Joule-Lenz Energy of Quantum Electron Transitions Compared with the Electromagnetic Emission of Energy
|
|
- April Hart
- 5 years ago
- Views:
Transcription
1 Joural of Modr Physics, 06, 7, Publishd Oli August 06 i SciRs Joul-Lz Ergy of Quatum Elctro Trasitios Compard with th Elctromagtic Emissio of Ergy Staisław Olszwski Istitut of Physical Chmistry, Polish Acadmy of Scics Kasprzaka, Warsaw, Polad Rcivd 4 July 06; accptd 9 August 06; publishd August 06 Copyright 06 by author ad Scitific Rsarch Publishig Ic This work is licsd udr th Crativ Commos Attributio Itratioal Lics (CC BY) Abstract I th first stp, th Joul-Lz dissipatio rgy spcifid for th lctro trasitios btw two ighbourig quatum lvls i th hydrog atom has b compard with th lctromagtic rgy of missio from a sigl lvl Both th lctric ad magtic vctors trig th Poitig vctor of th lctromagtic fild ar rfrrd to th o-lctro motio prformd alog a orbit i th atom I th xt stp, a similar compariso of missio rats is prformd for th harmoic oscillator Formally a full agrmt of th Joul-Lz ad lctromagtic xprssios for th rgy missio rats has b attaid Kywords Joul-Lz Ergy, Quatum Elctro Trasitios, ydrog Atom, Elctromagtic Ergy Emissio Itroductio Usually ay calculatio of th missio rat of rgy i th atom has as its backgroud a rathr complicatd statistical-ad-probabilistic thory This situatio sms to b ot chagd much sic th vry d of th itth ad bgiig of twtth ctury []-[3] I practic a idividual atomic systm has b vr cosidrd, but istad of it a smbl of th oscillatig atoms kow as th black body was xamid Rathr automatically th tmpratur paramtr importat for comparig th thortical rsults with xprimt has b ivolvd i such may-atomic calculatios Nxt th probabilistic approach to th missio itsity foud its justificatio, ad a rathr xtdd though complicatd applicatio, i quatum haics [4] [5] Mor rctly a approach to th tratmt of th rgy missio i a sigl atomic objct could b basd o ow to cit this papr: Olszwski, S (06) Joul-Lz Ergy of Quatum Elctro Trasitios Compard with th Elctromagtic Emissio of Ergy Joural of Modr Physics, 7,
2 S Olszwski th Joul-Lz law [6]-[0] For, wh th Bohr thory of th hydrog atom is tak as a xampl, ay atom has its lctro placd o a dfiit orbit which ca b approximatd by a circl Elctrically such a circular motio ca b rprstd by a currt havig a kow itsity For xampl, for th quatum stats ad + th currt itsity is rspctivly i = ad i+ =, () T T whr T ad T + ar th tim priods of th lctro circulatio about th proto uclus I th xt stp, th rgy diffrc btw lvls + ad, amly E = E E () provids us with th lctric pottial This lads to th lctric rsistac +, + E V = (3) V V V R = (4) i i i + whr th approximat rlatios i (4) hold i virtu of i i i + (5) valid for larg Th validity of (5) bcoms vidt if w apply Formula (7) i () For such larg w hav [] th rgy chag ( ) ( ) m m + m E = ; 3 = ( + ) + i th last stp of (6) th approximatio of larg is cosidrd Sic [] w obtai 3 3 π T =, 4 m ET π π R = = = = = ; i i m V E m h 3 4 this is a costat idpdt of Th sam valu of R ca b calculatd also for othr quatum systms tha th hydrog atom, s [6] [7] [0] A charactristic poit is that R is qual to a wll-kow rsult of xprimts do o th itgr quatum all ffct [] Th Joul-Lz law is rprstd by th wll-kow rlatio E = Ri (9) t whr t is th tim itrval cssary to produc th mittd rgy E I fact (9) implis that for i = i w hav or T π π π 3 4 π h h h E m m t = = = = = = T Ri m m m Morovr from (6) ad (0) w obtai m π E t = π h 3 4 m = = (6) (7) (8) (0) () 44
3 S Olszwski Thrfor th ratio (9) bcoms ( E) h t = () E 4 8 E m m = = 3 = 6 5 t h h π (3) Rsults similar to (0)-() ca b obtaid also for othr quatum systms tha th hydrog atom [6] [7] [0] Th pricipal aim of th papr is, i th first stp, to compar th ratio calculatd i (3) with th rat of rgy missio obtaid i trms of th lctromagtic thory Nxt, i ordr to compar th quatum missio with th classical missio rat, th proprtis of th harmoic oscillator missio ar also studid Filds Iducd by th Elctro Motio i th ydrog Atom Th lctric fild valu E actig o th lctro i th Bohr atom is wll kow: m E = E = = r Th last stp i (4) is attaid bcaus of th radius of th orbit which is [] (4) r = (5) m A lss-kow magtic fild omittd i th Bohr atomic modl [0] is iducd i th hydrog atom du to th circular lctro motio do with th frqucy Bcaus of th formula (s g [3]) th idtity btw (6) ad (7) combid with (7) givs π Ω = (6) T Ω =, (7) = = A charactristic poit is that wh xprssios for E ad = + c w obtai for th lctric compot of (9) E ar substitutd to th Lortz forc (8) F E [ v ], (9) = m r = = 6 4 m ad th sam valu is obtaid for th magtic compot of (9) m v = v = = c c c o coditio th vctor of th lctro vlocity havig th valu [] is ormal to v 3 6, (0) () = () 44
