Mathematical Methods of Physics

Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953.

Since the first volume of this work came out in Germany in 1924, this book has remained a classic in its field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volumes represent Richard Courant's second and final revision of 1953. It will be of interest to mathematicians and physicists.

In this paper we have tried to deduce the possible origin of particle and evolution of their intrinsic properties through spiral dynamics. We consider some of the observations which include exponential mass function of particles following a sequence when fitted on logarithmic potential spiral, inwardly rotating spiral dynamics in Reaction-Diffusion System, the separation of Electron’s Spin-Charge-Orbit into quasi-particles. The paper brings a picture of particles and their Anti Particles in spiral form and explains how the difference in structure varies their properties. It also explains the effects on particles in Accelerator deduced through spiral dynamics.

A superluminal quantum-vortex model of the electron and the positron is produced from a superluminal double-helix model of the photon during electron-positron pair production. The two oppositely-charged (with Q = ±e sqrt (2/α) = 16.6e) open-helix spin-½ half-photons compose the double-helix photon. These half-photons separate and curl up their separated superluminal single-helical trajectories to form an electrically-charged superluminal closed-helix spin-½ quantum-vortex electron model and a corresponding positron model. The helical radius and the Dirac equation's zitterbewegung angular frequency of the quantum vortex electron and positron models equal the helical radius and zitterbewegung angular frequency of the two spin-½ half-photons, each of energy E = mc^2 , that composed the double-helix photon model of energy E = 2mc^2 from which the electron and positron models were produced. The photon and electron models are also compatible when a photon of energy E > 2mc^2 produces a relativistic electron-positron pair. Implications of the quantum vortex electron model for electron stability are discussed.

A partial differential equation (PDE) is an equation containing an unknown function of two or more variables and its partial derivatives with respect to these variables. The order of a PDE is the order of the highest derivative present. Example. Consider ∂ 2 u ∂x∂y = 2x − y. The is a PDE of order two or a second-order PDE. Here u is the dependent variable while x and y are independnet variables. A solution of a PDE is any function which satisfies the equation identically. A general solution is a solution which contains a number of arbitrary independent functions equal to the order of the equation. A particular solution is one which can be obtained from the general solution by a particular choice of the arbitrary functions. A singular solution is one which can not be obtained from the general solution by a particular choice of the arbitrary functions. A boundary value problem (BVP) involving a PDE seeks a ll solutions of the equation which satisfy conditions known as boundary conditions Theorems relating to the existence and uniqueness of such solutions are called existence and uniqueness theorems. I. LINEAR PDES The general linear PDE of order two in two independent variable has the form Au xx + Bu xy + Cu yy + Du x + Eu y + F u = G (1) where A, B, ..., G may depend on x and y but not on u. A second-order equation with independent variables x and y which does not have the form (1) is called non-linear. If G = 0 identically the equation is called homogeneous while if G = 0 it is called it is called non-homogeneous. Generalizations to higher-order equations are easily made. Because of the nature of the solutions of (1), the equation is often classified as hyperbolic, parabolic or elliptic according to the B 2 − 4AC being greater than, equal to, or less than zero respectively. II. SOME IMPORTANT PDES A. Vibrating String Equation y tt = a 2 y xx This equation is applicable to the small transverse vibrations of a taut, flexible string such as a violin string, initially located at the x-axis and set into motion. The function y(x, t) is the displacement of any point x of the string at time t. The constant is defined as a 2 = τ µ where τ is the constant tension in the string and µ is the constant mass per unit length of the string. It is assumed that no external forces act on the string and that it vibrates only due to its elasticity. The equation can be easily generalized to higher dimensions.

The transform W (z) = z + 1 z where W (z) is a complex variable in the new space and z is the complex variable in the original space is called a Joukowski transform. Since dW dz = 1 − 1 z 2 = (z − 1)(z + 1) z 2 the mapping is conformal except at the points where z = 1 and z = −1, respectively. At these points W = 2 and W = −2, respectively. In fact, solving for z in terms of W we find z = W 2 ± 1 2 (W + 2)(W − 2) so that to any value of W = ±2 there corresponds two values of z. In fact the points W = 2 and W = −2 are called branch points and the transformation may be considered as a one to one mapping of the z-plane onto a Riemann surface, connected crosswise from W = −2 to W = 2. Now letting z = re iθ yields φ = r + 1 r cos θ, and ψ = r − 1 r sin θ which gives φ 2 a 2 + ψ 2 a 2 = 1 where a = r + 1 r and b = r − 1 r. This simply means that the circles r = c with c an arbitrary constant are mapped into ellipses. The unit circle r = 1 maps onto the line segment from W = −2 to W = 2. For every r = 1, the two circles with radii r and 1 r map onto the same ellipse, and the full W-plane corresponds to the exterior (or interior) of the unit circle |z| = 1.

