Dr. Sameen
Ahmed
KHAN
(Picture) Assistant Professor, Department of Mathematics and Sciences College of Arts and Applied Sciences (CAAS) Dhofar University (Logo) Post Box No. 2509, Postal Code 211 Salalah Dhofar Sultanate of Oman (National Emblem). (Progress Report). 
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[email protected] EMail (Logo): EMail (Logo): EMail (Logo): Fax: +968 Phone: +968, Extension 7297 (Office) Phone: +96823295XXX (Home) GSM: +9689953XXXX (An eight digit prime number from Oman Mobile, Logo) List of 38+ Articles from the Scopus: http://www.SCOPUS.com/authid/detail.url?authorId=8452157800 http://SameenAhmedKhan.webs.com/ http://www.imsc.res.in/~jagan/khancv.html http://sites.google.com/site/rohelakhan/ http://rohelakhan.webs.com/ http://www.du.edu.om/ 
Sameen Ahmed Khan
(See the Picture and the
Visiting Card). S/O Late Mr. Hamid Ahmed KHAN (Picture) and Late Mrs. Nighat Iqbal Khan (Picture) INDIA (National Emblem). 
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Teaching
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The quantum theory of chargedparticle beam dynamics is being developed essentially using an algebraic approach. On the basis of this theory, optics of the transport of nonrelativistic and relativistic chargedparticle beams through electromagnetic systems (of importance for chargedparticle beam devices, like electron microscopy, microelectronbeam lithography, etc., and accelerator design) is being analyzed systematically. The machinery of Lie algebraic methods is used primarily and this facilitates an easy passage from the quantum theory to the traditional classical theory (geometrical optics). Our results include the modifications of the paraxial properties and the aberration coefficients, with hbardependent contributions, for the various optical elements, like the magnetic round lenses, quadrupoles, etc., using the Schrödinger (nonrelativistic), KleinGordon and Dirac equations. For charged spinhalf particles, the Dirac equation leads to spinor contributions to the beam dynamics. We do hope that these quantum corrections, albeit small, would be of some practical significance in certain situations; it should, however, be emphasized that, in any case, it is certainly satisfying to understand the working of the traditional classical theory as an approximation of a proper quantum theory since after all any physical system is quantum mechanical at the fundamental level.
The application of the Wigner phasespace distribution for studying the quantum mechanics of chargedparticle beam transport through electromagnetic optical systems provides a natural link between the classical and the quantum descriptions. In this context, the relation between the transformation of the Wigner function of a chargedparticle optical system, corresponding to the associated scalar wave function, and the transformation of the classical phasespace of the system has been studied.
The application of the spinor beam optical formalism has been shown to lead to a fully quantum mechanical understanding of the dynamics of a spinhalf particle with anomalous magnetic moment, including the spin evolution, at the level of singleparticle dynamics. The general theory, developed for any magnetic optical element with straight axis, describes the the quantum mechanics of the orbital dynamics, the SternGerlach kicks and the ThomasBargmannMichelTelegdi (ThomasBMT) spin evolution.
In the paraxial régime of 3dim optics, two evolution
Hamiltonians are equivalent when one can be transformed to the other
modulo scale by similarity through an optical system. To determine the
equivalence sets of paraxial optical Hamiltonians one requires the orbit
analysis of the algebra sp(4,R) of 4×4 real Hamiltonian matrices.
Our strategy uses instead the isomorphic algebra so(3,2) of 5×5
matrices with metric (+1,+1,+1,−1,−1) to find 4 orbit
regions (strata), 6 isolated orbits at their boundaries, and 6
degenerate orbits at their common point. We thus resolve the
degeneracies of the eigenvalue classification.
Portions of my work would be concerned with the applications of the above formalism and related ideas to various problems such as developing a complete quantum mechanical treatment of high energy polarized beams of Dirac particles (electrons, protons, muons, ...), including polarization, radiation effects etc., studying the quantum mechanics of beam optical aberrations relevant for electron microscopy (from low voltage to high voltage regions) and microelectronbeam device technology, ..., etc.
Using the analogy of the Helmholtz equation with the KleinGordon equation and the PauliVillars approach to the KleinGordon equation a a formalism utilizing the powerful techniques of quantum mechanics has been developed for scalar optics including aberrations. This provides an alternative to the traditional squareroot approach and gives rise to wavelengthdependent contributions modifying the aberration coefficients.
