we investigate the classical dynamics of the yellow and green exciton series in cuprous oxide for two-dimensional orbits in the symmetry planes as well as fully three-dimensional orbits. Our analysis reveals substantial differences between the dynamics of the yellow and green exciton series. While it is mostly regular for the yellow series, large regions in phase space with classical chaos do exist for the green exciton series.

ALTERNATIVE EXPLANATION OF FILLING THE ELECTRONIC LAYERS OF ATOMS OF CHEMICAL ELEMENTS

The article provides a new approach to the formation of periods in Mendeleev's periodic system. Reconfiguration of the periods in the Mendeleev table using the newly proposed formula and the newly proposed quantum states for the outer electron shells of atoms of chemical elements is proposed. Purpose of work: Further development of the alternative theory of creation of electronic shells of atoms of chemical elements in D.I.Mendeleev's table. Results and discussions. The following order of formation of electron layers is proposed: principle quantum number (n), then the quantum state of the electrons forming the electronic configuration of the subperiods (first and second), and only then the remaining quantum orbitals (s, d, f and p). First and second new quantum states and a new formula for calculating the number of electrons in the outer electron shell of element periods and subperiods were proposed. The new level of electronic counting (II) and the new quantum number proposed by us allow to change the understanding of the internal structure of chemical elements without changing the general appearance and order of arrangement of chemical elements in D.I.Mendeleev's table itself, and therefore its contents. An important advantage of the proposed alternative model is that it takes into account and relies on already known, classical data for most of its main operational characteristics and is an extension of this important topic for classical chemistry theory. Therefore, the change in the number of periods, the introduction of a new quantum number in the form of D.I.Mendeleev's table, outwardly it does not differ much, but requires the necessary additional explanation. Concept. The proposed alternative approach to the structure and electronic structure of atomic shells of chemical elements in the periods and groups of D.I.Mendeleev's periodic system was recently developed by the authors and has good prospects for development primarily in the following directions: - identification of possible patterns of changes in the characteristics of chemical and physical properties of elements; - development of an accompanying model of the electronic structure of electronic shells of atoms of chemical elements of the proposed alternative model.

New definitions of the effective nuclear charge and its application to estimate the matrix element [formula omitted]

New definitions of the effective nuclear charge are proposed to evaluate the orbital wave function and the dipole matrix element ⟨n,l|r|n′,l′⟩ for the non-hydrogenic atoms or ions. It is found that the commonly used effective nuclear charge defined by the orbital energy is insufficiently precise to estimate the orbital wave functions and matrix elements ⟨n,l|r|n′,l′⟩. Instead, the effective nuclear charge defined by ⟨r⟩ or ⟨r2⟩ are more advantageous. It is shown that the effective nuclear charge method becomes increasingly precise to predict wave functions and matrix elements as the orbital angular momentum, the principle quantum number and the degree of ionization increase. Good accuracy is achieved when principal quantum numbers are identical n=n′ or when both orbital quantum numbers l and l′ are non-zero. When s-orbitals are involved (l or l′ equal to zero) the precision is decreasing.

Fully relativistic energies, transition properties, and lifetimes of lithium-like germanium

Abstract Employing two fully relativistic methods, the multi-reference configuration Dirac–Hartree–Fock (MCDHF) method and the relativistic many-body perturbation theory (RMBPT) method, we report energies and lifetime values for the lowest 35 energy levels of the (1s2)nl configurations (where the principal quantum number n = 2–6 and the angular quantum number l = 0, …, n–1) of lithium-like germanium (Ge XXX), as well as complete data on the transition wavelengths, radiative rates, absorption oscillator strengths, and line strengths between the levels. Both the allowed (E1) and forbidden (magnetic dipole M1, magnetic quadrupole M2, and electric quadrupole E2) ones are reported. The results from the two methods are consistent with each other and align well with previous accurate experimental and theoretical findings. We assess the overall accuracies of present RMBPT results to be likely the most precise ones to date. The present fully relativistic results should be helpful for soft x-ray laser research, spectral line identification, plasma modeling and diagnosing. The datasets presented in this paper are openly available at https://doi.org/10.57760/sciencedb.j00113.00135.

Existence of quantum states for Klein–Gordon particles based on exact and approximate scenarios

In the present study, Kummer’s eigenvalue spectra from a charged spinless particle located at spherical pseudo-dot of the form r 2 + 1/r 2 is reported. Here, it is shown how confluent hypergeometric functions have principal quantum numbers for considered spatial confinement. To study systematically both constant rest-mass, m 0 c 2 and spatial-varying mass of the radial distribution m 0 c 2 + S(r), the Klein–Gordon equation is solved under exact case and approximate scenario for a constant mass and variable usage, respectively. The findings related to the relativistic eigenvalues of the Klein–Gordon particle moving in the spherical space show the dependence of mass distribution, so it has been obtained that the energy spectra has bigger eigenvalues than m 0 c 2 = 1 fm−1 in exact scenario. Following analysis also shows eigenvalues satisfy the range of E < m 0 c 2 through approximate scenario.