This study is an investigation of heat flow initiated within a hollow infinite cylinder at an initial condition generated from the roots of any type of Bessel's functions. The internal
circumferential surface is kept at zero temperature, where the external circumferential surface
emits heat to the surrounding of zero temperature. The problem is solved by separation of variables using Maple 13 and the solutions are presented as graphical animations simulate changes in the shapes of the isothermal radiant surfaces as a function of time.

The Poincaré map is a method of converting a flow (continuous time) to a map (discrete time). In the simplest case, pick a time T 0 > 0 and define a map S T 0 : R n → R n : x → S T 0 (x) = Φ T 0 (x); this is called the T 0 shift map, and is most useful for non-autonomous systems with time-periodic forcing. Poincaré map II For autonomous systems, it is more useful to consider the case when trajectories repeatedly return to a region of phase space, for example, near a periodic orbit L In this context, the Poincaré map is sometimes called the first return map.

By the end of the nineteenth century theoretical physicists thought that soon they could pack up their bags and go home. They had developed a powerful mathematical theory, classical mechanics, which seemed to described just about all that they observed, with the exception of a few sticking points. In particular the classical world was ruled by Newtonian physics where matter (atoms) interacted with a radiation field (light) as described by Maxwell's equations. However as it turned out these sticking points were not smoothed over at all but rather were glimpses of the microscopic (and relativistic) world which was soon to be experientially discovered. As has been stated countless times, by the end of the first decade of the twentieth century quantum mechanics and relatively had appeared and would soon cause classical mechanics, with its absolute notions of space, time and determinacy to be viewed as an approximation. Historically the first notion of quantized energy came in 1900 with Planck's hypothesis that the energy contained in radiation could only be exchanged in discrete lumps, known as " quanta " , with matter. This in turn implies that the energy, E, of radiation is proportional to its frequency, ν, E = hν (1.1) with h a constant, Planck's constant. This allowed him to derive his famous formula for black body spectra which was in excellent agreement with experiment. From our perspective this is essentially a thermodynamic issue and the derivation is therefore out of the main theme of this course. The derivation of such a formula from first principles was one of the sticking points mentioned above. While Planck's derivation is undoubtedly the first appearance of quantum ideas it was not at all clear at the time that this was a fundamental change, or that there was some underlying classical process which caused the discreteness, or that it was even correct. However there were further experiments which did deeply challenge the continuity of the world and whose resolution relies heavy on Planck's notion of quanta.

In quantum mechanics the Pauli exclusion principle plays a crucial role in the description of nature (not least for the explaination of the Mendelejev's table of elements). This principle connects the symmetry or antisymmetry of the N −particle wave functions of identical particles with the spins, in particular with integer or half-integer spins, respectively. Our pourpose is to discuss the nature of this principle from a QFT point of view, in particular we will prove the Spin-statistics theorem, which is the rigorous statement of the Pauli principle, assuming the Wightman's axioms (to be honest the Wightman axioms for the Wightman's distributions). In order to do that, after a review of theWightaman's axioms, we will discuss first the free case, since this case is somehow special and we don't need many assumptions; then we will discuss the general case.

It is only test preparation practice for GAT test also academic test under K-12 students. these questions mostly asked in Graduate Assessment Test. So i gathered some questions and write there solutions to help other who are preparing for exams.

William Kingdon Clifford is famous for statements that he made in 1870 to the effect that matter is nothing but ripples, hills and bumps of space curved in a higher dimension and the motion of matter is nothing more than variations in that curvature. For having said this Clifford has been both hailed and condemned for having anticipated Einstein’s general theory of relativity. The standard view of historians, scholars and scientists is that his ideas were not tenable at the time, he never developed or wrote down his theory (if he ever had one), and he had no followers or students to carry on his work. Nothing is further from the truth on all three counts. Modern scholars have failed to recognize Clifford’s theory because they have only been looking for Einstein’s gravity theory in the 1870s, which is based on tensors, while tensor calculus was not developed until decades after Clifford’s untimely and
unfortunate early death from consumption in 1879. So they are wrong on both counts. Those scholars who have even bothered to look at the original literature on the subject have been unable to find Clifford’s published but incomplete theory because he was forced to develop his own mathematical system of biquaternions to model motion as it would appear in the higher-dimensional space. Nor was Clifford seeking a
new theory of gravity, but rather a more coherent and intuitive portrayal of his friend Maxwell’s new electromagnetic theory. However, Clifford believed that dynamical forces in three-dimensional space reduced to kinematical motions in four-dimensional space and therefore planned to include a new theory of gravity after he polished off electromagnetism. In other words, Clifford was trying to develop a unified field theory that would more consistently look like modern day unifications rather than just an earlier version of general relativity.