Diraclike form of the Maxwell equations is well known in literature. Starting with the Diraclike form of the Maxwell equations we build a formalism which provides a unified treatment of beam optics and polarization. The traditional results (including aberrations) of the scalar optics are modified by the wavelengthdependent contributions. Some of the wellknown results in polarization studies are realized as the leadingorder limit of a more general framework of our formalism.
We are also studying the Beam Halo Problem and building a diffractionbased model for the beam losses. In the proposed model we use the powerful machinery of the Quantumlike approaches.
I am also currently trying to analyze the bulk characteristics of the
beams using the powerful techniques of Statistical Mechanics.
Any physical system is quantum mechanical at the fundamental level. So, the proposed research would lead, first of all, to a better understanding of the quantum physics of beam dynamics. Besides this, of course, the results should lead to some insight into the solutions of some of the practical problems of beam dynamics; in the polarization analysis, for example. One immediate result shall be the generalization of the beamoptical form of the ThomasBMT equation to all orders. In our earlier paper the leading order approximation leads to the paraxial beamoptical form of the ThomasBMT equation.
The preliminary results of the proposed halo model are encouraging and further work is in progress.
This will enable us to arrive at the bulk characteristics of the beams
using a microscopic theory.
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List of 35+ Articles from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo).  List of 20+ Articles from the LANL EPrint archive (see the Atom Feeds). 
Some Publications from the American Mathematical Society (AMS, Logo) MathSciNet (Logo).  List of 250+ Articles from the Google Scholar (Logo). 
http://www.research.att.com/~njas/sequences/
A geometricarithmetic progression of primes is a set of k primes
(denoted by GAPk) of the form p_{1}*r ^{j} + j*d
for fixed p_{1}, r and d and consecutive j,
from j = 0 to k  1.
i.e, {p_{1}, p_{1}*r + d, p_{1}*r ^{2} + 2 d,
p_{1}* r ^{3} + 3 d, ...}.
For example 3, 17, 79 is a 3term geometricarithmetic progression
(i.e, a GAP3) with a = p_{1} = 3, r = 5 and d = 2.
A GAPk is said to be minimal if the minimal start p_{1} and
the minimal ratio r are equal, i.e, p_{1} = r = p, where p
is the smallest prime ≥ k.
Such GAPs have the form p*p ^{j} + j*d.
Minimal GAPs with different differences, d do exist. For example, the minimal GAP5
(p_{1} = r = 5) has the
possible differences, 84, 114, 138, 168, ... (see the Sequence A209204)
and the minimal
GAP6 (p_{1} = r = 7) has the possible differences,
144, 1494, 1740, 2040, .... (see the Sequence A209205).
The following article gives the conditions under which, a GAPk is a
set of k primes in geometricarithmetic progression.
Sameen Ahmed Khan,
Primes in GeometricArithmetic Progression,
19 pages,
LANL
EPrint Archive:
http://arxiv.org/abs/1203.2083/.
Bibliographic Code:
2012arXiv1203.2083K
(Friday the 09 March 2012).
The minimal possible difference in an APk is conjectured to be k# for all k > 7.
The exceptional cases (for k < = 7) are k = 2, k = 3, k = 5 and k = 7.
For k = 2, we have d = 1 and there is ONLY one AP2 with this difference: {2, 3}.
For k = 3, we have d = 2 and there is ONLY one AP3 with this difference: {3, 5, 7}.
For k = 4, we have d = 4# = 6 and AP4 is {5, 11, 17, 23} and is not unique.
The first primes is the Sequence A023271:
5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, ...
For k = 5, we have d = 3# = 6 and there is ONLY one AP5 with this difference: {5, 11, 17, 23, 29}.
For k = 6, we have d = 6# = 30 and AP6 is {7, 37, 67, 97, 127, 157} and is not unique.
The first primes is the Sequence A156204:
7, 107, 359, 541, 2221, 6673, 7457, 10103, 25643, 26861, 27337, 35051, 56149, ...
For k = 7, we have d = 5*5# = 150 and there is ONLY one AP7 with this difference:
{7, 157, 307, 457, 607, 757, 907}.
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List of 35+ Articles from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo).  List of 20+ Articles from the LANL EPrint archive (see the Atom Feeds). 
Some Publications from the American Mathematical Society (AMS, Logo) MathSciNet (Logo).  List of 250+ Articles from the Google Scholar (Logo). 
This page was born in March 1996 at the The Institute of Mathematical Sciences (MATSCIENCE/IMSc, Logo), Chennai (Madras) INDIA (National Emblem). 
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