Energy as Frequency, Relative Velocity and Lorentz Transformations

In quantum mechanics, energy is associated with frequency for both a particle with rest mass and a photon. In the case of the photon, we try to show that this result follows from special relativity (Lorentz transformation) which in the photon case gives rise to relative velocities. We consider a photon moving along the x axis and bouncing back and forth between two mirrors separated by L and moving to the right with speed v. We argue that Lorentz transformations for photons create the notion of relative velocity because E=|p|c, but there still exists a moving frame with velocity v. As a result, a relationship between the energies of the photon moving to the right and then to the left exists with each energy being multiplied by a respective time= 1/ relative speed. This suggests photon energy representing a frequency. . We also note that relative velocities from the Lorentz transformations are the opposite of those associated with the times the photon spends moving from one mirror to the other.. In other words, the photon moving to the right is associated with 1/c-v, but its time with 1/c-v because it looks as if the photon is moving more slowly when trying to hit a mirror that is moving away from it

Hypothetical Velocity and Photon Ray Theory for Reflection/Refraction from a Moving Surface Part II

In Part I, we argued that the hypothetical velocity approach of (1) and the photon ray theory of (2) were identical and based on the idea of a constant linked to the phase of the photon wavefunction -Et+p dot r= t |p| (- c/n + v cos(AA)) ((1)) where AA is the angle between an interface velocity vector and the p vector. In the case of refraction (or reflection) from a fixed surface, there is no velocity of the interface so v is zero, while for a moving mirror or interface (of media with two different indices of refraction) there is. Thus dr (vector) /dt, in this analysis, is not the velocity vector of the photon, but rather of a moving interface. The surface of the interface (or line in 2-dimensions) yields no impulse hit, so there is conservation of the component of momentum along this line. When an interaction occurs, t in ((1)) is the same for the incident ray and the reflected and refracted, so |p| (- c/n + v cos(AA)) should also be a conserved quantity, based on the photon phase. We show how this approach gives rise to Snell’s law, a relationship between the incident and reflected angle of light for a moving mirror and refraction of light for a moving interface with a different index of refraction. The point we make is that the phase of the photon remains constant in these interactions if r(vector)/t is considered to be the velocity of a moving mirror or interface. In other words, there seems to be a generalization of the Lorentz invariant -Et+ p dot r for r vector/t = velocity of a surface which gives rise to reflection or refraction. In other words, the reflected and refracted rays are equivalent to the incident one in that they all have the same phase: t |p| (- c/n + v cos(AA)). For Snell’s law, v=0, so the phase is -Et which is constant, but is usually associated with the photon retaining the same energy whether it reflects or refracts

Density Due to Collective versus Fluctuation Kinetic Energy in Classical, Classical Statistical and Quantum Mechanics

The purpose of this note is to consider spatial densities in the case of classical mechanics (1/v), quantum mechanics (W(x)*W(x) for bound states) and statistical mechanics exp(-V(x)/T) with respect to the effects of collective motion. In the classical mechanical case, with a density of 1/v, the entire motion is collective and seems to be modeled as a compressible gas with flux continuity. (It seems that this idea may even possibly lead to Newton´s second law.) In the case of quantum mechanics, high energy eigenvalue densities are supposed to approach the classical density near peak density points. We try to investigate why by considering root mean square velocities and what we think is fluctuation flux in a quantum system. For peak wavefunction values dW(x)/dx=0 and we try to show that these points represent places where there is no fluctuation flux. Thus, at such points, there is only collective motion and one would expect W(x1)W(x1) vave(x1) =W(x2)W(x2) vave(x2) between two such points x1 and x2, i.e. a continuity equation. For points in between, fluctuation flux mixes with collective velocity to create a more complicated pattern. In such high energy cases, cycling time 1/E from exp(iEt) is small, so the idea of a density at a point x at a time t with a small delta t seems to make sense. Actual results of the intersection of 1/v(x) and the quantum density are not actually at the peak, but the peak seems to be the approximating the match. (3) For low quantum energies, the cycling time 1/En is long compared to times required to traverse the system and fluctuation flux mixes with collective motion. For the quantum oscillator this leads to a Gaussian. We also examine the case of the classical statistical oscillator and argue that the fluctuation flux (the thermal kinetic energy) is mixed with collective kinetic energy which leads to higher interparticle spacing for places where the potential is higher

Classical Frequencies in Quantum Mechanics?

In this brief note, we investigate the role of classical frequencies in quantum mechanical problems. It is known through the correspondence principle that quantum mechanics matches classical physics for high energy levels. It has been shown in previous notes, however, that classical features are already present in all quantum energy levels as the Schrodinger equation may be written as: (-1/2m) [d/dx d/dx W(x)] / W(x) = .5m v(x)v(x) = E-V(x) , where W(x) is the wavefunction. Thus, if there is periodic behaviour associated with v(x), as in an oscillator, it should carry through into the quantum average kinetic energy and potential energy and finally into E, the average energy. This also suggests that the fp’s (W(x)= Sum over p fp exp(ipx)) are influenced by the classical frequency w

Linear low density polyethylene (LLDPE) films containing perylene dyes as stress-strain luminescent indicators