4 S Olszwski 3 Fild Valus Spcific for th O-Elctro Currt Prst i th Atom ad th Elctromagtic Rat of th Ergy Emissio Our aim is to costruct th Poytig vctor which provids us with th lctromagtic disps of rgy Th vctors E ad bcom slightly difffrt tha i Sctio bcaus thy rfr to th currt bhaviour of th lctro which is circulatig alog its orbit With th pottial V giv i (3) ad (6) ad qual to 3 m V =, (3) 3 th lctric fild o th orbit havig th lgth l = πr (4) attais th valu [6] E orbit πr π π 3 5 V m m m = = = This givs a lctric vctor dirctd alog th currt O th othr had th magtic fild dirctd ormally to th currt attais th valu [4] [6] 4 3 = i m m c orbit cr = Tcr = π = π This fild diffrs from that giv i (8) solly by th factor qual to π It should b otd that paramtr r trig (6) is th radius of th circular cross-sctio ara of th orbit assumd to b qual to th cross-sctio of th lctro microparticl cosidrd as a sphr [5] [7]: r (7) Th valu of th Poytig vctor maatig th rgy from th orbit is calculatd accordig to th formula [4] P c c S = Eorbit orbit Sorbit = Eorbit orbit Sorbit 4π 4π (8) c m m = π π = π π π m π whr (5) (6) Sorbit = πrπr = 4π rr (9) is th toroidal surfac of th orbit havig th lgth (4) ad th lgth of th cross-sctio circumfrc of th orbit is qual to π r (30) I ffct w obtai from (3), (6) ad (8) th rsult prcisly qual to Formula (3) calculatd from th Joul-Lz thory Sic (3) assumd th lctro trasitios solly btw th lvls +, (3) th idtity btw (3) ad (8) implis that th limitatio to trasitio (3) applis also to th lctromagtic rsult calculatd i (8) A problm may aris to what xtt th rgy rat (3), or (8), ca b radiatd as a lctromagtic wav A altativ bhaviour is that th rgy E is spt for a haical rarragmt of th lctro positio du to th trasitio procss A argumt for that is th prsc of th lctric forc E = π m 6 orbit 5 4 (3) 443
5 S Olszwski alog th orbit Th forc (3) multiplid by th orbit lgth calculatd i (4) givs which is prcisly th rgy le 6 4 m m = π = m π orbit E of th lctro trasitio obtaid i (6) (33) 4 Quatum ad Classical Emissio Rat Calculatd for th armoic Oscillator A atural tdcy is to compar th quatum rat of th rgy missio with th classical missio rat To this purpos th o-dimsioal harmoic oscillator has b chos as a suitabl objct of xamiatio Th classical rgy of th oscillator is E osc = ka, (34) a is th oscillator amplitud; m is th oscillator mass which togthr with th forc costat k rfrs to th circular frqucy of th oscillator k π ω = = ; (35) m T T is th oscillatio priod [8] Th quatum oscillator rgy is E = + ω ω (36) (th last stp holds for larg ) ad th chag of rgy du to trasitio btw th lvls + ad is E = ω (37) Accordig to th Joul-Lz approach to th quata [6]-[0] th missio rat btw th lvls + ad is so This givs ( E) ( ω) E = = t h h E ω hν = = = = ν =, t h h h T (38) (39) t = T, (40) bcaus th rfrc btw ω ad ν is ω = π ν (4) Th pottial V coctd with th rgy chag E is E ω V = = (4) If w ot that a maximal distac travlld by th lctro oscillator i o dirctio is l = a, (43) th lctric fild coctd with th oscillator paralll to its motio is V V ω E = = = (44) l a a Th lctric currt lt b cosidrd as rmaiig approximatly costat i cours of th oscillatio I this cas th magtic fild which is ormal to th currt [s (6)] is 444
