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The quantum harmonic oscillator is a fundamental piece of physics. In this paper, we present a self-contained full-fledged analytical solution to the quantum harmonic oscillator. To this end, we use an eight-step procedure that only uses standard mathematical tools available in natural science, technology, engineering and mathematics disciplines. This solution is accessible not only for physics students but also for undergraduate engineering and chemistry students. We provide interactive web-based graphs for the reader to observe the shape of the wave functions for an electron and a proton when both are subject to the same potential. Each of the eight steps in our solution procedure is treated as a separate problem in order to allow the reader to quickly consult any step without the need to review the entire article.

For scientists and engineers, the Laplacian operator is a fundamental tool that has made it possible to carry out important frontier studies involving wave propagation, potential theory, heat conduction, the distribution of stresses in a deformable solid and quantum mechanics. Knowing, understanding, and manipulating the Laplacian operator allows us to tackle complex and exciting physics, chemistry, and engineering problems. In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. Our derivation is self-contained and employs well-known mathematical concepts used in all science, technology, engineering, and mathematics (STEM) disciplines. Our lengthy but straightforward procedure shows that this fundamental tool in mathematics is not intractable but accessible to anyone who studies any of the STEM disciplines. We consider that this work may be helpful for students and teachers who wish to discuss the derivation of this vital tool from an elementary approach in their courses.

By using a remote plasma enhanced chemical vapor deposition system, we grow a silicon rich silicon nitride thin film on the surface of five different substrates: silicon wafer, fused silica, highly oriented pyrolytic graphite, muscovite mica and potassium chloride. By means of high-resolution transmission electron microscopy we studied the influence that each substrate has on the auto-formation of silicon quantum dots (≤4.2 nm) embedded in the grown film. We conjecture that the growth of the film is carried out by the formation of highly reactive intermediates that are chemisorbed on the substrate surface. We conclude proposing the hypothesis that the substrate surface profile has minimal influence on the growth of a silicon nitride thin film that can embed silicon quantum dots.

The optimal development of silicon photonics could improve the current state of microelectronics technology. However due to the multiple parameters involved in the elaboration processes, it is not a simple task to fabricate and reproduce silicon emitting structures. In this paper we discuss the role played by ammonia in depositing silicon nanoparticles (Si-NPs) embedded in silicon nitride matrix by remote plasma enhanced chemical vapor deposition (RPECVD). We found that by varying the ammonia flow, the deposition rate changes. It was observed by high resolution transmission electron microscopy (HRTEM) that higher density and a lower average size of Si-NPs is achieved by increasing the ammonia flow. We recognized a linear behavior between the ammonia flow used to deposit a sample and its maximum photoluminescence (PL) emission peak. Finally we propose a model that allows to predict the maximum PL peak for a given ammonia flow within the range of 403–540 nm.

The pulsed-laser photoacoustic-technique (PLPA) was implemented to characterize molecular orientational order and anisotropy in push–pull poled polymeric films as function of temperature and laser polarization. Traditionally, photoacoustic signals are considered to be directly proportional to the linear optical absorption in amorphous media. In this work, however, it is shown that photoacoustic signals can also be highly sensitive to the material anisotropy when convenient polarization dependent photoacoustic analyses are performed. Thus, variation of the molecular orientation in organic films, comprising rod-like polar chromophores, can be unambiguously monitored via rms-analyses performed on the amplitude of the generated opto-acoustical PLPA-signals as function of the incident laser polarization. This result can be useful for the characterization of organic-based nonlinear optical (NLO) poled films and, in general, in studies of anisotropic materials. In fact, in this work we were able to accurately determine the molecular order parameter ( ϕ ) of a NLO-active spin-coated polymeric film containing optically active push–pull chromophores. These molecules, previously oriented via an electrical-poling procedure, are capable to exhibit strong second harmonic generation (SHG) effects. The PLPA-measurements were systematically compared to the linear UV–vis optical absorbance spectra while heating the poled film sample in order to monitor the thermally induced molecular disorder, so that the order parameter may be photo-acoustically evaluated via the PLPA-signals generated from the poled to the unpoled film phase. These PLPA-experiments were performed taking into account the UV–vis reference spectra for calibration and comparison purposes in the evaluation of the order parameter. A significant advantage of the PLPA-technique over commonly used optical spectral methodologies is its convenient applicability in samples exhibiting poor or null optical transmission.

