ANALYSIS OF QUANTUM EIGENSTATES IN A 3-MODE SYSTEM
|
|
- Bryce Shepherd
- 5 years ago
- Views:
Transcription
1 AAYSIS OF QUATUM EIGESTATES I A 3-MODE SYSTEM SRIHARI KESHAVAMURTHY AD GREGORY S. EZRA Depatment of Chemisty, Bake aboatoy Conell Univesity, Ithaca, Y 14853, USA. Abstact. We study the quantum eigenstates of a thee degee of feedom spectoscopic Hamiltonian fo the H 2 O molecule. Using the classical esonance zones as a template, we ae able to undestand and oganize the enegy level spectum, and povide quantum numbe assignments fo the eigenstates. 1. Intoduction The poblem of the natue of highly excited ovibational states of polyatomic molecules is of cental significance fo molecula spectoscopy and chemical kinetics [1]. At high enegies, the taditional appoach of molecula spectoscopy, based on on a hamonic oscillatoigid oto model [2], beaks down due to stong intemode couplings. These mode couplings, which esult fom anhamonicities in the potential and/o otation-vibation inteaction, lead to intamolecula vibational enegy tansfe [3] and complicated enegy specta. A typical classical polyatomic molecule Hamiltonian is inheently a multi ( 3) mode, nonintegable system. Moeove, at high levels of intenal excitation, the semiclassical egime is appoached, and the geneal poblem of the classical-quantum coespondence fo nonintegable systems must be faced in ode to elate classical and quantum aspects [4]. Fo both = 2- and 3-mode systems, we can ecognize seveal mechanisms leading to complicated enegy level pattens. One possibility involves stong mixing of nea-degeneate manifolds of states by a single esonant coupling, so that the coesponding classical dynamics is still integable. Anothe possibility is stong state mixing due to the pesence of two o moe esonant couplings, so that the coesponding classical dynamics is nonintegable and we have the possibility of chaos. Finally, thee is the possibility of state mixing due to the phenomenon of dynamical tunneling [5], associated with accidental nea-degeneacies of levels. Model spectoscopic Hamiltonians descibing two modes coupled by a single esonant tem have been widely studied [6]. Fo such systems thee exists an additional constant of motion apat fom the enegy, and the educed phase space is a 2-dimensional sphee, the so-called polyad phase sphee. Kellman and cowokes have studied the bifucation stuctue of fixed points of the educed Hamiltonian on the polyad phase sphee, which coespond to the peiodic obits in the full phase space, and have investigated in detail the
2 2 elation between the classical phase space stuctue and the associated quantum mechanics [6]. Fo the case of a single esonant coupling, Rose and Kellman have extended this analysis to a multimode plana model of C 2 H 2 [7]. Multimode Hamiltonians with two o moe esonant couplings have not been studied in a systematic fashion. One eason is that, fo multidimensional systems, the stable and unstable manifolds of unstable peiodic obits cannot patition phase space into disjoint zones [8], theeby peventing staightfowad genealization of peiodic obit based appoaches. The classical mechanics of multimode systems is of couse consideably iche than that of 2-mode Hamiltonians: thee is the possibility of Anold diffusion [9], and, of moe immediate elevance to molecula poblems, the existence of intesections between esonance zones. The implications of this latte featue of the phase space stuctue of multi-mode systems have begun to be exploed fo molecula systems [10]. In the pesent pape we outline a qualitative appoach to analysis of enegy level pattens and wavefunctions in multimode systems, which is based on Chiikov esonance analysis [9]. The system studied is a 3-mode spectoscopic vibational Hamiltonian fo H 2 O due to Baggot [11]. We establish a diect coelation between featues of the associated eigenstates with the location of esonance zones in phase space. In paticula, we obseve pogessions of esonant eigenstates along classical esonance zones leading to mixed states in the vicinity of the esonance channel intesections. The classical esonance analysis theefoe povides a template fo analyzing the quantum states, and theeby facilitates an undestanding of the level spectum. 