6 S Olszwski i ν = = = = =, (45) cr Tcr Tc T sic th cross-sctio of th lctro currt is assumd to b idtical with th cross-sctio ara of th lctro microparticl, s (7) Th surfac ara of th sampl cotaiig th oscillator is o coditio th cotributio of th d aras of th sampl surfac qual to S = πrl = 4πra = 4π a, (46) πr = π 4π a (47) has b glctd bcaus (47) is a small umbr i compariso with S i (46) I cosquc, for th vctor ormal to vctor E th valu of th Poytig vctor bcoms 4π a P c c S ω 4π S = E = = ω = E 4π 4π a T 4πT T This is a rsult idtical with (38) o coditio Formula (39) is tak ito accout Accordig to th classical lctrodyamics [9] th missio rat of rgy from a classical oscillator is sic 4 4 de ω π a = p = 3 3 dt 3 c 3 T c p (48) (49) = a (50) is th dipol momt of th classical harmoic oscillator Formula (49) ca b compard with th quatum approach to th Joul-Lz missio rat of rgy [s (38)]: ( ω) E h h = = = t h T h T I th cas of vry small quatum systms th amplitud a i (49) ca b clos to its miimal lgth [0] ad th tim priod T ca approach its miimal siz [0] Th quality rquird btw (49) ad (5) lads to th rlatio a (5) (5) T (53) 4 π a h = 3 3 T c T Wh a ad T ar tak rspctivly from (5) ad (53), Formula (54) bcoms ( π) ( π) ( π) = = = h c 3 c c 3 c from which w hav th rlatio ( ) 3 c π 65 = = 3 α (54) (55) (56) 445
7 S Olszwski Th rsult obtaid i (56) diffrs by oly 0 prct from th rciprocal valu of th atomic costat qual to 37 5 Ratio of th Classical ad Quatum Emissio Rat Dfid by th Dampig Cofficit of th Classical Radiatio A attmpt of this Sctio is to dmostrat that th classical missio ca b cosidrd as a dampd quatum missio rat Th classical dampig cofficit of th oscillator is [9] γ 3 = ω (57) 3 O th othr had, th classical missio rat giv i (49) ca b modifid wh th amplitud a trig (49) is xprssd i trms of th oscillator rgy E [8]: E E a = k = ω mω = mω = (58) mω r, at th d of (58), th rgy E is rplacd by th approximat quatum formula for th oscillator rgy giv i (36) I ffct th classical missio rat i (49) bcoms 3 class ω η = ω a = ω = (59) c 3c mω 3c m Aothr trasformatio may cocr th quatum missio rat i (5): η quat As a rsult of (59) ad (60) w obtai th ratio E h π h ω = = = = t T T π ( π) class 3 η 4 ω ω 8π ω ω = : = = 4π = 4πγ = Tγ (6) quat η 3 c m π 3 c m 3 cmω ω which is proportioal to γ i (57) A multipl of th oscillatio tim priod T is th proportioality cofficit rprstig (6) i trms of γ Thrfor aothr way to writ (6) ca b η (60) class quat = Tγη (6) Lt us ot that E trig (36) ad (58) is proportioal to It is worth to ot that th Eisti cofficit A α of th missio probability ca b coupld with γ by th rlatio [0] so A hν = γ f hν (63) α A α, α = γ f (64), α Accordig to isbrg [0] [] w hav ( ) h h a, ν (, ) = ν (, ) = f, (65) πmω 8π m whr a is th quatum-thortical amplitud of th xpasio of th coordiat x x( t) = of a aharmoic oscillator; ω 0 is th circular frqucy of th harmoic oscillator For small pturbatio λ of th oscillator w hav [] ω = π ν, ω, (66) so Formula (65) givs 0 ( ) 0 hω hω h h = = f, (67) π m π m π m 8π m 0 ω0 ω0 446
8 S Olszwski or 4 = f (68), I ffct for α = tak i (63) w obtai from (64) ad (68): If γ is prstd, accordig to (6), i trms of th ratio of whr T is th oscillatio tim priod of th harmoic oscillator A = γ f, = 4 γ (69) class η ad quat η, w obtai class class 4 A η η = =, (70) quat quat T η T η 6 Rciprocal Valu of th Atomic Costat ad th Elctro Spi Th rciprocal valu α of th atomic costat (~37) approachd i (56) is importat i th tratmt of th lctro spi [0] [] [3] W show blow that th magtic fild itsity cssary to produc th lctro spi ca b obtaid approximatly as a rsult of a couplig of α with th radius r of th lctro microparticl, s (7) Accordig to th classical lctrodyamics [4] th magtic fild at a distat r from th ctr of th liar wir carryig a currt i is coupld with i ad r by th formula i = (7) cr If th currt i is flowig o a surfac of th coductor which is th lctro orbit, w ca assum that r = r which is both th radius of th lctro microparticl ad cross-sctio of th orbit Th fild bcoms i this cas [4] = = = Tcr c 4 3 m 3 3 π π whr th tim priod T of th lctro circulatio alog th orbit is tak from Formula (7) for = : 3 π T = T = 4 m Th ssc of th spi ffct is that th path of th spiig lctro circumvts th lctro orbit about α c = 37 tims durig th tim priod T idicatd i (73) I classical lctrodyamics this mas that th magtic fild producd i this way is α tims strogr tha that obtaid i (7): spi = c α = π = π, (7) (73) (74) (75) Th rsult i (75) diffrs solly by th factor of π from th magtic fild assumd to produc a spiig lctro particl i [0] [] [3]: 3 spi = A discrpacy btw (75) ad (76) ca b ascribd to som ucrtaity coctd with th calculatio of th radius r, s [4] 7 Coclusios Th aim of th papr was to gt mor isight ito a o-probabilistic dscriptio of th trasfr of rgy (76) 447
