Dr. Sameen
Ahmed
KHAN
(Picture) Associate 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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rohelakhan@yahoo.com E-Mail (Logo): E-Mail (Logo): E-Mail (Logo): Fax: +968- Phone: +968-23237227, Extension 7227 (Office) Phone: +968-23296XXX (Home) GSM: +968-9953XXXX (An eight digit prime number from Oman Mobile, Logo) List of 100+ Articles from the Scopus: http://www.SCOPUS.com/authid/detail.url?authorId=8452157800 https://orcid.org/0000-0003-1264-2302 http://SameenAhmedKhan.webs.com/ http://www.imsc.res.in/~jagan/khan-cv.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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(See the
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The quantum theory of charged-particle beam dynamics is being developed essentially using an algebraic approach. On the basis of this theory, optics of the transport of nonrelativistic and relativistic charged-particle beams through electromagnetic systems (of importance for charged-particle beam devices, like electron microscopy, microelectron-beam 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 hbar-dependent contributions, for the various optical elements, like the magnetic round lenses, quadrupoles, etc., using the Schrödinger (nonrelativistic), Klein-Gordon and Dirac equations. For charged spin-half 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 phase-space distribution for studying the quantum mechanics of charged-particle 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 charged-particle optical system, corresponding to the associated scalar wave function, and the transformation of the classical phase-space 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 spin-half particle with anomalous magnetic moment, including the spin evolution, at the level of single-particle dynamics. The general theory, developed for any magnetic optical element with straight axis, describes the the quantum mechanics of the orbital dynamics, the Stern-Gerlach kicks and the Thomas-Bargmann-Michel-Telegdi (Thomas-BMT) spin evolution.
In the paraxial régime of 3-dim 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 microelectron-beam device technology, ..., etc.
Using the analogy of the Helmholtz equation with the Klein-Gordon equation and the Pauli-Villars approach to the Klein-Gordon 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 square-root approach and gives rise to wavelength-dependent contributions modifying the aberration coefficients.
Dirac-like form of the Maxwell equations is well known in literature. Starting with the Dirac-like 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 wavelength-dependent contributions. Some of the well-known results in polarization studies are realized as the leading-order limit of a more general framework of our formalism.
We are also studying the Beam Halo Problem and building a diffraction-based model for the beam losses. In the proposed model we use the powerful machinery of the Quantum-like 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 beam-optical form of the Thomas-BMT equation to all orders. In our earlier paper the leading order approximation leads to the paraxial beam-optical form of the Thomas-BMT 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 43+ Articles from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo). | List of 24+ Articles from the LANL E-Print archive (see the Atom Feeds). |
Some Publications from the American Mathematical Society (AMS, Logo) MathSciNet (Logo). | List of 300+ Articles from the Google Scholar (Logo). |
http://www.research.att.com/~njas/sequences/
A geometric-arithmetic progression of primes is a set of k primes
(denoted by GAP-k) of the form p1*r j + j*d
for fixed p1, r and d and consecutive j,
from j = 0 to k - 1.
i.e, {p1, p1*r + d, p1*r 2 + 2 d,
p1* r 3 + 3 d, ...}.
For example 3, 17, 79 is a 3-term geometric-arithmetic progression
(i.e, a GAP-3) with a = p1 = 3, r = 5 and d = 2.
A GAP-k is said to be minimal if the minimal start p1 and
the minimal ratio r are equal, i.e, p1 = 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 GAP-5
(p1 = r = 5) has the
possible differences, 84, 114, 138, 168, ... (see the Sequence A209204)
and the minimal
GAP-6 (p1 = r = 7) has the possible differences,
144, 1494, 1740, 2040, .... (see the Sequence A209205).
The following article gives the conditions under which, a GAP-k is a
set of k primes in geometric-arithmetic progression.
Sameen Ahmed Khan,
Primes in Geometric-Arithmetic Progression,
19 pages,
LANL
E-Print Archive:
http://arxiv.org/abs/1203.2083/.
Bibliographic Code:
2012arXiv1203.2083K
(Friday the 09 March 2012).
The minimal possible difference in an AP-k 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 AP-2 with this difference: {2, 3}.
For k = 3, we have d = 2 and there is ONLY one AP-3 with this difference: {3, 5, 7}.
For k = 4, we have d = 4# = 6 and AP-4 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 AP-5 with this difference: {5, 11, 17, 23, 29}.
For k = 6, we have d = 6# = 30 and AP-6 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 AP-7 with this difference:
{7, 157, 307, 457, 607, 757, 907}.
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List of 43+ Articles from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo). | List of 24+ Articles from the LANL E-Print archive (see the Atom Feeds). |
Some Publications from the American Mathematical Society (AMS, Logo) MathSciNet (Logo). | List of 300+ Articles from the Google Scholar (Logo). |
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