2. Resonance analysis of the Baggot Hamiltonian We conside the classical analog of the spectoscopic Hamiltonian fo H 2 O deived by Baggot [11]. This 3 degee of feedom Hamiltonian descibes the two local OH stetching modes and the HOH bend, and includes two 2:1 stetch-bend esonant tems H 2:1, =1,2, a 1:1 stetch-stetch esonant tem H 1:1 and a 2:2 stetch-stetch esonant tem H 2:2 : H = H 0 + H 1:1 + H 2:2 + H 2:1, (1) whee H 0 is the zeoth ode Hamiltonian H 0 =Ω s I +Ω b I b +α s I 2 + α b Ib 2 + ɛ ss I 1 I 2 + ɛ sb I b I, (2) =1,2 and the esonant coupling tems ae =1,2 =1,2 =1,2 H 1:1 = β 12(I 1 I 2 ) 1/2 cos(θ 1 θ 2 ) H 2:2 = β 22 I 1 I 2 cos[2(θ 1 θ 2 )] (3) H 2:1 = β sb (I Ib 2 ) 1/2 cos(θ 2θ b ). In the above equations, (I, θ) ae the canonical action-angle vaiables fo the two stetches and the bend mode, espectively, with β 12 β 12 + λ (I 1 + I 2 )+λ I b. Values of the vaious paametes ae taken fom Baggot s fit to the vibational levels of H 2 O (Table 2 of efeence [11]). The 2:2 esonance coupling does not significantly affect the classical
3 phase space stuctues in the egime of inteest fo this pape and will be neglected in the classical esonance analysis. The complete Hamiltonian H, although nonintegable, has a constant of the motion in addition to the enegy: =2 I + I b, (4) =1,2 which is the classical analog of (twice) the polyad numbe, P = =1,2 n s + n b /2. Thus the dynamics is constained to the plane in action space = constant. Chiikov esonance analysis fo H 0 is staightfowad, and in standad fashion one can detemine the location of the esonance zones in action space [9]. Both the 1:1 and 2:1 esonance zones ae shown in Figue 1, pojected onto the (I 1,I 2 ) action plane (P = 8). It is impotant to note that, fo the Baggot Hamiltonian, as β 12 < 0andλ >0 the 1:1 esonance width actually deceases as the actions I 1,I 2 get lage. ote also that we do not explicitly conside seconday esonances since we ae inteested only in the lage-scale stuctue of the classical phase space (i.e., intesection of the lowest ode esonance zones). 3. Analysis and assignment of eigenstates and specta The quantum eigenfunctions and eigenenegies wee obtained by diagonalizing the Baggot Hamiltonian in the numbe basis n 1,n 2,n b, whee n 1,n 2 ae the quantum numbes fo the anhamonic local O-H stetch modes and n b denotes the bend quantum numbe. The Hamiltonian is block diagonal in the polyad numbe P, and the total numbe of states at fixed P is (P +1)(P+2)/2. To analyse the eigenstates, it is natual to poject them onto the (n 1,n 2 ) quantum numbe (action) plane. The physical points in the (n 1,n 2 ) lattice ae those points fo which P n 1 n 2 is nonnegative. The eigenstates ae epesented by plotting at evey physical lattice point a cicle with adius equal to the squae of the coefficient of the coesponding zeoth ode basis state in the eigenstate of inteest. In this pape we concentate on the P = 8 case. Thee ae a total of 45 states and fo this case the 1:1 and 2:1 esonance lines do not intesect in the physical action space. Moeove, the 1:1 esonance inteaction dominates in the sense that an appopiate zeothode pictue of the spectum is one in which only the 1:1 coupling is pesent, i.e., the bend quantum numbe n b is conseved. In this (classically integable) limit we have essentially a stack of independent 2-mode systems, labelled with the quantum numbe n b. States fo the 1:1 coupled local modes ae classified as local o nomal, accoding to whethe amplitude is situtated outside o inside the 1:1 esonance zone, espectively. In the absence of 2:1 couplings, each eigenstate has nonzeo amplitude only along the line I 1 + I 2 = constant. The nomal mode states fom pogessions along the 1:1 esonance line, whee all states in a given sequence have eithe a node o antinode along the diagonal. Figue 1a shows one paticula sequence in the case when only the 1:1 inteaction tem is pesent. Of key inteest is the effect of the 2:1 esonant couplings on this zeoth-ode pictue; the 2:1 couplings couple togethe stacks of states with diffeent values of n b, theeby manifesting the tue multidimensional natue of the system. Figue 1b shows a sequence of states fo the fully coupled system coesponding to those of Fig 1. It is clea that, in the nonintegable case, one of the states in the sequence is significantly distoted by the 2:1 esonant tem (this state is maked with thick cicles). Fo lage values of the polyad numbe P, we obseve pogessions of esonant states along the 2:1 esonance channels. 3
4 4 8 a b 6 I I I 1 Figue 1. case (b). omal mode sequence along the 1:1 esonant line fo the integable case (a) and nonintegable E, cm β n b Figue 2. (a) Evolution of the enegy spectum as β β sb is tuned on and (b) associated states gouped by n b., and stand fo nomal mode, nonesonant local mode doublets and esonant local mode doublets espectively.
