Using Quantum Mechanics in Simple Systems Chapter 15
|
|
- Pauline May
- 5 years ago
- Views:
Transcription
1 /16/17 Qutiztio rises whe the loctio of prticle (here electro) is cofied to dimesiolly smll regio of spce qutum cofiemet. Usig Qutum Mechics i Simple Systems Chpter 15 The simplest system tht c be cosidered to study the effect of cofiemet is to cosider sigle electro system restricted to spce i sigle dimesio. To uderstd the chrcteristics of prticle i oe dimesiol bo system it is ecessry to solve the SE. Solvig Schrodiger equtio (SE) yields s which c be used to obti vlues for mesurble qutities like eergy. The prticle i 1D bo Uderstdig of the solutio to the prticle i 1-D bo cse, would llow us to gi some isights ito the ture of qutum systems. Tht is, the ide of qutum behvior d the pperce of discrete sttes with specific eergies. 1D bo system: A coceptully simple system where electro cofied to spce i directio of legth d completely free (of zero potetil eergy) to move i tht spce of legth but ot llowed to move outside the spce of legth. origi = The prticle cot leve the bo, i eergy terms, the prticle is icpble of overcomig eormously lrge eergy brrier for > >. Tht is, for > > the potetil eergy V =, d for < <, V =. 1
2 /16/17 Solvig for s of strtig with SE Usig SE; The potetil eergy outside the lie is mkig the prticle totlly cofied withi the lie, cot peetrte the potetil brrier. (V iside = for coveiece, c be y costt.) Iside the bo, V() = H d ( ) k ( ) d Solvig the SE would yield s s solutios for llowed sttes. Solutios to this differetil equtio re of the bove form: Solvig the SE would yield s, wvefuctios s solutios for llowed sttes. SE is lier, the of is solutio the b is solutio, b = costt. The lierity of the SE is importt d the cosequece is ot trivil. This lierity is i qutum mechicl mplitude. Determiig k, A d B For to be well behved fuctio, for > >. At = d = ; the wve fuctio is ot llowed to cquire vlue d lso cot become ifiite. Thus () = d () = (boudry coditios). Apply BC; A geerl solutio of SE is ot ormlized fuctio. Lierity llows ormliztio of the wve fuctio. which mkes, A si k =
3 /16/17 Also; A, if ot there is o fuctio t ll; which mkes, si( k) k k Where = 1,, d. ( ) = Asi = qutum umber Ech defies stte. A wvefuctio is ssocited with stte boud stte, meig the prticle s eistece is cofied/boud ito the bo. Qutum umbers re the outcome of the pplictio of boudry coditios i the solutio of the differetil equtios. For the prticle i 1D bo, stte is ssocited with sigle qutum umber,. A sigle qutum umber is geerted becuse there is oly oe set of boudry coditios ivolved here. Geerlly ech set of boudry coditios led to qutum umber. A geerl solutio of SE is ot ormlized fuctio. Normliztio to fid the Amplitude A. ( ) A =? =Asi ( ) d = 1 = 1,, d. ( ) d A si d = 1 A si d 1 A 1 A si Complete set of wvefuctios: defies stte with the ssocited stte fuctio. For prticle i ifiitely deep potetil well there re ifiite umber of boud sttes. Prticle i ever free i.e. lwys cofied ito the bo. Full wve fuctio is of the form: spce * time 3
4 /16/17 Well behved s. The solutios of the prticle i bo (SE) forms complete set of fuctios. Note; the wve fuctios (solutios of SE equtio) bove re eige fuctios of the totl eergy opertor. Wve fuctios tht re eige fuctios of certi opertor re ot ecessrily be eige fuctios of other opertor; they my be eige fuctios of the other opertor or my be ot. Stdig wves Note withi ; #odes = -1 The time evolvig wve fuctio, - stdig wves. The time evolvig wve fuctio, - stdig wves. Full wve fuctio - form: spce * time * * * Some trjectories of prticle i bo ccordig to Newto's lws of clssicl mechics (A), d ccordig to the Schrödiger equtio of qutum mechics (B-F). I (B-F), the horizotl is is positio, d the verticl is is the rel prt (blue) d imgiry prt (red) of the wvefuctio. The sttes (B,C,D) re eergy eigesttes, but (E,F) re ot. The time vritio of the wve fuctios will be ectly s previously see. = 8 Oe prt show. bo 4
