Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork
|
|
- Bryan Moore
- 5 years ago
- Views:
Transcription
1 Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: June June To be dscussed on Tuesday of Week-II. Classwok. Deve all the quantum numbes I G J PC n the t channel of the followng eactons (a)! and K K! K K (b) N! N, N! N and KN! KN (c) N! N and N! N (d)! Notaton: =( +,, 0 ); =( +,, 0 ) ; K =(K +,K 0 ) ; N =(p, n).. Assume that the Regge exchange fom a SU() octet and a SU() snglet wth the couplng fo the octet and the snglet beng dffeent. Consde a vecto and a tenso nonet (octet plus snglet). Fom the dualty hypothess and the absence of double chage meson, fnd the combnaton of octet-snglet tenso that decouples fom. Use the SU() Clebsch-Godan coeffcents fom Rev.Mod.Phys. 6 (964) 005. What ae the quak content and the K K couplngs of these states?. Assumng deal mxng fo the vecto and tenso, deve the exchange degeneacy elatons comng dualty and the absence of esonance n the followng eactons (a)! (b) K K! K K (c) KN! KN (d)! (and! ) 4. Deve a Loentz-covaant bass, the sospn decomposton and the cossng popetes fo the followng eactons (a) N! N and KN! KN (b) NN! NN (c)!! and B! J/ K (d)! (e) N! N and N! N (use F µ = µ k k µ )
2 Soluton. The lst of exchanges havng only I =0, s pesented on Table. Notaton: sgnatue =( ) J and natualty = P ( ) J. In the quak model, P =( )`+ and C =( )`+S, hence 0, (,, 5,...) + and (0,, 4,...) + ae fobdden n the quak model. Let s efe to these quantum numbes as exotc". Table : Regge Tajectoes I G J PC I G J PC f + (0,, 4,...) f (,, 5,...) ! (,, 5,...) 0 +! + (0,, 4,...) ++ (0,, 4,...) ++ a (,, 5,...) (,, 5,...) ++ + (0,, 4,...) (0,, 4,...) (,, 5,...) + 0 h (,, 5,...) h + (0,, 4,...) (0,, 4,...) + + (,, 5,...) + + b (,, 5,...) b + (0,, 4,...) + (a) fo : G =+, =+and ( ) I =+(Bose symmety) ) f + and. fo K K: =+and ( ) I =+) f +,,! and. (b) fo : G =, =+, I =; fo NN: I =0, and no exotc ). fo K K: =+; fo NN: I =0, and no exotc ) f +,,! and. (c) fo and 0 : C = ; fo NN: I =0, and no exotc. )! ±, ±, b and h. fo + : I =; fo NN: I =0, and no exotc. ) a ±, ±, b and. (d) fo : G = )a ±, ±,! ± and h ± Table : Exchanges (a) +! + f + ± 0 0! 0 0 f + K + K! K + K f + ±! + ± K + K 0! K 0 K + (b) p! n p p! 0 n + p! p f + ± + n! n f + + K p! K p 0 p p ( + a+ ) K + n! K 0 p ( a+ ) K p! K p f + ± + ±! K n! K n f + ±! (c) p! p (! + )+(h + b +! ) p! 0 p (! + )+(h + b +! ) p! + n ( + )+(b a ) n! p ( )+b a ) (d) + 0! 0 + ( + )+( + ) + +! + + (! + h + )+(! + h + ) + +! + + f +. Fo a geneal teatment of exchange degeneacy usng goup theoy, see Ref. [].
