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About eigen-problem of single photon Hamiltonian

Nenad V. Delić, Svetlana Pelemiš, Jovan P. Šetrajčić

Published May 1, 2008• 1 citation
Publisher

Abstract

Altered state of consciousness is a consequence of quantum change which is presented by photons. There is not exist adequate theory, to describe this phenomenon. Linearized single photon Hamiltonian is used for the analysis of its features in coordinate systems. It can be concluded that photon behavior is equally determined by its translation characteristics (impulse components) and its rotation characteristics expressed by way of spin operators.

DOI
10.1109/icmel.2008.4559240

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Altered state of consciousness is a consequence of quantum change which is presented by photons. There is not exist adequate theory, to describe this phenomenon. Linearized single photon Hamiltonian is used for the analysis of its features in coordinate systems. It can be concluded that photon behavior is equally determined by its translation characteristics (impulse components) and its rotation characteristics expressed by way of spin operators.
1. The analysis of single photon behaving in coordinate systems of various geometries has shown the following: · In Cartesian coordinate system the components of single photon eigen-vector are progressive and regressive plane waves. In other words, the photon behaving is characterized by coordinates and momentum, only. · In cylindrical coordinate system the components of single photon eigen-vector of are progressive and regressive plane waves, but only along z axis. In comparison to the results of analysis in Cartesian coordinates the fact that in x y planes photon oscillations are attenuated according to ­1/2 law represents a generalization. This conclusion has been based on asymptotic behaving of Bessel functions. · The most authentic result is obtained by the analysis in spherical coordinates. Photon states do not depend on angles, and they are damped according to r­1 law, where r is radius of the sphere. It has also turned out that orbital momentum of a free photon is equal to zero (this is understandable considering the fact that it moves along a straight line). The important element determining the behavior of a free photon is spin. One components of photon eigen-vector corresponds to spin Sz = 1/2 projection, while the other component corresponds to spin Sz = ­1/2 projection. This was concluded on the basis of the fact that its eigen-vector components are Bessel functions having indices +1/2 and ­1/2. The last result shows that linearization of photon Hamiltonian gives more complete picture of single photon than Kline-Gordon's approach. Concluding the exposed analysis we shall try to connect the results obtained in series of experimental investigation of photon orbital momentum (Beth, 1936; Leach, et al., 2002; Allen, et al., 1992; Allen, 1966; He, et al., 1995; Friese, et al., 1996; Markoski, et al., 2008; van Enk & Nienhuis, 2007; Santamato, et al., 1988; O'Neil, et al., 2002; Volke-Sepulveda, et al., 2002). We shall not describe all quoted experiments. Instead of it we shall describe the essential idea: the orbital momentum of photon was determines from the changes of
Abstract A system of 4 point masses (without spin) is considered. The potential energy is assumed to be a function of the distances between the particles only. It is possible to express the kinetic energy relative to the center of mass as the kinetic energy of 3 "reduced" masses. The different possibilities for that procedure are discussed for systems with an arbitrary number of particles. The positions of the 3 reduced masses are described by 9 coordinates. 6 of them, the "internal" coordinates, are needed to determine the internal structure of the system, and 3, the "external" coordinates, for its orientation in space. The classical Hamiltonian in such a set of coordinates is derived. In a first step the Hamiltonian is expressed in terms of linear and angular momenta. Then these quantities are determined as functions of the coordinates and the conjugate momenta by infinitesimal trans-formations. From the classical Hamiltonian we get the quantum mechanical one by the well-known translation rule. It can be expressed in terms of the linear and angular momentum operators quite analogous to the classical case. For the solution of the time-independent Schrödinger equation the rotational symmetry of the system is used. The wave function is expanded in terms of the eigenfunctions of the total angular momentum. This leads to a system of coupled differential equations for the expansion-coefficients which depend on the 6 internal coordinates only. The further symmetries of the system (reflection and particle exchange) appear in symmetry properties of these functions. Differences and similarities in the solution theory for the 4-body-problem with respect to the 3-body-problem are discussed.
When light scatters from an object, it can impart some physical quantity such as momentum or angular momentum. This can act as a measurement on the photon, which collapses on to an eigenstate of the measurement operator. However the corresponding operator is not the same as that describing the total linear or angular momentum in free space. Optical eigenmodes provide a powerful method to describe this interaction by expanding the field as a linear combination of some basis modes and examining the eigenvalues and eigenvectors of the quadratic measure in question. We extend this to the quantum case by writing the quantum operator corresponding to a given measurement such as energy, momentum or angular momentum as a superposition of creation and annihilation operators for each eigenmode. Upon measurement we find that the possible states of a single photon are simply the classical eigenmodes of the measurement. As an application, we examine the force and torque on a general, possibly anisotropic, material. By looking at eigenvalues of the measurement operator we show that the amount of a given quantity transferred in an interaction with matter is not in general the expected amount which a photon carries in free space, even at the single photon level. In particular the difference in linear or angular momentum from before and after is in general not equal to ~k or ~ which are the eigenvalues of these quantities in free space.
For a linear system, the response to a stimulus is often superposed by its responses to other decomposed stimuli. In quantum mechanics, a state is the superposition of multiple eigenstates. Here, by taking advantage of the phase difference, a common feature as we identified in data sets, we propose eigen component analysis (ECA), an interpretable linear learning model that incorporates the principle of quantum mechanics into the design of algorithm design for feature extraction, classification, dictionary and deep learning, and adversarial generation, etc. The simulation of ECA, possessing a measurable class-labelclass\text{-}labelclass-label H\mathcal{H}H, on a classical computer outperforms the existing classical linear models. Eigen component analysis network (ECAN), a network of concatenated ECA models, enhances ECA and gains the potential to be not only integrated with nonlinear models, but also an interface for deep neural networks to implement on a quantum computer, by analogizing a data set as recordings of quantum states. Therefore, ECA and ECAN promise to expand the feasibility of linear learning models, by adopting the strategy of quantum machine learning to replace heavy nonlinear models with succinct linear operations in tackling complexity.