5 E, cm β n b Figue 3. Same as figue 2 with a diffeent egion of the enegy spectum. We now examine the changes in the level spectum fo the 1:1 coupling case when the 2:1 couplings ae tuned on. Figues 2 and 3 show the pats of the spectum which undego the most significant changes fo P = 8. ocal mode (±) doublets situated in the vicinity of the phase space egion coesponding to the 2:1 esonance zone become 2:1 esonant doublets unde the influence of the 2:1 couplings. Most of the mixing occus between the n b =2,4 and 6 manifolds of states, which coelates well with location of the 2:1 esonance zone. All of the avoided cossings seen in Figues 2 and 3 ae simple two state cossings. Fo P = 8, one can theefoe undestand the mixing of states fom diffeent n b manifolds via simple two state analysis. The assignment of the spectum is staightfowad, due to the appoximate consevation of n b, and states ae assigned as nomal mode, nonesonant local mode doublets, o 2:1 esonant local mode doublets. The simple two state scenaio beaks down at highe values of P whee one sees complicated mixings involving thee o moe states. In such cases it becomes moe useful to oganize the spectum on the basis of the quantum numbes associated with esonant sequences as opposed to the n b gouping. The details of the analysis fo highe values of P will be epoted in a futue publication [12]. 4. Acknowledgements This eseach was suppoted by SF gant CHE Refeences 1. C. E. Hamilton, J.. Kinsey and R. W. Field, Ann. Rev. Phys. Chem. 37, 493 (1986); M. E. Kellman, in Molecula Dynamics and Spectoscopy by Stimulated Emission Pumping, edited by H.-. Dai and R. W. Field, (Wold Scientific, 1995).
6 6 2. D. Papousek and M. R. Aliev, Molecula Vibational-Rotational Specta (Elsevie, 1982). 3. K.K. ehmann, G. Scoles and B.H. Pate, Ann. Rev. Phys. Chem. 45, 241 (1994). 4. M. Gutzwille, Chaos in Classical and Quantum Mechanics (Spinge, 1990). 5. M.J. Davis and E.J. Helle, J. Phys. Chem. 75, 246 (1981). 6.. Xiao and M. E. Kellman, J. Chem. Phys. 90, 6086 (1989); Z. i,. Xiao and M. E. Kellman, J. Chem. Phys. 92, 2251 (1990);. Xiao and M. E. Kellman, J. Chem. Phys. 93, 5805 (1990). 7. J. Rose and M.E. Kellman (pepint). 8. R. E. Gillilan and G. S. Eza, J. Chem. Phys. 94, 2648 (1991). 9. B. V. Chiikov, Phys. Rep. 52, 263 (1979). 10. K.M. Atkins and D. ogan, Phys. ett. 162, 255 (1992); C.C. Matens, J. Stat. Phys. 68, 207 (1992). 11. J. E. Baggot, Mol. Phys. 65, 739 (1988). 12. S. Keshavamuthy and G. S. Eza, to be submitted.