Single-electron capture from helium targets by heavy nuclei of charges 1–7

Single-electron capture by multiply charged nuclei from helium atoms is studied by means of the prior form of the four-body boundary-corrected continuum intermediate state (BCIS-4B) method. Computations concern total cross sections for the state-selective (Qnlm) and state-summed (Qnl,Qn,QΣ) populations at 20–3000 keV/amu. These refer to the collisions of the type AZP++He(1s2)→A(ZP−1)+(nlm)+He+(1s). Here, the projectile AZP+ covers the ions H+, He2+, Li3+, Be4+, B5+, C6+ and N7+. The reported findings are tabulated for each value of the quantum numbers {n,l,m}. The maximum values nmax of the principal quantum number n are 4 (H+, He2+, Li3+), 5 (Be4+), 6 (B5+) and 7 (C6+, N7+). The sum over all the n states is truncated by selecting an appropriate cutoff value nmax and by subsequently applying the Oppenheimer n−3 scaling rule to approximately include the contributions from n>nmax. The obtained results for QΣ and for some final n−specific states favorably describe the corresponding experimental data. This qualifies the prior form of the BCIS-4B method for further useful applications in several interdisciplinary fields, including astrophysics, thermonuclear fusion, plasma physics and ion therapy in medicine.

Advantages of Ultrathin AlN/GaN/AlN Quantum Wells for Excitonic Population Distribution and Transition Features Studied by Phononic–Excitonic–Radiative Model

Excitons are expected to be a high‐efficiency emission source in UV light‐emitting devices. However, the damping of the excitonic laser oscillation has been reported under conditions where the excitonic states are expected to be populated in the conventional theory. In order to understand the exciton dynamics under the thermal nonequilibrium state, a theoretical model including various energy species in semiconductors such as electrons, phonons, and photons is required. Herein, a 2D phononic–excitonic–radiative model is constructed to analyze the exciton dynamics in a 2D system. 2D excitons with four principal quantum number states and the continuum in the lowest energy level of the AlN/GaN/AlN quantum wells are considered. It is found that the 2D phonon significantly augments the excitation transition rate. When the high recombination rate corresponding to stimulated emission is considered, the exciton binding energy of 108 meV is not enough to reduce the population in the high‐order discreet states and the continuum states, while the binding energy of 215 meV corresponding to the one monolayer GaN has an advantage of reducing these populations. The analysis of population flux has an advantage in discussing the increase in the kinetic energy transfer to the 1S exciton.

Demonstration of a tunable interface between Rydberg atoms and superconducting microwave circuits with differential polarizability nulling

Microwave-dressed Rydberg atoms in pairs of states that are optically accessible and engineered so that the frequency of the transition between them is tunable while exhibiting a low sensitivity to stray electric fields have been interfaced with a superconducting coplanar waveguide (CPW) microwave resonator. This was achieved with helium atoms in triplet Rydberg states with principal quantum numbers n=55 and 56. The atoms were dressed with two strong microwave fields—a control field and a differential polarizability nulling field. The control field allowed the atoms to be coupled to the resonator using the two-color two-photon transition between the equal parity 1s55sS13≡|55s〉 and 1s56sS13≡|56s〉 Rydberg states, with one photon supplied by it and the other provided by the resonator. The nulling field was detuned from the field-free 1s55sS13→1s55pPJ3≡|55p〉 transition to admix |55p〉 character into the |55s〉 state and eliminate the differential static electric dipole polarizability of the |55s〉 and |56s〉 states. The experiments were performed with atoms located ∼300µm above a λ/4 niobium nitride superconducting CPW resonator operated at temperatures between 3.59 and 3.73K. The measured ac and dc Stark shifts of the Rydberg states at this atom-circuit interface in the presence of the dressing fields are in good quantitative agreement with the results of Floquet calculations. The differential polarizability nulling fields used in the experiments allowed the difference in the Stark shifts of the |55s〉 and |56s〉 states over the range of dc fields offset from the residual stray field by ±65 mV/cm to be reduced from 2π×2.1 MHz to 2π×125 kHz. Published by the American Physical Society 2024

State-selective electron capture in Ar16++H(1s) collisions for charge-exchange recombination spectroscopy

Abstract State-selective electron capture in collisions of Ar16+ ions with ground-state hydrogen atoms has been modeled using the two-center wave-packet convergent close-coupling approach. The partially stripped He-like projectile ion is represented using a model potential. Experimental measurements are not available for this collision system and to date, only the classical trajectory Monte Carlo (CTMC) method has been applied to calculate cross sections. The calculated total electron-capture cross section (TECS) is in good agreement with the previous CTMC results at the low energies but slightly larger at higher energies. This is likely because, in this work, we account for capture into highly excited states, which contribute significantly to the TECS at the intermediate energies. The n-resolved electron-capture cross sections have also been presented for capture into states with n = 6 − 19 , where n is the final-state principal quantum number. The most important of these are the cross sections for capture into the n = 14 − 17 states, which are used in charge-exchange recombination spectroscopy techniques. For these cross sections, a significant difference is observed between the present and previously published data. The cross sections differ by an order of magnitude in the 10−60 keV u−1 energy range. The agreement between the calculations is observed at the energies above 70 keV u−1. The n ℓ -resolved electron-capture cross sections have also been presented at 15, 60, 100 and 200 keV u−1 projectile energies, where ℓ is the final-state angular momentum quantum number.