3 We assume that the esdues obey a SU() symmety: R ac(t) /h8y a I a I a ; 8 Yc I c Ic 8 Y R I R I R, () whee Y = Y and I = I. The hypechage Y s the stangeness and I s the sospn pojecton. The Clebsch-Godan coeffcents fo SU() ae lsted n Ref. []. Note the exta mnus sgn fo the + and K. The fou couplngs ae 8V, 8T, V and T fo the octet/ snglet fo the tenso and vecto tajectoes. The absence of sospn meson n + K +! K + + and n + +! + + lead to + K +! K + + : + +! + + : 8 0 T s T + 5 8T s 8T 6 8T s 8T 8V s 8V =0 (a) 8V s 8T =0 (b) We combne them to get 8V = 8V = T and (/5) 8T =(/8) T, I choose by conventon 8 T = Let us defne the octet-snglet mxng f cos T f 0 = sn T 5 V. () sn T cos T f8 f (4) The states ae f 8 = 8; 000 and f = ; 000. The notaton s R; YII. Let us mpose that the f 0 couplng to + + vanshes sn T 5 8T! + cos T 8 T! =0. (5) Wth the elaton between the couplngs, we obtan sn T = p cos T o tan T =/. The quak content ae then 0 uū+d d 0 q q 0 p uū+d d p s A q q 6 uū+d s s d+s s A (6) p The couplngs ae f = f K + K + = f 0 K + K + = 5 8T cos T + 8 T sn T = 5 8T (7a) 0 8T cos T + 8 T sn T = 5 8T (7b) 0 8T sn T + 8 T cos T = p 5 8T (7c). In ths secton, I only wote the elatve sgn, not the elatve magntude gven by SU() and SU() Clebsch-Godan coeffcents. In the + +! + + case we obtan 0= f + (t)s f + (t) (t)s (t). (8) Snce ths elaton s vald n a ange of s and t, we obtan (t) = f+ (t) and (t) = f + (t). Fo patcles wth spn, one can epeat the agument wth specfc combnaton of helcty ampltudes
4 havng good natualty. Hence we obtan EXD elatons between exchanges wth the same natualty. In the case of + +! + + case we obtan fo the natual exchanges 0=! (t)s! (t) (t)s h + (t)s h + (t) (t)s (t) (9a) =(! a (t)) s N h+ (t) (t) s EN(t), (9b) and fo the unnatual exchanges 0=! + (t)s! + (t) a (t)s a h (t)s h (t) + (t)s + (t) (9c) =! a (t)+ h (t) s U (t), (9d) In the eacton + +! + +, the exchanges pck up a sgn equal to PC, we obtan 0=(! a (t)) s N h+ (t) (t) s EN(t) 0=! a (t) h (t)+ + (t) s U (t) (9e) (9f) Thee ae then EXD elaton between exchanges wth same natualty, same PC, same G paty and opposte sgnatue. The Regge tajectoes ae ndcated on Fg.. The exchange degeneacy elatons ae summazed n Table and n Fg. 7 J Ê w Ù a Ú f p Ø b Ï h h ÙÚ Ê Ù Ê Ï ÚÙ Ê ØÚ ÏÙ M HGeV L Fgue : Regge tajectoes.the sold lnes ae N (t) =0.9(t m )+and U (t) =0.7(t m )+0. Table : Exchange degneacy elaton + +! + + f+ = f + = + + K + K 0! K 0 K + a+ = = K + K K + K 0 K + K +! K + K + f+ =! K + K + f + =! K + K + K + n! K + n a+ = pp = pp pp! K + p! K + p f+ =! f + pp = + +! + + h+ = h = ! + + h = + h + + = a+ =! + + =! + + a =!+ a + + =!