Chem 453/544 Fall /08/03. Exam #1 Solutions
Chem 453/544 Fall 3 /8/3 Exam # Solutions. ( points) Use the genealized compessibility diagam povided on the last page to estimate ove what ange of pessues A at oom tempeatue confoms to the ideal gas law
More informationThe geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
More information= e2. = 2e2. = 3e2. V = Ze2. where Z is the atomic numnber. Thus, we take as the Hamiltonian for a hydrogenic. H = p2 r. (19.4)
Chapte 9 Hydogen Atom I What is H int? That depends on the physical system and the accuacy with which it is descibed. A natual stating point is the fom H int = p + V, (9.) µ which descibes a two-paticle
More informationLong-range stress re-distribution resulting from damage in heterogeneous media
Long-ange stess e-distibution esulting fom damage in heteogeneous media Y.L.Bai (1), F.J.Ke (1,2), M.F.Xia (1,3) X.H.Zhang (1) and Z.K. Jia (1) (1) State Key Laboatoy fo Non-linea Mechanics (LNM), Institute
More informationyou of a spring. The potential energy for a spring is given by the parabola U( x)
Small oscillations The theoy of small oscillations is an extemely impotant topic in mechanics. Conside a system that has a potential enegy diagam as below: U B C A x Thee ae thee points of stable equilibium,
More informationBIFURCATION ANALYSIS FOR A QUASI-LINEAR CENTRIFUGAL PENDULUM ABSORBER
11 th Intenational Confeence on Vibation Poblems Z. Dimitovová et al. (eds.) Lisbon, Potugal, 9-12 Septembe 213 BIFURCATION ANALYSIS FOR A QUASI-LINEAR CENTRIFUGAL PENDULUM ABSORBER Eugen B. Keme* 1, Mikhail
More informationarxiv: v1 [physics.gen-ph] 18 Aug 2018
Path integal and Sommefeld quantization axiv:1809.04416v1 [physics.gen-ph] 18 Aug 018 Mikoto Matsuda 1, and Takehisa Fujita, 1 Japan Health and Medical technological college, Tokyo, Japan College of Science
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
More informationPreliminary Exam: Quantum Physics 1/14/2011, 9:00-3:00
Peliminay Exam: Quantum Physics /4/ 9:-: Answe a total of SIX questions of which at least TWO ae fom section A and at least THREE ae fom section B Fo you answes you can use eithe the blue books o individual
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationNuclear and Particle Physics - Lecture 20 The shell model
1 Intoduction Nuclea and Paticle Physics - Lectue 0 The shell model It is appaent that the semi-empiical mass fomula does a good job of descibing tends but not the non-smooth behaviou of the binding enegy.
More informationHydroelastic Analysis of a 1900 TEU Container Ship Using Finite Element and Boundary Element Methods
TEAM 2007, Sept. 10-13, 2007,Yokohama, Japan Hydoelastic Analysis of a 1900 TEU Containe Ship Using Finite Element and Bounday Element Methods Ahmet Egin 1)*, Levent Kaydıhan 2) and Bahadı Uğulu 3) 1)
More informationAPPLICATION OF MAC IN THE FREQUENCY DOMAIN
PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he
More informationProblem Set 10 Solutions
Chemisty 6 D. Jean M. Standad Poblem Set 0 Solutions. Give the explicit fom of the Hamiltonian opeato (in atomic units) fo the lithium atom. You expession should not include any summations (expand them
More informationNuclear size corrections to the energy levels of single-electron atoms
Nuclea size coections to the enegy levels of single-electon atoms Babak Nadii Nii a eseach Institute fo Astonomy and Astophysics of Maagha (IAAM IAN P. O. Box: 554-44. Abstact A study is made of nuclea
More informationCalculation of Quark-antiquark Potential Coefficient and Charge Radius of Light Mesons
Applied Physics Reseach ISSN: 96-9639 Vol., No., May E-ISSN: 96-9647 Calculation of Quak-antiquak Potential Coefficient and Chage Radius of Light Mesons M.R. Shojaei (Coesponding autho ) Depatment of Physics
More informationMATH 415, WEEK 3: Parameter-Dependence and Bifurcations
MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationNuclear models: Shell model
Lectue 3 Nuclea models: Shell model WS0/3: Intoduction to Nuclea and Paticle Physics,, Pat I Nuclea models Nuclea models Models with stong inteaction between the nucleons Liquid dop model α-paticle model
More informationEFFECTS OF FRINGING FIELDS ON SINGLE PARTICLE DYNAMICS. M. Bassetti and C. Biscari INFN-LNF, CP 13, Frascati (RM), Italy
Fascati Physics Seies Vol. X (998), pp. 47-54 4 th Advanced ICFA Beam Dynamics Wokshop, Fascati, Oct. -5, 997 EFFECTS OF FRININ FIELDS ON SINLE PARTICLE DYNAMICS M. Bassetti and C. Biscai INFN-LNF, CP
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationIdentification of the degradation of railway ballast under a concrete sleeper
Identification of the degadation of ailway ballast unde a concete sleepe Qin Hu 1) and Heung Fai Lam ) 1), ) Depatment of Civil and Achitectual Engineeing, City Univesity of Hong Kong, Hong Kong SAR, China.