Linear low density polyethylene (LLDPE) films containing different concentrations of perylene (P) chromophores have been prepared by solution-casting and compression-moulding. The light emission features of the films depend on P concentration and polymer films deformation. A well-defined band attributed to the formation of micro/ nano-structured perylene chromophoric aggregates is observed with more than 0.1 wt.% of P in the film. The occurrence of this phenomenon changed the emission colour of the films from a bright blue (non- interacting dyes) to low intense pale yellow (interacting dyes) colour. During film drawing the LLDPE macromolecules reorganization is able to break the P supramolecular organization, leading to the prevalence of the blue emission of monomeric P. The optical behaviour of perylene dyes acting as internal stress- strain luminescent indicators for polyethylene films, provides a powerful tool to detect macromolecular organization

New Insights Into Amyloid Formation and Structure by Innovative Atomic Force Microscopy Methods

Today, more than 40 million people worldwide are affected by neurodegenerative disorders. Onset of these diseases is associated with insoluble fibrillar protein aggregates, termed amyloids. The molecular origin and the link between amyloid formation and disease aetiology are unclear and there are not are available therapies for these disorders. Strong evidence links propensity of proteins to misfolding and aggregation to the pathological biology implicated in the onset of these diseases. Despite its importance, unraveling amyloids properties and formation is still a formidable experimental challenge, mainly because of their nanoscale dimensions and their heterogeneous and transient nature. Therefore, the investigation of the misfolding of monomers and oligomers into fibrils and their mechanical and structural properties is central to understand their stability and toxicity in the body and to design new therapeutic strategies to the amyloid diseases problem. The main objective of this PhD thesis was the biophysical investigation of amyloids structure and formation at the single scale aggregate. This objective was pursed mainly by the use of both conventional and innovative Atomic Force Microscopy (AFM) methodologies, such as peak force quantitative nanomechanical mapping (PF-QNM) and infrared nanospectroscopy (nanoIR). These methods were assessed to resolve the complex and heterogeneous energy landscape of proteins aggregation and to provide direct information on aggregates properties. Initially, we used AFM to compare the kinetics of aggregation of wild type and mutated forms of huntingtin and a-synuclein. In the first case, we focused on the effects of N-terminal post-translation modifications in the fibrillization. We demonstrated that a phosphorylation of the protein significantly slowed down the aggregation. In the latter case, we investigated wild type and H50Q mutated form of a-synuclein. We demonstrated the strict link between the disease and the mutation, which enhanced aggregation. Successively, we studied the early stages of a-synuclein fibrillization and we showed that the amyloid assembly proceed directly through the formation of single monomeric strands, which hierarchically assembly into the mature fibrils. Moreover, we performed force spectroscopy experiments, which confirmed the non-mature structure of this species and enabled studying their force of interaction with surfaces. Successively, PF-QNM was applied to investigate the mechanical properties of single aggregates forming during fibrillization of amyloid proteins. We demonstrated that β-sheet content is a major factor determining their intrinsic stiffness. Finally, nanoIR was applied to investigate at the nanoscale the misfolding process and the structure of the species present during the aggregation. The technique proved to be ideal to characterize individually the amyloid species formed by the Josephin domain of ataxin-3. For the first time, we were able to link their nanomechanical properties and secondary structure at the nanoscale. Innovative AFM-based techniques enabled correlating morphological and ultrastructural properties of amyloids at the single aggregate scale. Thus, they represent a future fruitful avenue to unravel protein misfolding process and the mechanisms of amyloid formation. The comprehension of these fundamental processes could allow the design of pharmacological approaches to contrast the onset of amyloid diseases

Nanoscale spatially resolved infrared spectra from single microdroplets

Droplet microfluidics has emerged as a powerful platform allowing a large number of individual reactions to be carried out in spatially distinct microcompartments. Due to their small size, however, the spectroscopic characterisation of species encapsulated in such systems remains challenging. In this paper, we demonstrate the acquisition of infrared spectra from single microdroplets containing aggregation-prone proteins. To this effect, droplets are generated in a microfluidic flow-focussing device and subsequently deposited in a square array onto a ZnSe prism using a micro stamp. After drying, the solutes present in the droplets are illuminated locally by an infrared laser through the prism, and their thermal expansion upon absorption of infrared radiation is measured with an atomic force microscopy tip, granting nanoscale resolution. Using this approach, we resolve structural differences in the amide bands of the spectra of monomeric and aggregated lysozyme from single microdroplets with picolitre volume.Comment: 5 pages, 3 Figure

Aggregation-Induced Emission of Tetraphenylethylene in Styrene-Based Polymers

In the present work, the preparation of different styrene-based polymer films containing small amounts of TPE and the evaluation of their photoluminescent behaviour is reported. When TPE is dispersed in a poor solvent or in a glassy PS matrix, the arrested intramolecular rotations of its aryls favour the strong emission of light centred at about 455-460 nm. Conversely, TPE fluorescence significantly weakens to a faint signal when good solvents or viscous but not glassy polymer matrices are used. Near-field optical microscopy correlates the fluorescence behaviour with the different matrix morphologies. These results should be able to be used for developing a new tool for polymer traceability