5 4. (a) Fo pon-nucleon scatteng, the covaant bass s [] h hn j (p 4 ) b (p ) N (p ) a (p ) =ū(p 4 ) A ba j +(p/ + p/ )Bj ba u(p ) (0a) A ba j = ba ja (+) + abc ( c ) j A ( ) (0b) A (±) (,t)=±a (±) (,t) (0c) B (±) (,t)= B (±) (,t) (0d) The cossng vaable s =(s u)/ wth s =(p + p ) and u =(p p 4 ). To deve the cossng elaton, use C nvaance T = C TC, v = Cū T, v =ū T C, C µc = T µ and C 5C =+ 5 T and take the tanspose complexe conjugate. Fo kaon-nucleon scatteng, the covaant bass s h hn j (p 4 )K l (p ) N (p )K k (p ) =ū(p 4 ) A lk j +(p/ + p/ )Bj lk u(p ) () and the sospn decomposton s A (0) and A () have sospn 0 and n the t A lk j = lk ja (0) +( a ) j ( a ) kl A () () channel. The cossng elatons ae A (0) (,t)=+a (0) (,t) B (0) (,t)= B (0) (,t) (a) A () (,t)= A () (,t) B () (,t)=+b () (,t) (b) (b) In nucleon-nucleon scatteng thee ae fve ndependent Loentz stuctues. One possble soluton s to use a t channel base 5X hn j (p 4 )N l (p ) N (p )N k (p ) = (A n ) lk A A j ū n u ū n u 4 (4) The ndex A s a collectve epesentaton of Loentz ndces. The tenso stuctues ae n= = = 5 µ = µ µ 4 = 5 µ µ 5 = [ µ, ] (5a) In ths base, the scala ampltudes A n have good t channel quantum numbes. One could also use a s channel base 5X hn j (p 4 )N l (p ) N (p )N k (p ) = (B n ) lk A A j ū n u ū n u 4 (6) Fetz denttes elate the two bass. The tansfomaton s B /4 /4 /4 /4 /4 A B BB B 4 A = / 0 / A B / 0 / 0 / C BA / 0 / A 4 A B 5 /4 /4 /4 /4 /4 A 5 The sospn decomposton s the same as n KN scatteng and the cossng elatons ae n= (7) A (0,,) (,t)=+a (0,,) (,t) (8a) A (0,,) (,t)=+a (0,,) (,t) (8b) A (0,,) (,t)= A (0,,) (,t) (8c) A (0,,) 4 (,t)= A (0,,) 4 (,t) (8d) A (0,,) 5 (,t)=+a (0,,) 5 (,t) (8e) A and A 4 pck up a mnus sgn because they coespond to negatve sgnatue exchanges (vecto and axal-vecto exchange). That s a good coss-check of the method. 5
6 (c) The eactons nvolve a vecto wth momentum p V and polazaton tenso µ (p V, ) and thee pseudoscala wth momenta p,,. We need a Lev-Cvta tenso fo paty (an odd numbe of unnatual paty mesons) f paty s conseved. If paty s not conseved (weak decay) thee ae two addtonal stuctues. In the paty consevng decay!! the Loentz stuctue s h a (p ) b (p ) a (p )!(p V, ) = A abc (,t)" µ (p V, )p p µ p. (9) The only sospn stuctue s A abc (,t)=" abc A(,t). Two pons ae always n sospn. The scala functon s odd unde cossng (p,! p, f t =(p V p ) ), A(,t)= A(,t), snce only vecto ae allowed. The decay B! J/ K can volate paty. Thee ae then thee Loentz stuctues h (p )K(p )J/ (p V ) B(p, ) = A (,t)" µ (p V, )p p µ p + A (,t) µ (p V, )(p p ) µ + A (,t) µ (p V, )(p + p ) µ (0) Isospn s not conseved, so the sospn stuctue s elevant. Ths base s elavant to study cossng unde p,! p,. We obtan A (,t)= A (,t), A (,t)=+a (,t) and A (,t)= A (,t). So the exchanges (o esonances) n the channel ae ( =+, = ) n A, ( =+, =+)n A and ( =, = ) n A. (d) Thee ae fou ndependent stuctues. Wth the notaton P =(p + p )/, (k, (k, ), they ae h d (p ) c (k, ) a (p ) b (k, ) = A abcd (,t) The sospn decomposton s A abcd = ac bda (0) + ab cd ) and + A abcd (,t) P P + A abcd (,t)[k P + P k ] + A abcd 4 (,t) k k. () ad bc A () + ab cd + ad bc A () () The Loentz and sospn bases ae chosen to have good popetes unde cossng the two pons (o the two s). We obtan, fo =,,, 4 A (0,) (,t)=+a (0,) (,t) A () (,t)= A () (,t) () (e) The momenta ae ( ) (k)+n(p )! (q)+n(p ) and p =(p +p )/. Use F µ = µ k k µ. Paty eques a 5 o an " µ. The matx element s hn j (p ) a (q) (k)n (p ) = X n (A n ) a j M n (4) We found n the notaton of Ref. [4] M = 5 µ F µ (5a) M = 5 q µ p F µ (5b) M = 5 µ q F µ (5c) M 4 = µ q F µ (5d) M 5 = 5 µ k F µ (5e) M 6 = 5 q µ k F µ (5f) 6
