arXiv subjects

Sameen Ahmed Khan

Publications and source records attributed to Sameen Ahmed Khan.

At least 19 recordsLinked to original sources

arXiv2024

A new matrix representation of the Maxwell equations based on the Riemann-Silberstein-Weber vector for a linear inhomogeneous medium

We derive a new eight dimensional matrix representation of the Maxwell equations for a linear homogeneous medium and extend it to the case of a linear inhomogneous medium. This derivation starts ab initio with the Maxwell equations and uses arguments based on the algebra of the Pauli matrices. This process leads automatically to the matrix representation based on the Riemann-Silberstein-Weber (RSW) vector. The new representation for the homogeneous medium is a direct sum of four Pauli matrix blocks. This aspect of the new representation should make it suitable for studying the propagation of electromagnetic waves in a linear inhomogeneous medium adopting the techniques of quantum mechanics treating the inhomogeneity as a perturbation. The new representation is used to rederive the Mukunda-Simon-Sudarshan matrix substitution rule for transition from the Helmholtz scalar wave optics to the Maxwell vector wave optics.

math-ph↗

arXiv2021

Quantum mechanics of round magnetic electron lenses with Glaser and power law models of $B(z)$

Scalar theory of quantum electron beam optics, at the single-particle level, derived from the Dirac equation using a Foldy-Wouthuysen-like transformation technique is considered. Round magnetic electron lenses with Glaser and power law models for the axial magnetic field $B(z)$ are studied. Paraxial quantum propagator for the Glaser model lens is obtained in terms of the well known fundamental solutions of its paraxial equation of motion. In the case of lenses with the power law model for $B(z)$ the well known fundamental solutions of the paraxial equations, obtained by solving the differential equation, are constructed using the Peano-Baker series also. Quantum mechanics of aberrations is discussed briefly. Role of quantum uncertainties in aberrations, and in the nonlinear part of the equations of motion for a nonparaxial beam, is pointed out. The main purpose of this article is to understand the quantum mechanics of electron beam optics though the influence of quantum effects on the optics of present-day electron beam devices might be negligible.

quant-ph↗

arXiv2020

On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential

The Tsallis $q$-exponential function $e_q(x) = (1+(1-q)x)^{\frac{1}{1-q}}$ is found to be associated with the deformed oscillator defined by the relations $\left[N,a^\dagger\right] = a^\dagger$, $[N,a] = -a$, and $\left[a,a^\dagger\right] = ϕ_T(N+1)-ϕ_T(N)$, with $ϕ_T(N) = N/(1+(q-1)(N-1))$. In a Bargmann-like representation of this deformed oscillator the annihilation operator $a$ corresponds to a deformed derivative with the Tsallis $q$-exponential functions as its eigenfunctions, and the Tsallis $q$-exponential functions become the coherent states of the deformed oscillator. When $q = 2$ these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when $q = 1$ this deformed oscillator is a canonical boson oscillator, when $1 < q < 2$ its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when $q \longrightarrow 2$ it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state.

math-ph↗

arXiv2019

Quantum mechanics of bending of a nonrelativistic charged particle beam by a dipole magnet

Quantum mechanics of bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet is studied in the paraxial approximation. The transfer map for the position and momentum components of a particle of the beam between two transverse planes at different points on the curved optic axis of the system is derived starting with the nonrelativistic Schrödinger equation. It is found that the quantum transfer map contains the classical transfer map as the main part and there are tiny quantum correction terms. The negligibly small quantum corrections explain the remarkable success of classical mechanics in charged particle beam optics.

quant-ph↗

arXiv2017

A Statistical Approach to Prime Gaps and Andrica's Conjecture

We examine the prime gaps using a statistical approach. It is first shown that the Andrica's conjecture is true for half or more cases. Using the arguments of averages, it is further shown that Andrica's conjecture is true. We further obtain a precise bound for the Andrica's expression.