Information-theoretic measures and Compton profile of H atom under finite oscillator potential

Information-theoretic measures for nl (2L) states of a H atom (with n=1−10 and l=0−2 , where n and l denote principal and angular momentum quantum numbers) have been investigated within a quantum dot by utilizing the Ritz variational principle, with the help of a Slater-type basis set. A well-established two-parameter (depth and width) model of finite oscillator potential is used to simulate the dot environment. The variationally optimized position (r)-space wave function is utilized to determine the momentum (p)-space wave function, leading to the generation of p-space radial density distribution. We explore the impact of cavity parameters on quantum information theoretic measures, such as Shannon (S) and Fisher information (I) entropy, in the ground as well as the excited state. The results of S were also used to test the Bialynicki–Birula–Mycielski inequality, related to the entropic uncertainty principle for the confined H atom. Some simple new fitting laws pertaining to S and I have been proposed. Furthermore, the p-space radial density is employed to derive the Compton profile of the confined H atom. Possible tunability of S,I and Compton profiles with respect to the parameters is noted.

The wide spectral range characteristics and dynamic evolution of laser-produced tin plasmas

The energy levels and transitions of highly charged Sn ions are key data for analyzing emission spectra, radiative opacity, and conversion efficiency of laser-produced Sn plasmas. Temporally–resolved spectra in the 6–45-nm range were acquired via high-precision, spatio–temporally resolved techniques. In addition, the Cowan code, based on Hartree–Fock multi-configurational interactions, was used to perform structural calculations for Sn4+–Sn13+ ion spectra. Emission spectra in the 11–18-nm range were primarily attributed to Sn8+–Sn12+ ions, with transitions involving principal quantum numbers Δn=0 and Δn=1. Whereas, emission spectra in the 6–11-nm and 18–45-nm ranges were mainly attributed to Sn9+–Sn13+and Sn4+– Sn9+ions, with transitions involving principal quantum numbers Δn=1,2. Furthermore, Segmented simulations were conducted based on the assumptions of a normalized Boltzmann distribution among the excited states and a steady-state collisional-radiative model to explain the characteristics of the radiation spectra and variations in state parameters variations during laser-produced plasma expansion from the core to the outer periphery. A specific analysis was conducted on the temporal evolution of the electron temperature and density, while the charge state of the plasma was analyzed over the 6–18-nm range. Based on the spectral diagnostics, this work focused on the impact of non-uniformity in the expansion of the Sn plasma.

Measurements of cesium PJ-series quantum defect with the microwave spectroscopy.

High-precision microwave spectroscopy has been used to measure the transition frequency of nS1/2 → nPJ (n is the principle quantum number) and further the quantum defect of nPJ states in a standard cesium magneto-optical trap. A microwave field with 30-μs duration coupling the nS1/2 → nP1/2,3/2 transition yields a narrow linewidth microwave spectroscopy with the linewidth approaching the Fourier limit. After carefully compensating the stray electric and magnetic field and using the diluted atomic gas, we extract improved quantum defects of nPJ state, δ0(nP1/2) = 3.59159091(19), δ2(nP1/2) = 0.36092(35) and δ0(nP3/2) = 3.55907153(25), δ2(nP3/2) = 0.37344(47).

Spatiotemporal optical vortices with controllable radial and azimuthal quantum numbers

Optical spatiotemporal vortices with transverse photon orbital angular momentum (OAM) have recently become a focal point of research. In this work we theoretically and experimentally investigate optical spatiotemporal vortices with radial and azimuthal quantum numbers, known as spatiotemporal Laguerre-Gaussian (STLG) wavepackets. These 3D wavepackets exhibit phase singularities and cylinder-shaped edge dislocations, resulting in a multi-ring topology in its spatiotemporal profile. Unlike conventional ST optical vortices, STLG wavepackets with non-zero p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{\\boldsymbol{p}}}}}}$$\\end{document} and l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{\\boldsymbol{l}}}}}}$$\\end{document} values carry a composite transverse OAM consisting of two directionally opposite components. We further demonstrate mode conversion between an STLG wavepacket and an ST Hermite-Gaussian (STHG) wavepacket through the application of strong spatiotemporal astigmatism. The converted STHG wavepacket is de-coupled in intensity in space-time domain that can be utilized to implement the efficient and accurate recognition of ultrafast STLG wavepackets carried various p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{\\boldsymbol{p}}}}}}$$\\end{document} and l.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{\\boldsymbol{l}}}}}}.$$\\end{document} This study may offer new insights into high-dimensional quantum information, photonic topology, and nonlinear optics, while promising potential applications in other wave phenomena such as acoustics and electron waves.