9 S Olszwski btw two quatum lvls A suitabl situatio for discussio is th cas wh th lvls ar ighbourig i thir mutual positio of th rgy stats Th th rgy chag ( E ) btw th lvls, ad th tim itrval t cssary to attai E, satisfy a vry simpl formula E t = h; (77) s [6]-[0] I th papr, Formula (77) fids its coutrparts supplid by th lctromagtic thory of missio Two physical objcts, amly th hydrog atom ad lctro harmoic oscillator, wr studid Th cas of th lctro oscillator allowd us to prform a mor dirct compariso of th quatum approach to th missio rat with th classical lctromagtic thory It occurs that th classical rat is qual to th quatum rat multiplid by th Bor dampig cofficit ad a itrval of tim, s (6) Rfrcs [] Plack, M (90) Acht Vorlsug ubr Thortisch Physik S irzl, Lipzig [] Eisti, A (97) Physikalisch Zitschrift, 8, [3] Bohr, N (967) O th Quatum Thory of Li Spctra I: Va dr Ward, BL, Ed, Sourcs of Quatum Mchaics, Dovr Publicatios, Nw York, [4] Bth, (933) Quathaik dr Ei- ud Zwi-Elktroproblm I: Gigr, ad Schl, K, Eds, adbuch dr Physik, Vol 4, Part, Sprigr, Brli, [5] Codo, EU ad Shortly, G (970) Th Thory of Atomic Spctra Cambridg Uivrsity Prss, Cambridg, UK [6] Olszwski, S (05) Joural of Modr Physics, 6, [7] Olszwski, S (06) Joural of Modr Physics, 7, [8] Olszwski, S (06) Joural of Modr Physics, 7, [9] Olszwski, S (06) Joural of Modr Physics, 7, [0] Olszwski, S (06) Rviws i Thortical Scic, 4, [] Sommrfld, A (93) Atombau ud Spktrallii 5th Editio, Vol, Viwg, Brauschwig [] MacDoald, A (989) Quatum all Effct A Prspctiv Kluwr, Milao [3] Slatr, JC (967) Quatum Thory of Molculs ad Solids Vol 3, McGraw-ill, Nw York [4] Lass, (950) Vctor ad Tsor Aalysis McGraw-ill, Nw York [5] Matvv, AN (964) Elctrodyamics ad th Thory of Rlativity Izd Wyzszaja Szkola, Moscow (I Russia) [6] Grir, W (998) Classical Elctrodyamics Sprigr, Nw York [7] Ladau, LD ad Lifshits, EM (969) Mchaics Elctrodyamics Izd Nauka, Moscow (I Russia) [8] Sommrfld, A (943) Mchaik Akadmisch Vrlagsgsllschaft, Lipzig [9] Bor, M (933) Optik Sprigr, Brli [0] Va dr Ward, BL (967) Itroductio I: Va dr Ward, BL, Ed, Sourcs of Quatum Mchaics, Dovr Publicatios, Nw York [] isbrg, W (95) Zitschrift fur Physik, 33, [] Olszwski, S (04) Joural of Modr Physics, 5, [3] Olszwski, S (04) Joural of Modr Physics, 5, [4] Olszwski, S (06) Joural of Modr Physics, 7,
10 Submit or rcommd xt mauscript to SCIRP ad w will provid bst srvic for you: Accptig pr-submissio iquiris through , Facbook, LikdI, Twittr, tc A wid slctio of jourals (iclusiv of 9 subjcts, mor tha 00 jourals) Providig 4-hour high-quality srvic Usr-fridly oli submissio systm Fair ad swift pr-rviw systm Efficit typsttig ad proofradig procdur Display of th rsult of dowloads ad visits, as wll as th umbr of citd articls Maximum dissmiatio of your rsarch work Submit your mauscript at:
Blackbody Radiation. All bodies at a temperature T emit and absorb thermal electromagnetic radiation. How is blackbody radiation absorbed and emitted?
All bodis at a tmpratur T mit ad absorb thrmal lctromagtic radiatio Blackbody radiatio I thrmal quilibrium, th powr mittd quals th powr absorbd How is blackbody radiatio absorbd ad mittd? 1 2 A blackbody
More informationSession : Plasmas in Equilibrium
Sssio : Plasmas i Equilibrium Ioizatio ad Coductio i a High-prssur Plasma A ormal gas at T < 3000 K is a good lctrical isulator, bcaus thr ar almost o fr lctros i it. For prssurs > 0.1 atm, collisio amog
More informationcoulombs or esu charge. It s mass is about 1/1837 times the mass of hydrogen atom. Thus mass of electron is
1 ATOMIC STRUCTURE Fudamtal Particls: Mai Fudamtal Particl : (a) Elctro: It is a fudamtal particl of a atom which carris a uit gativ charg. It was discovrd by J.J. Thomso (1897) from th studis carrid out
More informationBlackbody Radiation. All bodies at a temperature T emit and absorb thermal electromagnetic radiation. How is blackbody radiation absorbed and emitted?