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationRight-handed screw dislocation in an isotropic solid
Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More informationAnnihilation of Relativistic Positrons in Single Crystal with production of One Photon
Annihilation of Relativistic Positons in Single Cystal with poduction of One Photon Kalashnikov N.P.,Mazu E.A.,Olczak A.S. National Reseach Nuclea Univesity MEPhI (Moscow Engineeing Physics Institute),
More informationQUASI-STATIONARY ELECTRON STATES IN SPHERICAL ANTI-DOT WITH DONOR IMPURITY * 1. INTRODUCTION
ATOMIC PHYSICS QUASI-STATIONARY ELECTRON STATES IN SPHERICAL ANTI-DOT ITH DONOR IMPURITY * V. HOLOVATSKY, O. MAKHANETS, I. FRANKIV Chenivtsi National Univesity, Chenivtsi, 581, Ukaine, E-mail: ktf@chnu.edu.ua
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationFrom Gravitational Collapse to Black Holes
Fom Gavitational Collapse to Black Holes T. Nguyen PHY 391 Independent Study Tem Pape Pof. S.G. Rajeev Univesity of Rocheste Decembe 0, 018 1 Intoduction The pupose of this independent study is to familiaize
More information13. Adiabatic Invariants and Action-Angle Variables Michael Fowler
3 Adiabatic Invaiants and Action-Angle Vaiables Michael Fowle Adiabatic Invaiants Imagine a paticle in one dimension oscillating back and foth in some potential he potential doesn t have to be hamonic,
More informationRADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION
RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION JINGYI CHEN AND YUXIANG LI Abstact. We show that a smooth adially symmetic solution u to the gaphic Willmoe suface equation is eithe
More informationPhysical Chemistry II (Chapter 4 1) Rigid Rotor Models and Angular Momentum Eigenstates
Physical Chemisty II (Chapte 4 ) Rigid Roto Models and Angula Momentum Eigenstates Tae Kyu Kim Depatment of Chemisty Rm. 30 (tkkim@pusan.ac.k) http://cafe.nave.com/moneo76 SUMMAR CHAPTER 3 A simple QM
More informationON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS
ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS L. MICU Hoia Hulubei National Institute fo Physics and Nuclea Engineeing, P.O. Box MG-6, RO-0775 Buchaest-Maguele, Romania, E-mail: lmicu@theoy.nipne.o (Received
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationPHYSICS 4E FINAL EXAM SPRING QUARTER 2010 PROF. HIRSCH JUNE 11 Formulas and constants: hc =12,400 ev A ; k B. = hf " #, # $ work function.
PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Fomulas and constants: hc =1,4 ev A ; k B =1/11,6 ev/k ; ke =14.4eVA ; m e c =.511"1 6 ev ; m p /m e =1836 Relativistic enegy - momentum elation E = m c 4 + p c ;
More informationBLACK HOLES IN STRING THEORY
Black holes in sting theoy N Sadikaj & A Duka Pape pesented in 1 -st Intenational Scientific Confeence on Pofessional Sciences, Alexande Moisiu Univesity, Dues Novembe 016 BLACK OLES IN STRING TEORY NDRIÇIM
More informationMEASURING CHINESE RISK AVERSION
MEASURING CHINESE RISK AVERSION --Based on Insuance Data Li Diao (Cental Univesity of Finance and Economics) Hua Chen (Cental Univesity of Finance and Economics) Jingzhen Liu (Cental Univesity of Finance
More informationDiffusion and Transport. 10. Friction and the Langevin Equation. Langevin Equation. f d. f ext. f () t f () t. Then Newton s second law is ma f f f t.