A highly-conjugated polar dye with three aromatic rings connected with azo groups was prepared and it was incorporated in polycinnamate. It showed a third-order nonlinear susceptibility of 8 × 10-10 esu determined by a Z-scan technique. The unpoled and poled films show the same susceptibility indicating the polymer film could not be poled. The open aperture Z-scan showed negligible two-photon absorption at 1064 nm.

In order to study the effect of main-chain conformation on the second-order non-linear optical properties, two novel isomeric polymers, consisting of hexa-2,4-diynylene-1,6-dioxydinaphthoates main-chains containing a polar azo dye, Disperse Red 19, were synthesized and characterized. The 1,2- and 3,2-polymers had a T g of 134 and 143°C, respectively. Their spin-coated films were exposed to corona poling, and the orientation behavior was studied. The order parameters of the two polymers were similar, around 0.2. The 3,2-polymer underwent cross-linking during poling. Neither polymer did cross-link when irradiated by UV light during poling, indicating that their chemical structures are not apt to cross-link.

In order to study the relationships between the chemical structures of polymer main chains and second order nonlinear optical properties, novel polymers with hexa-2,4-diynylene-1,6-dioxynaphthoate as a main chain and common azo dyes (Disperse Red 1 and 13) in the side chains, were synthesized and characterized. The polymers were soluble in common organic solvents such as chloroform, dimethylformamide, N-methylpyrrolidone, etc, and gave films with excellent optical quality by spin coating. Apparent orientation parameters of around 0.25 were observed by the absorption decrease by Corona poling. The steric effect of chlorine atom of the dye on poling was not observed. On heating at 100 °C after poling the order parameters did not return to zero but reached a constant. It is thought that some of the chromophores remain orientated permanently.

The poling and orientation parameters of the four polymers para and meta benzoates and cinnamates containing Disperse Red 19 dye were studied. The para polymers were found to be orientated more efficiently, and their stabilities were much better than the meta polymers. The photobleaching takes place when irradiated at temperatures around Tg, but did not occur when irradiated under the poling conditions. After the photo cross‐linking at around Tg the orientation maintenance of the para polymers were found to be almost indefinite. The stability of the para polymers is considered to be due to their stable main chain conformation when poled.

Rigid-rod aromatic polyimides (PIs) constantly attract wider interest because of their unique combination of properties, such as the excellent thermo-oxidative stability and mechanical properties, good dielectric strength and dimensional stability.[1-4] Additionally in recent years, PIs have been considered as one of the best materials for gas separation membranes due to their reasonable permeability to CO2 and high selectivity against CH4.[5] However, synthesis and processing of these polymers are generally very difficult because of their limited solubility and infusibility.[6] Considerable efforts have been made to improve the solubility through the synthesis of new diamine or dianhydride monomers. The common strategy consists in the incorporation of bulky lateral substituents,[4,7-15] flexible alkyl side chains,[16,17] non-coplanar biphenylene moieties,[18] and kinked units,[19-22] into rigid polymer backbones.

Predesigned push-pull azo-dye polymers were homogeneously dispersed within a SiO2sol-gel matrix synthesized via the sonogel (SG) route. High-quality spin-coated films were obtained with these hybrid structures in the liquid sol-phase. The spectroscopic UV- Vis analyses reveal the appropriate insertion of these organic compounds within the highly pure SG-environment whereas the thermal (DSC) analysis and photoacoustic measurements evidence the thermomechanical stability of the amorphous hybrid layers. As the optical attenuation, refractive index and film thickness values of the obtained films are adequate for opto-electronic applications; these hybrid films were implemented to fabricate optical waveguiding prototypes. In this sense, functional planar waveguides were fabricated for nonlinear optical (NLO) applications after performing a molecular ordering via a corona DC-poling procedure in order to achieve a macroscopic polar order (ferroelectric and noncentrosymmetric arrangement of the organic chromophores). The poled films were then able to exhibit stable NLO-waveguiding effects as excited with a Nd:YAG laser system in order to generate second harmonic waves travelling within the planar layer.