7 The last M 5,6 ae zeo fo photopoducton. The sospn decomposton s (A n ) a j = A(+) a j + A ( ) [ a, ] j + A (0) a j (6) Fnally the cossng popetes ae A (0,+) (,t)=+a (0,+) (,t) A ( ) (,t)= A ( ) (,t) =,, 4 (7a) A (0,+) (,t)= A (0,+) (,t) A ( ) (,t)=+a ( ) (,t) (7b) Refeences [] J. Mandula, J. Weyes and G. Zweg, Ann. Rev. Nucl. Pat. Sc. 0, 89 (970). [] P. S. J. McNamee and F. Chlton, Rev. Mod. Phys. 6, 005 (964). [] G. F. Chew, M. L. Goldbege, F. E. Low and Y. Nambu, Phys. Rev. 06, 7 (957). [4] F. A. Beends, A. Donnache and D. L. Weave, Nucl. Phys. B 4, (967). 7
Set of square-integrable function 2 L : function space F
Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,
More informationChapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.
Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,
More informationStructure of Hadrons. quarks d (down) s (strange) c (charm)
quaks Flavo A t t 0 S B T Q(e) Mc 2 (GeV) u (up) 1 3 1 2-1 2 0 0 0 0 2 3 0.002-0.008 d (down) 1 3 1 2 1 2 0 0 0 0-1 3 0.005-0.015 s (stange) 1 3 0 0-1 0 0 0-1 3 0.1-0.3 c (cham) 1 3 0 0 0 1 0 0 2 3 1.0-1.6
More informationLecture 22 Electroweak Unification
Lectue 22 Electoweak Unfcaton Chal emons EW unfcaton Hggs patcle hyscs 424 Lectue 22 age 1 2 Obstacles to Electoweak Unfcaton Electomagnetsm and the weak foce ae exactly the same, only dffeent... 1. Whle
More informationThermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering
Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs
More informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton
More informationEnergy in Closed Systems
Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and
More informationPHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
More informationEngineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems
Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,
More informationTest 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?
Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What
More informationSupersymmetry in Disorder and Chaos (Random matrices, physics of compound nuclei, mathematics of random processes)
Supesymmety n Dsoe an Chaos Ranom matces physcs of compoun nucle mathematcs of anom pocesses Lteatue: K.B. Efetov Supesymmety n Dsoe an Chaos Cambge Unvesty Pess 997999 Supesymmety an Tace Fomulae I.V.
More information24-2: Electric Potential Energy. 24-1: What is physics
D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a
More informationPhysics 202, Lecture 2. Announcements
Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn
More informationV. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.
Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum
More informationMechanics Physics 151
Mechancs Physcs 151 Lectue 18 Hamltonan Equatons of Moton (Chapte 8) What s Ahead We ae statng Hamltonan fomalsm Hamltonan equaton Today and 11/6 Canoncal tansfomaton 1/3, 1/5, 1/10 Close lnk to non-elatvstc
More informationTensor. Syllabus: x x
Tenso Sllabus: Tenso Calculus : Catesan tensos. Smmetc and antsmmetc tensos. Lev Vvta tenso denst. Pseudo tensos. Dual tensos. Dect poduct and contacton. Dads and dadc. Covaant, Contavaant and med tensos.