Diffusion and Tanspot 10. Fiction and the Langevin Equation Now let s elate the phenomena of ownian motion and diffusion to the concept of fiction, i.e., the esistance to movement that the paticle in the
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation
More informationRotor Blade Performance Analysis with Blade Element Momentum Theory
Available online at www.sciencediect.com ScienceDiect Enegy Pocedia 5 (7 ) 3 9 The 8 th Intenational Confeence on Applied Enegy ICAE6 Roto Blade Pefomance Analysis with Blade Element Momentum Theoy Faisal
More informationPearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms
Peason s Chi-Squae Test Modifications fo Compaison of Unweighted and Weighted Histogams and Two Weighted Histogams Univesity of Akueyi, Bogi, v/noduslód, IS-6 Akueyi, Iceland E-mail: nikolai@unak.is Two
More informationOn the Sun s Electric-Field
On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a
More informationAnalysis of high speed machining center spindle dynamic unit structure performance Yuan guowei
Intenational Confeence on Intelligent Systems Reseach and Mechatonics Engineeing (ISRME 0) Analysis of high speed machining cente spindle dynamic unit stuctue pefomance Yuan guowei Liaoning jidian polytechnic,dan
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationSTABILITY AND PARAMETER SENSITIVITY ANALYSES OF AN INDUCTION MOTOR
HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY VESZPRÉM Vol. 42(2) pp. 109 113 (2014) STABILITY AND PARAMETER SENSITIVITY ANALYSES OF AN INDUCTION MOTOR ATTILA FODOR 1, ROLAND BÁLINT 1, ATTILA MAGYAR 1, AND
More informationAbsorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere
Applied Mathematics, 06, 7, 709-70 Published Online Apil 06 in SciRes. http://www.scip.og/jounal/am http://dx.doi.og/0.46/am.06.77065 Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in
More informationTwo-level quantum dot in the Aharonov Bohm ring. Towards understanding phase lapse *
Mateials Science-Poland, Vol. 5, No. 4, 007 Two-level quantum dot in the Ahaonov Bohm ing. Towads undestanding phase lapse * P. STEFAŃSKI ** Institute of Molecula Physics of the Polish Academy of Sciences,
More informationThe Poisson bracket and magnetic monopoles
FYST420 Advanced electodynamics Olli Aleksante Koskivaaa Final poject ollikoskivaaa@gmail.com The Poisson backet and magnetic monopoles Abstact: In this wok magnetic monopoles ae studied using the Poisson
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More information20th Century Atomic Theory - Hydrogen Atom
0th Centuy Atomic Theoy - Hydogen Atom Ruthefod s scatteing expeiments (Section.5, pp. 53-55) in 1910 led to a nuclea model of the atom whee all the positive chage and most of the mass wee concentated
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationCHAPTER 25 ELECTRIC POTENTIAL
CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When
More informationPerturbation to Symmetries and Adiabatic Invariants of Nonholonomic Dynamical System of Relative Motion
Commun. Theo. Phys. Beijing, China) 43 25) pp. 577 581 c Intenational Academic Publishes Vol. 43, No. 4, Apil 15, 25 Petubation to Symmeties and Adiabatic Invaiants of Nonholonomic Dynamical System of
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationPhysics 221 Lecture 41 Nonlinear Absorption and Refraction
Physics 221 Lectue 41 Nonlinea Absoption and Refaction Refeences Meye-Aendt, pp. 97-98. Boyd, Nonlinea Optics, 1.4 Yaiv, Optical Waves in Cystals, p. 22 (Table of cystal symmeties) 1. Intoductoy Remaks.