We investigate the formation of dark-state polaritons in an ensemble of degenerate two-level atoms admitting electromagnetically induced transparency. Using a generalization of microscopic equation-of-motion technique, multiple collective polariton modes are identified depending on the polarizations of two coupling fields. For each mode, the polariton dispersion relation and composition are obtained in a closed form out of a matrix eigenvalue problem for arbitrary control field strengths. We illustrate the algorithm by considering the F g = 2 → F e = 1 transition of the D 1 line in 87 Rb atomic vapor. In addition, an application of dark-state polaritons to the frequency and/or polarization conversion, using D 1 and D 2 transitions in cold Rb atoms, is given.

We begin with a lightening review of the relevant concepts of special relativity. The basic postulate of relativity is that the laws of physics are the same in all inertial reference frames. The theory of special relativity tells us how things look to observers who are moving relative to each other. Consider the first observer who sits in an inertial frame S with space-time coordinates (ct, x, y, z) while the second observer sits in an inertial frame S with space-time coordinates (ct , x , y , z). If S is taken to be moving with speed v in the x-direction relative to S then the coordinate systems are related by the Lorentz boost x = γ x − v c ct andct = γ ct − v c x while y = y and z = z with c being the speed of light and γ being the ubiquitous factor γ = 1 1 − v 2 /c 2. A. Four-Vectors To make things easier and useful, the space-time coordinates can be packaged in 4-vectors with indices running from µ = 0 to µ = 3 X µ = (ct, x, y, z), µ = 0, 1, 2, 3. Note that the index is a superscript rather than a subscript. This will bear great importance in the discussion which follows. A general Lorentz transformation is a linear map from X to X of the form (X) µ = Λ µ ν X ν. Here Λ is a 4 × 4 which obeys the matrix equation Λ T ηΛ = η ⇐⇒ Λ ρ µ η ρσ Λ σ ν = η µν where η µν = diag(+1, −1, −1, −1). The solutions to Λ T ηΛ = η ⇐⇒ Λ ρ µ η ρσ Λ σ ν = η µν fall into two classes. The first class is simply rotations. Given a 3×3 rotation matrix R obeying R T R = 1, we can construct a Lorentz transformation Λ by embedding R in the spatial part    1 0 0 0 0 0 R 0    These transformations describe how to relate the coordinates of two observers who are rotated with respect to each other. The second class is that of Lorentz boosts. These are transformations appropriate for observers moving relative to each other. The Lorentz transformation x = γ x − v c ct andct = γ ct − v c x

Notion of the stationary quantum state is recalled. The classical way of finding stationary states for the Schrödinger equation is analysed. Attention is paid to the difference between the notion of physical stationary state and mathematical notion of function with separated variables. The mathematical basis of some general method for solving the Dirac equation is presented. The problem of finding stationary solutions for the case of the Dirac equation is analysed. A conclusion appears that there are no stationary states for the Dirac equation in a homogeneous electric field. Results of our paper [1] are confronted with some papers of other authors dealing with the problem of a homogeneous or constant electric field.

Clebsch parameterization: Basic properties and remarks on its applications J. Math. Phys. 50, 113101 (2009) Studies of perturbed three vortex dynamics J. Math. Phys. 48, 065402 (2007) Point vortex motion on the surface of a sphere with impenetrable boundaries Phys. Fluids 18, 036602 (2006) Schouten tensor and bi-Hamiltonian systems of hydrodynamic type J. Math. Phys. 47, 023504 (2006) Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors A differential-algebraic approach to studying the Lax integrability of the generalized Riemann type hydrodynamic hierarchy is revisited and its new Lax representation is constructed in exact form. The bi-Hamiltonian integrability of the generalized Riemann type hierarchy is discussed by means of the gradient-holonomic and symplectic methods and the related compatible Poissonian structures for N = 3 and N = 4 are constructed. C