More information19 The Born-Oppenheimer Approximation
9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A
More informationDYNAMICS VECTOR MECHANICS FOR ENGINEERS: Kinematics of Rigid Bodies in Three Dimensions. Seventh Edition CHAPTER
Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Fednand P. Bee E. Russell Johnston, J. Lectue Notes: J. Walt Ole Teas Tech Unvest Knematcs of Rgd Bodes n Thee Dmensons 003 The McGaw-ll Companes, nc.
More informationCapítulo. Three Dimensions
Capítulo Knematcs of Rgd Bodes n Thee Dmensons Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd
More informationOptimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis
Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each
More informationPhysics 11b Lecture #2. Electric Field Electric Flux Gauss s Law
Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same
More informationGenerating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences
Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu
More informationChapter Fifiteen. Surfaces Revisited
Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)
More informationRanks of quotients, remainders and p-adic digits of matrices
axv:1401.6667v2 [math.nt] 31 Jan 2014 Ranks of quotents, emandes and p-adc dgts of matces Mustafa Elshekh Andy Novocn Mak Gesbecht Abstact Fo a pme p and a matx A Z n n, wte A as A = p(a quo p)+ (A em
More information1. A body will remain in a state of rest, or of uniform motion in a straight line unless it
Pncples of Dnamcs: Newton's Laws of moton. : Foce Analss 1. A bod wll eman n a state of est, o of unfom moton n a staght lne unless t s acted b etenal foces to change ts state.. The ate of change of momentum
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 25
ECE 6345 Sprng 2015 Prof. Davd R. Jackson ECE Dept. Notes 25 1 Overvew In ths set of notes we use the spectral-doman method to fnd the nput mpedance of a rectangular patch antenna. Ths method uses the
More informationUNIT10 PLANE OF REGRESSION
UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationLearning the structure of Bayesian belief networks
Lectue 17 Leanng the stuctue of Bayesan belef netwoks Mlos Hauskecht mlos@cs.ptt.edu 5329 Sennott Squae Leanng of BBN Leanng. Leanng of paametes of condtonal pobabltes Leanng of the netwok stuctue Vaables:
More informationPHY126 Summer Session I, 2008
PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment
More informationPhysics Exam II Chapters 25-29
Physcs 114 1 Exam II Chaptes 5-9 Answe 8 of the followng 9 questons o poblems. Each one s weghted equally. Clealy mak on you blue book whch numbe you do not want gaded. If you ae not sue whch one you do
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationIntegral Vector Operations and Related Theorems Applications in Mechanics and E&M
Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More information2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles
/4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationIf there are k binding constraints at x then re-label these constraints so that they are the first k constraints.
Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationCOMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS
ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant
More informationReview of Vector Algebra and Vector Calculus Operations
Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost
More informationgravity r2,1 r2 r1 by m 2,1
Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of
More information8 Baire Category Theorem and Uniform Boundedness
8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal
More informationMultistage Median Ranked Set Sampling for Estimating the Population Median
Jounal of Mathematcs and Statstcs 3 (: 58-64 007 ISSN 549-3644 007 Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame Al-Oma and Kamaulzaman Ibahm
More informationP 365. r r r )...(1 365
SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty
More information(8) Gain Stage and Simple Output Stage
EEEB23 Electoncs Analyss & Desgn (8) Gan Stage and Smple Output Stage Leanng Outcome Able to: Analyze an example of a gan stage and output stage of a multstage amplfe. efeence: Neamen, Chapte 11 8.0) ntoducton
More informationRemember: When an object falls due to gravity its potential energy decreases.
Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee
More informationGENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS
#A39 INTEGERS 9 (009), 497-513 GENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS Mohaad Faokh D. G. Depatent of Matheatcs, Fedows Unvesty of Mashhad, Mashhad,
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationSrednicki Chapter 34
Srednck Chapter 3 QFT Problems & Solutons A. George January 0, 203 Srednck 3.. Verfy that equaton 3.6 follows from equaton 3.. We take Λ = + δω: U + δω ψu + δω = + δωψ[ + δω] x Next we use equaton 3.3,
More informationPart V: Velocity and Acceleration Analysis of Mechanisms
Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton.