More informationEN40: Dynamics and Vibrations. Midterm Examination Thursday March
EN40: Dynamics and Vibations Midtem Examination Thusday Mach 9 2017 School of Engineeing Bown Univesity NAME: Geneal Instuctions No collaboation of any kind is pemitted on this examination. You may bing
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In Chaptes 2 and 4 we have studied kinematics, i.e., we descibed the motion of objects using paametes such as the position vecto, velocity, and acceleation without any insights
More informationLecture 7: Angular Momentum, Hydrogen Atom
Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In chaptes 2 and 4 we have studied kinematics i.e. descibed the motion of objects using paametes such as the position vecto, velocity and acceleation without any insights as to
More information5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS
5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,
More informationarxiv:gr-qc/ v1 1 Sep 2005
Radial fall of a test paticle onto an evapoating black hole Andeas Aste and Dik Tautmann Depatment fo Physics and Astonomy, Univesity of Basel, 456 Basel, Switzeland E-mail: andeas.aste@unibas.ch June
More informationStudy on Application of New Theorem of Kinetic Energy
Intenational Jounal of pplied Physics and Mathematics Study on pplication of New Theoem of Kinetic Enegy Yong Yang * Tongchuan Banch of Shanxi Radio and TV Univesity, Tongchuan Polytechnics, Tongchuan
More informationAnalytic Evaluation of two-electron Atomic Integrals involving Extended Hylleraas-CI functions with STO basis
Analytic Evaluation of two-electon Atomic Integals involving Extended Hylleaas-CI functions with STO basis B PADHY (Retd.) Faculty Membe Depatment of Physics, Khalikote (Autonomous) College, Behampu-760001,
More informationOLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.
OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationBifurcation Analysis for the Delay Logistic Equation with Two Delays
IOSR Jounal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 5 Ve. IV (Sep. - Oct. 05), PP 53-58 www.iosjounals.og Bifucation Analysis fo the Delay Logistic Equation with Two Delays
More informationPrecessing Ball Solitons as Self-Organizing Systems during a Phase Transition in a Ferromagnet
Applied Mathematics,, 4, 78-8 http://dxdoiog/46/am4a Published Online Octobe (http://wwwscipog/jounal/am) Pecessing Ball Solitons as Self-Oganiing Systems duing a Phase Tansition in a Feomagnet V V Niet
More informationCOUPLED MODELS OF ROLLING, SLIDING AND WHIRLING FRICTION
ENOC 008 Saint Petesbug Russia June 30-July 4 008 COUPLED MODELS OF ROLLING SLIDING AND WHIRLING FRICTION Alexey Kieenkov Ins ti tu te fo P ob le ms in Me ch an ic s Ru ss ia n Ac ad em y of Sc ie nc es
More informationMany Electron Atoms. Electrons can be put into approximate orbitals and the properties of the many electron systems can be catalogued
Many Electon Atoms The many body poblem cannot be solved analytically. We content ouselves with developing appoximate methods that can yield quite accuate esults (but usually equie a compute). The electons
More informationMechanics Physics 151
Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus
More information5.111 Lecture Summary #6 Monday, September 15, 2014
5.111 Lectue Summay #6 Monday, Septembe 15, 014 Readings fo today: Section 1.9 Atomic Obitals. Section 1.10 Electon Spin, Section 1.11 The Electonic Stuctue of Hydogen. (Same sections in 4 th ed.) Read
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More informationMany Electron Theory: Particles and holes. Unitary transformations.
Many Electon Theoy: Paticles and holes. Unitay tansfomations. Continued notes fo a wokgoup Septembe-Octobe 00. Notes pepaed by Jan Lindebeg, Septembe 00 Heny Eying Cente fo Theoetical Chemisty Equivalencies
More informationChapter 6: Rotational and Rovibrational Spectra. A) General discussion of two- body problem with central potential
Fall 4 Chapte 6: Rotational and Rovibational Specta... 75 Diffeent Appoximations... 8 Spectum fo Hamonic Oscillato + Rigid Rotato... 8 Polyatomic Molecules... 84 Hamonic Oscillato + Rigid Roto Model to
More informationTeachers notes. Beyond the Thrills excursions. Worksheets in this book. Completing the worksheets
Beyond the Thills excusions Teaches notes Physics is the science of how the wold (and Univese) woks. Luna Pak Sydney is a lage hands-on physics laboatoy full of fee falling objects, otating systems and