Ever since Werner Heisenberg's 1927 paper on uncertainty, there has been considerable hesitancy in simultaneously considering positions and momenta in quantum contexts, since these are incompatible observables. But this persistent discomfort with addressing positions and momenta jointly in the quantum world is not really warranted, as was first fully appreciated by Hilbrand Groenewold and José Moyal in the 1940s. While the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely understood nor widely known-much less generally accepted-until the late 20th century. R. Feynman When Feynman first unlocked the secrets of the path integral formalism and presented them to the world, he was publicly rebuked [8]. "It was obvious," Bohr said, "that such trajectories violated the uncertainty principle." However, in this case, 1 Bohr was wrong. Today path integrals are universally recognized and widely used as an alternative framework to describe quantum behavior, equivalent to-although conceptually distinct from-the usual Hilbert space framework, and therefore completely in accord with Heisenberg's uncertainty principle. The different points of view offered by the Hilbert space and path integral frameworks combine to provide greater insight and depth of understanding. N. Bohr Similarly, many physicists hold the conviction that classical-valued position and momentum variables should not be simultaneously employed in any meaningful formula expressing quantum behavior, simply because this would also seem to violate the uncertainty principle (see Appendix Dirac). However, they are also wrong. Quantum mechanics (QM) can be consistently and autonomously formulated in phase space,

A model of a thin straight strip with a uniformly curved section and with boundary requirements zeroing at the edges a linear superposition of the wave function and its normal derivative ͑Robin boundary condition͒ is analyzed theoretically within the framework of the linear Schrödinger equation and is applied to the study of the processes in the bent magnetic multilayers, superconducting films and metallic ferrite-filled waveguides. In particular, subband thresholds of the straight and curved parts of the film are calculated and analyzed as a function of the Robin parameter 1 / ⌳, with ⌳ being an extrapolation length entering Robin boundary condition. For the arbitrary Robin coefficients which are equal on the opposite interfaces of the strip and for all bend parameters the lowest-mode energy of the continuously curved duct is always smaller than its straight counterpart. Accordingly, the bound state below the fundamental propagation threshold of the straight arms always exists as a result of the bend. In terms of the superconductivity language it means an increased critical temperature of the curved film compared to its straight counterpart. Localized-level dependence on the parameters of the curve is investigated with its energy decreasing with increasing bend angle and decreasing bend radius. Conditions of the bound-state existence for the different Robin parameters on the opposite edges are analyzed too; in particular, it is shown that the bound state below the first transverse threshold of the straight arm always exists if the inner extrapolation length is not larger than the outer one. In the opposite case there is a range of the bend parameters where the curved film cannot trap the wave and form the localized mode; for example, for the fixed bend radius the bound state emerges from the continuum at some nonzero bend angle that depends on the difference of the two lengths ⌳ at the opposite interfaces. Various transport properties of the film such as interference blockade of the current flow at some special energies is also discussed with the special attention being paid to the transformation from the Dirichlet to the Neumann case as the extrapolation length ⌳ sweeps the positive axis.

Constitutive criteria for the existence of secondary flows, similarities and analogies, developments in the history of transversal flows, recent research on secondary flows of dilute solutions in rotating pipes and channels and related drag reduction are reviewed in depth as well as research concerning fundamental aspects of transversal flows and industrial applications relevant to secondary flows.

The principles to use variables and mathematical methodologies in physics are addressed. A set of refined definitions with designated variables are used to derive the Velocity and Acceleration Theories in Distance Field and Vector Space. Mathematical methodologies such as Linear Algebra and Vector Calculus are used systematically in a step by step derivation process. The proof of the theories can be easily achieved by substitution of the designated variables with a set of parameters that matches the same assumptions and conditions in every step of the derivation process.

The main intension of this paper is to extract new and further general analytical wave solutions to the (2 þ 1)dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation in the sense of conformable derivative by implementing the advanced exp ðÀϕ ðξÞÞ-expansion method. This method is a particular invention of the generalized exp ðÀϕ ðξÞÞ-expansion method. By the virtue of the advanced exp ðÀϕ ðξÞÞ-expansion method, a series of kink, singular kink, soliton, combined soliton, and periodic wave solutions are constructed to our preferred space time-fractional (2 þ 1)-dimensional AKNS equation. An extensive class of new exact traveling wave solutions are transpired in terms of, hyperbolic, trigonometric, and rational functions. To express the underlying propagated features, some attained solutions are exhibited by making their three-dimensional (3D), twodimensional (2D) combined, and 2D line plot with the help of computational packages MATLAB. All plots are given to show the proper wave features through the founded solutions to the studied equation with particular preferring of the selected parameters. Moreover, it may conclude that the attained solutions and their physical features might be helpful to comprehend the water wave propagation in water wave mechanics.

Abstract: We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable $ z $(spectral parameter) and the other a recurrence relation in $ n $(the lattice variable). For the Jacobi weight

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In this paper we reproduce the continuum model of electro-osmotic oscillations at a non-charged porous membrane and study their onset with a focus on the singular nature of this transmission (singular Hopf bifurcation), resulting in a rapid transformation of the oscillations, harmonic at the onset, into the jerky ones of relaxation type.