All bodis at a tmpratur T mit ad absorb thrmal lctromagtic radiatio Blackbody radiatio I thrmal quilibrium, th powr mittd quals th powr absorbd How is blackbody radiatio absorbd ad mittd? 1 2 A blackbody
More information1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
More informationIdeal crystal : Regulary ordered point masses connected via harmonic springs
Statistical thrmodyamics of crystals Mooatomic crystal Idal crystal : Rgulary ordrd poit masss coctd via harmoic sprigs Itratomic itractios Rprstd by th lattic forc-costat quivalt atom positios miima o
More informationElectromagnetic radiation and steady states of hydrogen atom
Elctromagtic radiatio ad stady stats of hydrog atom Jiaomig Luo Egirig Rsarch Ctr i Biomatrials, Sichua Uivrsity, 9# Wagjiag Road, Chgdu, Chia, 610064 Abstract. Elctromagtic phoma i hydrog atom ar cotrolld
More informationDTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
More informationECE594I Notes set 6: Thermal Noise
C594I ots, M. odwll, copyrightd C594I Nots st 6: Thrmal Nois Mark odwll Uivrsity of Califoria, ata Barbara rodwll@c.ucsb.du 805-893-344, 805-893-36 fax frcs ad Citatios: C594I ots, M. odwll, copyrightd
More informationA Simple Proof that e is Irrational
Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural
More informationz 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
More informationAPPENDIX: STATISTICAL TOOLS
I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.
More informationOn the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
More informationMILLIKAN OIL DROP EXPERIMENT
11 Oct 18 Millika.1 MILLIKAN OIL DROP EXPERIMENT This xprimt is dsigd to show th quatizatio of lctric charg ad allow dtrmiatio of th lmtary charg,. As i Millika s origial xprimt, oil drops ar sprayd ito
More informationNEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES
Digst Joural of Naomatrials ad Biostructurs Vol 4, No, March 009, p 67-76 NEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES A IRANMANESH a*, O KHORMALI b, I NAJAFI KHALILSARAEE c, B SOLEIMANI
More informationChapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering
haptr. Physical Problm for Fast Fourir Trasform ivil Egirig Itroductio I this chaptr, applicatios of FFT algorithms [-5] for solvig ral-lif problms such as computig th dyamical (displacmt rspos [6-7] of
More informationThey must have different numbers of electrons orbiting their nuclei. They must have the same number of neutrons in their nuclei.
37 1 How may utros ar i a uclus of th uclid l? 20 37 54 2 crtai lmt has svral isotops. Which statmt about ths isotops is corrct? Thy must hav diffrt umbrs of lctros orbitig thir ucli. Thy must hav th sam
More informationDigital Signal Processing, Fall 2006
Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti
More informationPhysics 2D Lecture Slides Lecture 14: Feb 3 rd 2004
Bria Wcht, th TA is back! Pl. giv all rgrad rqusts to him Quiz 4 is This Friday Physics D Lctur Slids Lctur 14: Fb 3 rd 004 Vivk Sharma UCSD Physics Whr ar th lctros isid th atom? Early Thought: Plum puddig
More informationTriple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical
More informationDiscrete Fourier Transform (DFT)
Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial
More informationPURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
More informationHadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms
Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of
More informationProbability & Statistics,
Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said
More informationH2 Mathematics Arithmetic & Geometric Series ( )
H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic
More informationln x = n e = 20 (nearest integer)
H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77
More informationBipolar Junction Transistors
ipolar Juctio Trasistors ipolar juctio trasistors (JT) ar activ 3-trmial dvics with aras of applicatios: amplifirs, switch tc. high-powr circuits high-spd logic circuits for high-spd computrs. JT structur:
More informationEE 232 Lightwave Devices Lecture 3: Basic Semiconductor Physics and Optical Processes. Optical Properties of Semiconductors
3 Lightwav Dvics Lctur 3: Basic Smicoductor Physics ad Optical Procsss Istructor: Mig C. Wu Uivrsity of Califoria, Brly lctrical girig ad Computr Scics Dpt. 3 Lctur 3- Optical Proprtis of Smicoductors
More information5.1 The Nuclear Atom
Sav My Exams! Th Hom of Rvisio For mor awsom GSE ad lvl rsourcs, visit us at www.savmyxams.co.uk/ 5.1 Th Nuclar tom Qustio Papr Lvl IGSE Subjct Physics (0625) Exam oard Topic Sub Topic ooklt ambridg Itratioal
More informationLECTURE 13 Filling the bands. Occupancy of Available Energy Levels
LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad
More informationDiscrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform
Discrt Fourir Trasform Dfiitio - T simplst rlatio btw a lt- squc x dfid for ω ad its DTFT X ( ) is ω obtaid by uiformly sampli X ( ) o t ω-axis btw ω < at ω From t dfiitio of t DTFT w tus av X X( ω ) ω
More informationPhysics of the Interstellar and Intergalactic Medium
PYA0 Sior Sophistr Physics of th Itrstllar ad Itrgalactic Mdium Lctur 7: II gios Dr Graham M. arpr School of Physics, TCD Follow-up radig for this ad t lctur Chaptr 5: Dyso ad Williams (lss dtaild) Chaptr
More information10. Joint Moments and Joint Characteristic Functions
0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi
More informationStatistics 3858 : Likelihood Ratio for Exponential Distribution
Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai
More informationJournal of Modern Applied Statistical Methods
Joural of Modr Applid Statistical Mthods Volum Issu Articl 6 --03 O Som Proprtis of a Htrogous Trasfr Fuctio Ivolvig Symmtric Saturatd Liar (SATLINS) with Hyprbolic Tagt (TANH) Trasfr Fuctios Christophr
More informationBohr type models of the atom give a totally incorrect picture of the atom and are of only historical significance.