More informationThe Fourier Transform
e Processng ourer Transform D The ourer Transform Effcent Data epresentaton Dscrete ourer Transform - D Contnuous ourer Transform - D Eamples + + + Jean Baptste Joseph ourer Effcent Data epresentaton Data
More informationHamiltonian multivector fields and Poisson forms in multisymplectic field theory
JOURNAL OF MATHEMATICAL PHYSICS 46, 12005 Hamltonan multvecto felds and Posson foms n multsymplectc feld theoy Mchael Foge a Depatamento de Matemátca Aplcada, Insttuto de Matemátca e Estatístca, Unvesdade
More informationChapter I Vector Analysis
. Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw
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 informationPhysics 207 Lecture 16
Physcs 07 Lectue 6 Goals: Lectue 6 Chapte Extend the patcle odel to gd-bodes Undestand the equlbu of an extended object. Analyze ollng oton Undestand otaton about a fxed axs. Eploy consevaton of angula
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationTian Zheng Department of Statistics Columbia University
Haplotype Tansmsson Assocaton (HTA) An "Impotance" Measue fo Selectng Genetc Makes Tan Zheng Depatment of Statstcs Columba Unvesty Ths s a jont wok wth Pofesso Shaw-Hwa Lo n the Depatment of Statstcs at
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.
GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationThe Greatest Deviation Correlation Coefficient and its Geometrical Interpretation
By Rudy A. Gdeon The Unvesty of Montana The Geatest Devaton Coelaton Coeffcent and ts Geometcal Intepetaton The Geatest Devaton Coelaton Coeffcent (GDCC) was ntoduced by Gdeon and Hollste (987). The GDCC
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 informationAppendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk
Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating
More informationThe Unique Solution of Stochastic Differential Equations With. Independent Coefficients. Dietrich Ryter.
The Unque Soluton of Stochastc Dffeental Equatons Wth Independent Coeffcents Detch Ryte RyteDM@gawnet.ch Mdatweg 3 CH-4500 Solothun Swtzeland Phone +4132 621 13 07 SDE s must be solved n the ant-itô sense
More informationBeyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationA Study of C-Reducible Finsler Space. With Special Cases
Matheatcs Today Vol.27(June-2011)( Poc. of Maths Meet 2011) 47-54 ISSN 0976-3228 A Study of C-Reducble Fnsle Space Wth Specal Cases D. Pooja S. Saxena, D. Puneet Swaoop, E. Swat Swaoop Abstact The noton
More informationPhysics 1501 Lecture 19
Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason
More informationError Bars in both X and Y
Error Bars n both X and Y Wrong ways to ft a lne : 1. y(x) a x +b (σ x 0). x(y) c y + d (σ y 0) 3. splt dfference between 1 and. Example: Prmordal He abundance: Extrapolate ft lne to [ O / H ] 0. [ He
More informationCSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.
3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty.
More informationiclicker Quiz a) True b) False Theoretical physics: the eternal quest for a missing minus sign and/or a factor of two. Which will be an issue today?