math.GM↗

arXiv2013

Coordinate Geometric Generalization of the Spherometer and Cylindrometer

Spherometer is an instrument widely used for measuring the radius of curvature of a spherical surface. Cylindrometer is a modified spherometer, which can measure the radii of both spherical and cylindrical surfaces. Both of these instruments are based on a geometric relation unique to circles and spheres, from Euclidean geometry. A more general understanding is obtained using coordinate geometry. The coordinate geometric approach also enables a generalization of the spherometer and cylindrometer to devices, which can handle aspherical surfaces. Here, we present the newly developed coordinate geometric approach and its applications.

physics.gen-ph↗

arXiv2013

Coordinate Geometric Approach to Spherometer

The spherometer used for measuring radius of curvature of spherical surfaces is explicitly based on a geometric relation unique to circles and spheres. We present an alternate approach using coordinate geometry, which reproduces the well-known result for the spherometer and also leads to a scheme to study aspherical surfaces. We shall also briefly describe some of the modified spherometers.

physics.gen-ph↗

arXiv2012

Primes in Geometric-Arithmetic Progression

A geometric-arithmetic progression of primes is a set of $k$ primes (denoted by GAP-$k$) of the form $p_1 r^j + j d$ for fixed $p_1$, $r$ and $d$ and consecutive $j$, {\it i.e}, $\{p_1, \, p_1 r + d, \, p_1 r^2 + 2 d, \, p_1 r^3 + 3 d, \,...}$. We study the conditions under which, for $k \ge 2$, a GAP-$k$ is a set of $k$ primes in geometric-arithmetic progression. Computational data (along with the MATHEMATICA codes) containing progressions up to GAP-13 is presented. Integer sequences for the sets of differences $d$ corresponding to the GAPs of orders up to 11 are also presented.

math.NT↗

arXiv2010

The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel

The order of the set of equivalent resistances, A(n) of n equal resistors combined in series and in parallel has been traditionally addressed computationally, for n up to 22. For larger n there have been constraints of computer memory. Here, we present an analytical approach using the Farey sequence with Fibonacci numbers as its argument. The approximate formula, A(n) ~ 2.55^n, obtained from the computational data up to n = 22 is consistent with the strict upper bound, A(n) ~ 2.618^n, presented here. It is further shown that the Farey sequence approach, developed for the A(n) is applicable to configurations other than the series and/or parallel, namely the bridge circuits and non-planar circuits. Expressions describing set theoretic relations among the sets A(n) are presented in detail. For completeness, programs to generate the various integer sequences occurring in this study, using the symbolic computer language MATHEMATCA, are also presented.

physics.gen-ph↗

arXiv2002

Wavelength-Dependent Effects in Maxwell Optics

We present a new formalism for light beam optics starting with an exact eight-dimensional matrix representation of the Maxwell equations. The Foldy-Wouthuysen iterative diagonalization technique is employed to obtain a Hamiltonian description for a system with varying refractive index. Besides, reproducing all the traditional quasiparaxial terms, this method leads to additional contributions, which are dependent on the wavelength, in the optical Hamiltonian. This alternate prescription to obtain the aberration expansion is applied to the axially symmetric graded index fiber. This results in the wavelength-dependent modifications of the paraxial behaviour and the aberration coefficients. Furthermore it predicts a wavelength-dependent image rotation. In the low wavelength limit our formalism reproduces the Lie algebraic formalism of optics. The Foldy-Wouthuysen technique employed by us is ideally suited for the Lie algebraic approach to optics. The present study further strengthens the close analogy between the various prescription of light and charged-particle optics. All the associated machinery used in this formalism is described in the text and the accompanying appendices.

physics.optics↗

arXiv2002

Analogies between light optics and charged-particle optics

The close analogy between geometrical optics and the classical theories of charged-particle beam optics have been known for a very long time. In recent years, quantum theories of charged-particle beam optics have been presented with the very expected feature of wavelength-dependent effects. With the current development of non-traditional prescriptions of Helmholtz and Maxwell optics respectively, accompanied with the wavelength-dependent effects, it is seen that the analogy between the two systems persists. A brief account of the various prescriptions and the parallel of the analogies is presented.