VISUAL PHYSICS ONLIN BOHR MODL OF TH ATOM Bhr typ mdls f th atm giv a ttally icrrct pictur f th atm ad ar f ly histrical sigificac. Fig.. Bhr s platary mdl f th atm. Hwvr, th Bhr mdls wr a imprtat stp
More informationCircular Array of Tapered Nylon Rod Antennas: A Computational Study
tratioal Joural of Elctroics ad Commuicatio Egirig. SSN 974-266 Volum 4, Numbr (2), pp.3-38 tratioal Rsarch Publicatio Hous http://www.irphous.com Circular Array of Taprd Nylo Rod Atas: A Computatioal
More information8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions
Solutios for HW 8 Captr 5 Cocptual Qustios 5.. θ dcrass. As t crystal is coprssd, t spacig d btw t plas of atos dcrass. For t first ordr diffractio =. T Bragg coditio is = d so as d dcrass, ust icras for
More information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
More informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More informationTime : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120
Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,
More informationPartition Functions and Ideal Gases
Partitio Fuctios ad Idal Gass PFIG- You v lard about partitio fuctios ad som uss ow w ll xplor tm i mor dpt usig idal moatomic diatomic ad polyatomic gass! for w start rmmbr: Q( N ( N! N Wat ar N ad? W
More informationDiscrete Fourier Transform. Nuno Vasconcelos UCSD
Discrt Fourir Trasform uo Vascoclos UCSD Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if
More informationFigure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor
.8 NOISE.8. Th Nyquist Nois Thorm W ow wat to tur our atttio to ois. W will start with th basic dfiitio of ois as usd i radar thory ad th discuss ois figur. Th typ of ois of itrst i radar thory is trmd
More information(Reference: sections in Silberberg 5 th ed.)
ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists
More informationLectures 9 IIR Systems: First Order System
EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work
More informationFolding of Hyperbolic Manifolds
It. J. Cotmp. Math. Scics, Vol. 7, 0, o. 6, 79-799 Foldig of Hyprbolic Maifolds H. I. Attiya Basic Scic Dpartmt, Collg of Idustrial Educatio BANE - SUEF Uivrsity, Egypt hala_attiya005@yahoo.com Abstract
More informationThe Interplay between l-max, l-min, p-max and p-min Stable Distributions
DOI: 0.545/mjis.05.4006 Th Itrplay btw lma lmi pma ad pmi Stabl Distributios S Ravi ad TS Mavitha Dpartmt of Studis i Statistics Uivrsity of Mysor Maasagagotri Mysuru 570006 Idia. Email:ravi@statistics.uimysor.ac.i
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationOn a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.
O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck
More informationChapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
More informationA Mathematical Study of Electro-Magneto- Thermo-Voigt Viscoelastic Surface Wave Propagation under Gravity Involving Time Rate of Change of Strain
Thortical Mathmatics & Applicatios vol.3 o.3 3 87-6 ISSN: 79-9687 (prit) 79-979 (oli) Sciprss Ltd 3 A Mathmatical Study of Elctro-Magto- Thrmo-Voigt Viscolastic Surfac Wav Propagatio udr Gravity Ivolvig
More informationMONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx
MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of
More informationWashington State University
he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us
More informationChapter (8) Estimation and Confedence Intervals Examples
Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit
More informationSolution to 1223 The Evil Warden.
Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud
More informationSOLUTIONS TO CHAPTER 2 PROBLEMS
SOLUTIONS TO CHAPTER PROBLEMS Problm.1 Th pully of Fig..33 is composd of fiv portios: thr cylidrs (of which two ar idtical) ad two idtical co frustum sgmts. Th mass momt of irtia of a cylidr dfid by a
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationNET/JRF, GATE, IIT JAM, JEST, TIFR
Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k. kφ kφ lφ kφ lφ (a) ψ (b) ψ kφ (c) ψ
More informationOutline. Ionizing Radiation. Introduction. Ionizing radiation
Outli Ioizig Radiatio Chaptr F.A. Attix, Itroductio to Radiological Physics ad Radiatio Dosimtry Radiological physics ad radiatio dosimtry Typs ad sourcs of ioizig radiatio Dscriptio of ioizig radiatio
More informationElectronic Supplementary Information
Elctroic Supplmtary Matrial (ESI) for Joural of Matrials Chmistry A. This joural is Th Royal Socity of Chmistry 2016 Elctroic Supplmtary Iformatio Photolctrochmical Watr Oxidatio usig a Bi 2 MoO 6 / MoO
More information1 of 42. Abbreviated title: [SAP-SVT-Nmsm-g & 137] - Updated on 31 July, 09. Shankar V.Narayanan
1 of 4 ONE EQUATION ad FOUR Subatomic Particls ad thir FOUR Itractios icludig (g &17) factors with Spac Vortx Thory (A No matrial shll modl) (Part 1 of ) (Th cotts of this txt ar th sam as i Subatomic
More informationRestricted Factorial And A Remark On The Reduced Residue Classes
Applid Mathmatics E-Nots, 162016, 244-250 c ISSN 1607-2510 Availabl fr at mirror sits of http://www.math.thu.du.tw/ am/ Rstrictd Factorial Ad A Rmark O Th Rducd Rsidu Classs Mhdi Hassai Rcivd 26 March
More informationOption 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.
Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt
More informationTopological Insulators in 2D and 3D
Topological Isulators i D ad 3D 0. Elctric polarizatio, Chr Numbr, Itgr Quatum Hall Effct I. Graph - Halda modl - Tim rvrsal symmtry ad Kramrs thorm II. D quatum spi Hall isulator - Z topological ivariat
More informationA Strain-based Non-linear Elastic Model for Geomaterials
A Strai-basd No-liar Elastic Modl for Gomatrials ANDREW HEATH Dpartmt of Architctur ad Civil Egirig Uivrsity of Bath Bath, BA2 7AY UNITED KINGDOM A.Hath@bath.ac.uk http://www.bath.ac.uk/ac Abstract: -
More informationELECTRON-MUON SCATTERING
ELECTRON-MUON SCATTERING ABSTRACT Th lctron charg is considrd to b distributd or xtndd in spac. Th diffrntial of th lctron charg is st qual to a function of lctron charg coordinats multiplid by a four-dimnsional
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationChapter 4 - The Fourier Series
M. J. Robrts - 8/8/4 Chaptr 4 - Th Fourir Sris Slctd Solutios (I this solutio maual, th symbol,, is usd for priodic covolutio bcaus th prfrrd symbol which appars i th txt is ot i th fot slctio of th word
More informationUNIT 2: MATHEMATICAL ENVIRONMENT
UNIT : MATHEMATICAL ENVIRONMENT. Itroductio This uit itroducs som basic mathmatical cocpts ad rlats thm to th otatio usd i th cours. Wh ou hav workd through this uit ou should: apprciat that a mathmatical
More informationInternational Journal of Advanced and Applied Sciences
Itratioal Joural of Advacd ad Applid Scics x(x) xxxx Pags: xx xx Cotts lists availabl at Scic Gat Itratioal Joural of Advacd ad Applid Scics Joural hompag: http://wwwscic gatcom/ijaashtml Symmtric Fuctios
More information10. Excitons in Bulk and Two-dimensional Semiconductors
Excitos i Bulk ad Two-dimsioal Smicoductors Th Wair modl drivd i th prvious chaptr provids a simpl framwork for th iclusio of xcitos i th optical proprtis of smicoductors I this chaptr w will valuat th
More informationTime Dependent Solutions: Propagators and Representations
Tim Dpdt Solutios: Propagators ad Rprstatios Michal Fowlr, UVa 1/3/6 Itroductio W v spt most of th cours so far coctratig o th igstats of th amiltoia, stats whos tim dpdc is mrly a chagig phas W did mtio
More informationAvailable online at Energy Procedia 4 (2011) Energy Procedia 00 (2010) GHGT-10
Availabl oli at www.scicdirct.com Ergy Procdia 4 (01 170 177 Ergy Procdia 00 (010) 000 000 Ergy Procdia www.lsvir.com/locat/procdia www.lsvir.com/locat/xxx GHGT-10 Exprimtal Studis of CO ad CH 4 Diffusio
More information1 of 46. Abbreviated title: [SAP-SVT-Nmsm-g & 137] - Updated on 07 Oct, 09. Shankar V.Narayanan
1 of 46 Subatomic Particls ad thir FOUR Itractios icludig (g &17) (p&) factors with Spac Vortx Thory (A No matrial shll modl) (Part 1 of ) (Th cotts of this txt ar th sam as i ONE EQUATION ad FOUR Subatomic
More informationEmpirical Study in Finite Correlation Coefficient in Two Phase Estimation
M. Khoshvisa Griffith Uivrsity Griffith Busiss School Australia F. Kaymarm Massachustts Istitut of Tchology Dpartmt of Mchaical girig USA H. P. Sigh R. Sigh Vikram Uivrsity Dpartmt of Mathmatics ad Statistics
More informationInformation Theory Model for Radiation
Joural of Applied Mathematics ad Physics, 26, 4, 6-66 Published Olie August 26 i SciRes. http://www.scirp.org/joural/jamp http://dx.doi.org/.426/jamp.26.487 Iformatio Theory Model for Radiatio Philipp
More informationQuantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points)
Chm 5 Problm St # ANSWER KEY 5 qustios, poits Qutum Mchics & Spctroscopy Prof. Jso Goodpstr Du ridy, b. 6 S th lst pgs for possibly usful costts, qutios d itgrls. Ths will lso b icludd o our futur ms..