More informationThe main paradox of KAM-theory for restricted three-body problem (R3BP, celestial mechanics)
The main paadox of KAM-theoy fo esticted thee-body poblem (R3BP celestial mechanics) Segey V. Eshkov Institute fo Time Natue Exploations M.V. Lomonosov's Moscow State Univesity Leninskie goy 1-1 Moscow
More informationChapter Sixteen: Electric Charge and Electric Fields
Chapte Sixteen: Electic Chage and Electic Fields Key Tems Chage Conducto The fundamental electical popety to which the mutual attactions o epulsions between electons and potons ae attibuted. Any mateial
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More informationAccurate evaluation of lowest band gaps in ternary locally resonant phononic crystals
Vol 15 No 8, August 6 c 6 Chin. Phys. Soc. 19-1963/6/15(8)/1843-6 Chinese Physics and IOP Publishing Ltd Accuate evaluation of lowest band gaps in tenay locally esonant phononic cystals Wang Gang( ) a)b),
More informationPendulum in Orbit. Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ (December 1, 2017)
1 Poblem Pendulum in Obit Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 08544 (Decembe 1, 2017) Discuss the fequency of small oscillations of a simple pendulum in obit, say,
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More information1 Dark Cloud Hanging over Twentieth Century Physics
We ae Looking fo Moden Newton by Caol He, Bo He, and Jin He http://www.galaxyanatomy.com/ Wuhan FutueSpace Scientific Copoation Limited, Wuhan, Hubei 430074, China E-mail: mathnob@yahoo.com Abstact Newton
More informationCircular Orbits. and g =
using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is
More informationRecent Advances in Chemical Engineering, Biochemistry and Computational Chemistry
Themal Conductivity of Oganic Liquids: a New Equation DI NICOLA GIOVANNI*, CIARROCCHI ELEONORA, PIERANTOZZI ARIANO, STRYJEK ROAN 1 DIIS, Univesità Politecnica delle ache, 60131 Ancona, ITALY *coesponding
More informationEXAM NMR (8N090) November , am
EXA NR (8N9) Novembe 5 9, 9. 1. am Remaks: 1. The exam consists of 8 questions, each with 3 pats.. Each question yields the same amount of points. 3. You ae allowed to use the fomula sheet which has been
More informationFORMULA FOR SPIRAL CLOUD-RAIN BANDS OF A TROPICAL CYCLONE. Boris S. Yurchak* University of Maryland, Baltimore County, Greenbelt, Maryland =, (2)
PF.3 FORMULA FOR SPIRAL CLOUD-RAIN BANDS OF A TROPICAL CYCLONE Bois S. Yuchak* Univesity of Mayland, Baltimoe County, Geenbelt, Mayland. INTRODUCTION Spial cloud-ain bands (SCRB) ae some of the most distinguishing
More informationPARTICLE TRANSPORT AND DEPOSITION IN POROUS STRUCTURES: EFFECTS OF PARTICLE PROPERTIES, POROSITY AND REYNOLDS NUMBER
ISTP-16, 2005, PRAGUE 16 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA PARTICLE TRANSPORT AND DEPOSITION IN POROUS STRUCTURES: EFFECTS OF PARTICLE PROPERTIES, POROSITY AND REYNOLDS NUMBER S. Kuz*,
More informationPHYS 705: Classical Mechanics. Small Oscillations
PHYS 705: Classical Mechanics Small Oscillations Fomulation of the Poblem Assumptions: V q - A consevative system with depending on position only - The tansfomation equation defining does not dep on time
More informationrt () is constant. We know how to find the length of the radius vector by r( t) r( t) r( t)
Cicula Motion Fom ancient times cicula tajectoies hae occupied a special place in ou model of the Uniese. Although these obits hae been eplaced by the moe geneal elliptical geomety, cicula motion is still
More informationAnalytical Solutions for Confined Aquifers with non constant Pumping using Computer Algebra
Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) Analytical Solutions fo Confined Aquifes with non constant Pumping using Compute Algeba
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes.
More informationDouble folding analysis of 3 He elastic and inelastic scattering to low lying states on 90 Zr, 116 Sn and 208 Pb at 270 MeV
Double folding analysis of 3 He elastic and inelastic scatteing to low lying states on 90 Z, 116 Sn and 08 Pb at 70 MeV Mawa N. El-Hammamy Physics Depatment, Faculty of Science, Damanhu Univesity, Egypt
More informationBasic Bridge Circuits
AN7 Datafoth Copoation Page of 6 DID YOU KNOW? Samuel Hunte Chistie (784-865) was bon in London the son of James Chistie, who founded Chistie's Fine At Auctionees. Samuel studied mathematics at Tinity
More informationDonnishJournals
DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş
More information