physics.optics↗

arXiv2002

An alternate way to obtain the aberration expansion in Helmholtz Optics

Exploiting the similarities between the Helmholtz wave equation and the Klein-Gordon equation, the former is linearized using the Feschbach-Villars procedure used for linearizing the Klein-Gordon equation. Then the Foldy-Wouthuysen iterative diagonalization technique is applied to obtain a Hamiltonian description for a system with varying refractive index. Besides reproducing all the traditional quasiparaxial terms, this method leads to additional terms, which are dependent on the wavelength, in the optical Hamiltonian. This alternate prescription to obtain the aberration expansion is applied to the axially symmetric graded index fiber. This results in the wavelength-dependent modification of the paraxial behaviour and the aberration coefficients. Explicit expression for the modified coefficients of the aberration to third-order are presented. Sixth and eighth order Hamiltonians are derived for this system.

physics.optics↗

arXiv2002

Foldy-Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams

The Foldy-Wouthuysen iterative diagonalization technique is applied to the Helmholtz equation to obtain a Hamiltonian description of the propagation of a monochromatic quasiparaxial light beam through a system in which the refractive index $n(x,y,z)$ varies about a background value $n_0$ such that $|n(x,y,z)-n_0| \ll n_0$. This technique is presented as an alternative to the conventional method of series expansion of the radical. Besides reproducing all the traditional quasiparaxial terms, this method leads to additional terms in the optical Hamiltonian.

physics.optics↗

arXiv2002

Maxwell Optics: II. An Exact Formalism

We present a formalism for light optics starting with the Maxwell equations and casting them into an exact matrix form taking into account the spatial and temporal variations of the permittivity and permeability. This $8 \times 8$ matrix representation is used to construct the optical Hamiltonian. This has a close analogy with the algebraic structure of the Dirac equation, enabling the use of the rich machinery of the Dirac electron theory. We get interesting wavelength-dependent contributions which can not be obtained in any of the traditional approaches.

physics.optics↗

arXiv2002

Maxwell Optics: III. Applications

A new formalism of beam-optics and polarization has been recently presented, based on an exact matrix representation of the Maxwell equations. This is described in Part-I and Part-II. In this Part, we present the application of the above formalism to the specific example of the axially symmetric graded index fiber. This formalism leads to the wavelength-dependent modifications of the six aberrations present in the traditional prescriptions and further gives rise to the remaining three aberrations permitted by the axial symmetry. Besides, it also gives rise to a wavelength-dependent image rotation. The three extra aberrations and the image rotation are not found in any of the traditional approaches.

physics.optics↗

arXiv2002

Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium

Matrix representations of the Maxwell equations are well-known. However, all these representations lack an exactness or/and are given in terms of a {\em pair} of matrix equations. We present a matrix representation of the Maxwell equation in presence of sources in a medium with varying permittivity and permeability. It is shown that such a representation necessarily requires $8 \times 8$ matrices and an explicit representation for them is presented.

physics.optics↗

arXiv2001

The world of synchrotrons

A summary of results on synchrotron radiation is presented along with notes on its properties and applications. Quantum aspects are briefly mentioned. Synchrotron radiation facilities are described briefly with a detailed coverage to the accelerator programmes in India. The relocated and other upcoming synchrotrons are also described in some detail.

physics.acc-ph↗

arXiv2001

Quantum mechanical formalism of particle beam optics

A general procedure for construction of the formalism of quantum beam optics for any particle is reviewed. The quantum formalism of spin-1/2 particle beam optics is presented starting {\em ab initio} with the Dirac equation. As an example of application the case of normal magnetic quadrupole lens is discussed. In the classical limit the quantum formalism leads to the well-known Lie algebraic formalism of classical particle beam optics.

physics.acc-ph↗