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationNarayana IIT Academy
INDIA Sc: LT-IIT-SPARK Dat: 9--8 6_P Max.Mars: 86 KEY SHEET PHYSIS A 5 D 6 7 A,B 8 B,D 9 A,B A,,D A,B, A,B B, A,B 5 A 6 D 7 8 A HEMISTRY 9 A B D B B 5 A,B,,D 6 A,,D 7 B,,D 8 A,B,,D 9 A,B, A,B, A,B,,D A,B,
More informationPPS (Pottial Path Spac) i y i l j Vij (2) H x PP (Pottial Path ra) (gravity-typ masur) i i i j1 cij (1) D j j c ij ij 4)7) 8), 9) D j V ij j i 198 1)1
1 2 3 1 (68-8552 4 11) E-mail: taimoto@ss.tottori-u.ac.jp 2 (68-8552 4 11) 3 (657-851 1-1) Ky Words: accssibility, public trasportatio plaig, rural aras, tim allocatio, spac-tim prism 197 Hady ad Nimir
More informationOn the Reformulated Zagreb Indices of Certain Nanostructures
lobal Joural o Pur ad Applid Mathmatics. ISSN 097-768 Volum, Numbr 07, pp. 87-87 Rsarch Idia Publicatios http://www.ripublicatio.com O th Rormulatd Zagrb Idics o Crtai Naostructurs Krthi. Mirajar ad Priyaa
More informationSOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C
Joural of Mathatical Aalysis ISSN: 2217-3412, URL: www.ilirias.co/ja Volu 8 Issu 1 2017, Pags 156-163 SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C BURAK
More information2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005
Mark Schm 67 Ju 5 GENERAL INSTRUCTIONS Marks i th mark schm ar plicitly dsigatd as M, A, B, E or G. M marks ("mthod" ar for a attmpt to us a corrct mthod (ot mrly for statig th mthod. A marks ("accuracy"
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationS- AND P-POLARIZED REFLECTIVITIES OF EXPLOSIVELY DRIVEN STRONGLY NON-IDEAL XENON PLASMA
S- AND P-POLARIZED REFLECTIVITIES OF EXPLOSIVELY DRIVEN STRONGLY NON-IDEAL XENON PLASMA Zaporozhts Yu.B.*, Mitsv V.B., Gryazov V.K., Riholz H., Röpk G. 3, Fortov V.E. 4 Istitut of Problms of Chmical Physics
More informationReview Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
More informationPeriodic Structures. Filter Design by the Image Parameter Method
Prioic Structurs a Filtr sig y th mag Paramtr Mtho ECE53: Microwav Circuit sig Pozar Chaptr 8, Sctios 8. & 8. Josh Ottos /4/ Microwav Filtrs (Chaptr Eight) microwav filtr is a two-port twork us to cotrol
More information3.1 Atomic Structure and The Periodic Table
Sav My Exams! Th Hom of Rvisio For mor awsom GSE ad lvl rsourcs, visit us at www.savmyxams.co.uk/ 3. tomic Structur ad Th Priodic Tabl Qustio Par Lvl IGSE Subjct hmistry (060) Exam oard ambridg Itratioal
More informationNormal Form for Systems with Linear Part N 3(n)
Applid Mathmatics 64-647 http://dxdoiorg/46/am7 Publishd Oli ovmbr (http://wwwscirporg/joural/am) ormal Form or Systms with Liar Part () Grac Gachigua * David Maloza Johaa Sigy Dpartmt o Mathmatics Collg
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationω (argument or phase)
Imagiary uit: i ( i Complx umbr: z x+ i y Cartsia coordiats: x (ral part y (imagiary part Complx cougat: z x i y Absolut valu: r z x + y Polar coordiats: r (absolut valu or modulus ω (argumt or phas x
More informationKey words Non-uniform; specific energy; critical; gradually-varied steady flow; water surface profiles
Chaptr NON-UNIFORM FLOW 4.. Itroductio 4.. Gradually-varid stady 4.3. Typs of watr surfac profils 4.4. Drawig watr surfac profils Summary Likig up with Chaptr, dalig with uiform i op chals, it may b otd
More information3.4 Properties of the Stress Tensor
cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato
More informationSuperfluid Liquid Helium
Surfluid Liquid Hlium:Bo liquid ad urfluidity Ladau thory: two fluid modl Bo-iti Codatio ad urfluid ODLRO, otaou ymmtry brakig, macrocoic wafuctio Gro-Pitakii GP quatio Fyma ictur Rfrc: Thory of quatum
More informationHomotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
More informationReliability of time dependent stress-strength system for various distributions
IOS Joural of Mathmatcs (IOS-JM ISSN: 78-578. Volum 3, Issu 6 (Sp-Oct., PP -7 www.osrjourals.org lablty of tm dpdt strss-strgth systm for varous dstrbutos N.Swath, T.S.Uma Mahswar,, Dpartmt of Mathmatcs,
More informationA Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction
Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.
More informationChemical Physics II. More Stat. Thermo Kinetics Protein Folding...
Chmical Physics II Mor Stat. Thrmo Kintics Protin Folding... http://www.nmc.ctc.com/imags/projct/proj15thumb.jpg http://nuclarwaponarchiv.org/usa/tsts/ukgrabl2.jpg http://www.photolib.noaa.gov/corps/imags/big/corp1417.jpg
More information