Clce Quz I egsteed my quz tansmtte va the couse webste (not on the clce.com webste. I ealze that untl I do so, my quz scoes wll not be ecoded. a Tue b False Theoetcal hyscs: the etenal quest fo a mssng
More informationBayesian Assessment of Availabilities and Unavailabilities of Multistate Monotone Systems
Dept. of Math. Unvesty of Oslo Statstcal Reseach Repot No 3 ISSN 0806 3842 June 2010 Bayesan Assessment of Avalabltes and Unavalabltes of Multstate Monotone Systems Bent Natvg Jøund Gåsemy Tond Retan June
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Summer 2014 Fnal Exam NAME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationGroupoid and Topological Quotient Group
lobal Jounal of Pue and Appled Mathematcs SSN 0973-768 Volume 3 Numbe 7 07 pp 373-39 Reseach nda Publcatons http://wwwpublcatoncom oupod and Topolocal Quotent oup Mohammad Qasm Manna Depatment of Mathematcs
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationStellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory:
Stella Astophyscs Ovevew of last lectue: We connected the mean molecula weght to the mass factons X, Y and Z: 1 1 1 = X + Y + μ 1 4 n 1 (1 + 1) = X μ 1 1 A n Z (1 + ) + Y + 4 1+ z A Z We ntoduced the pessue
More informationEE 5337 Computational Electromagnetics (CEM)
7//28 Instucto D. Raymond Rumpf (95) 747 6958 cumpf@utep.edu EE 5337 Computatonal Electomagnetcs (CEM) Lectue #6 TMM Extas Lectue 6These notes may contan copyghted mateal obtaned unde fa use ules. Dstbuton
More informationLarge scale magnetic field generation by accelerated particles in galactic medium
Lage scale magnetc feld geneaton by acceleated patcles n galactc medum I.N.Toptygn Sant Petesbug State Polytechncal Unvesty, depatment of Theoetcal Physcs, Sant Petesbug, Russa 2.Reason explonatons The
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 informationAPPLICATIONS OF SEMIGENERALIZED -CLOSED SETS
Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,
More informationLINEAR MOMENTUM. product of the mass m and the velocity v r of an object r r
LINEAR MOMENTUM Imagne beng on a skateboad, at est that can move wthout cton on a smooth suace You catch a heavy, slow-movng ball that has been thown to you you begn to move Altenatvely you catch a lght,
More informationApplication of Complex Vectors and Complex Transformations in Solving Maxwell s Equations
Applcaton of Complex Vectos and Complex Tansfomatons n Solvng Maxwell s Equatons by Payam Saleh-Anaa A thess pesented to the Unvesty of Wateloo n fulfllment of the thess equement fo the degee of Maste
More information4 SingularValue Decomposition (SVD)
/6/00 Z:\ jeh\self\boo Kannan\Jan-5-00\4 SVD 4 SngulaValue Decomposton (SVD) Chapte 4 Pat SVD he sngula value decomposton of a matx s the factozaton of nto the poduct of thee matces = UDV whee the columns
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 information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationCOORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS
Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate
More informationScalars and Vectors Scalar
Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg
More informationCorrespondence Analysis & Related Methods
Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationAs is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.
Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationIsotope Effect in Nuclear Magnetic Resonance Spectra of Germanium Single Crystals.
Isotope Effect n Nuclea Magnetc Resonance Specta of Gemanum Sngle Cystals. Theoetcal goup: B.Z.Maln, S.K.San, Kazan State Unvesty, Russa Expemenal goups: S.V.Vehovs, A.V.Ananyev, A.P.Geasheno Insttute
More informationSOME NEW SELF-DUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15. KEYWORDS: automorphisms, construction, self-dual codes
Факултет по математика и информатика, том ХVІ С, 014 SOME NEW SELF-DUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15 NIKOLAY I. YANKOV ABSTRACT: A new method fo constuctng bnay self-dual codes wth
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationCOLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017
COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER 1 016/017 PROGRAMME SUBJECT CODE : Foundaton n Engneeng : PHYF115 SUBJECT : Phscs 1 DATE : Septembe 016 DURATION :
More information9/12/2013. Microelectronics Circuit Analysis and Design. Modes of Operation. Cross Section of Integrated Circuit npn Transistor
Mcoelectoncs Ccut Analyss and Desgn Donald A. Neamen Chapte 5 The pola Juncton Tanssto In ths chapte, we wll: Dscuss the physcal stuctue and opeaton of the bpola juncton tanssto. Undestand the dc analyss
More information