5 /16/17 Clcultio of eergy of sttes 1D Bo: Eergy opertor = H H ( ) E( ) H ( ) E ( ) Time idepedet Schrodiger Equtio Key to clcultio of observbles H ( ) E ( ) d d si md md d si md m H E Note eigeeergies re of precise vlues. Mesurig the eergy of eigestte lwys produce the sme vlue. Eergy qutized d icreses with. Lowest eergy o-zero! Zero poit eergy. BC dicttes the umber of wve legths betwee two cosecutive sttes is /. Wve vector: me k Eergy Stbility h 16 8 m h 9 8 m h 4 8 m h 1 8 m 5
6 /16/17 Prity of fuctios The eigefuctios re symmetricl. = 1, 3 mirror imge reltioship, eve prity, eve fuctio =, 4 iverted reltioship, odd prity, odd fuctio Fuctios with differet prities re lwys orthogol. So itegrls ivolvig products of fuctios with differet prities re zero. Mthemticlly useful 1 me ( V( )) h 1 me h 1 me me h h me k For to be well behved fuctio, t > >, d t = d = the wve fuctio is ot llowed to become ifiite. Thus () = d () = (boudry coditios). The boudry coditio tured the eigefuctios of SE from trvellig wves to stdig wves ssocite with discrete vlues for eergy; h E 8m 6
7 /16/17 Probbility desity = Probbility i spce d * d d = d * *( ) ( ) d The qutiztio of eergy levels rises s turl outcome of the solutio of SE, s opposed to rbitrry ssumptio (Bohr model). The wvefuctios ssocited with the eergy levels re turl cosequece of the mthemticl solutio of the SE. SE of y system, more complicted th the 1D prticle i bo would hve the sme geerl properties d chrcteristics. It is the cofiemet of the prticle by wy of trppig the prticle i potetil well tht leds to qutiztio! As icreses qutum system morphs ito clssicl system. =5 Further liftig the cofiemet ( >> ) mkes qutum system rech it s clssicl limit - correspodece priciple. Prticle foud with ~sme probbility every where. = Correspodece Priciple. =1 7
8 /16/17 Rmifictio of completeess of the set of wve fuctios The solutios of the prticle i bo (eigefuctios ) forms complete set (of fuctios). for ll. Ay wve fuctio f() tht stisfies the requiremets of cceptble well behved wve fuctio c be costructed from the members of the complete set of fuctios. The costructed wve fuctio would be sum with ech term i the sum is ssocited with coefficiet (i.e. weight/mplitude) tht is mesure of the cotributio by tht member. Ay f() is epded i terms of the ormlized eigefuctios. si f( ) b b si 1 1 ormlized set of eigefuctios (Bsis set - bsis) The set of epsio coefficiet b is the represettio i f() from the bse fuctio. Determitio of Epsio coefficiets; b m f( ) b 1 * pre-multiply by d itegrte; * * mf ( d ) m b d b d b b * m m m m m f( ) d * m m overlp itegrl Eergy opertor for prticle i 3D bo: E KEPE m y z H m y z 8
9 /16/17 D d 3D boes I SE: Etedig the results from 1-D to -D/3-D is deceptively esy. For bo of size b c with BCs, Divisio by (,y,z), the time idepedet SE; Solutios re of form; d E = E +E y +E z Seprtio of vribles H h X( ) E The wve equtio where c be writte s sum of terms tht do ot shre coordites c be resolved ito the set (DE with three vribles trsforms to three DEs, ech with oe vrible); Solutios ~ 1D solutios!! Y( y) Z( z) si y y si b b z z si c c E E y z 8m h 8mb h z 8mc y E, y, z, y, z 8 y y z z ( yz,, ) si si si bc b c h y z 8m b c 9
10 /16/17 If the totl eergy c be writte s sum of idepedet vribles correspodig to differet degrees of freedom, the wve fuctio is product of idividul terms, ech correspodig to oe of the degrees of freedom. For cube of side ; E, y, z, y, z 8 y z ( yz,, ) si si si h y z 8m h E 8m, y, z y z y z 3 Severl i combitios c yield the sme E j,k,l - degeerte sttes Leds to the cocept of degeercy. Degeercy is the umber of wys system c chieve certi specified eergy. Cosolidtio of QM Postultes with solutios of Prticle i Bo For the prticle i 3D bo, stte is ssocited with three qutum umbers. Three qutum umbers re geerted becuse there re three sets of boudry coditios ivolved i 3D bo. More th oe eigefuctio (stte) would be ssocited with the sme eigevlue (eergy) i degeerte sttes. I geerl c be comple fuctio with terms crryig i = (-1); If is rel the use; (, td ) * (, t) (, t) d 1
11 /16/17 Note: eigevlues from opertor re of precise vlues. Mesurig the vlue of property ssocited with the opertor of eige stte lwys produce the sme vlue. i.e. Wht if the stte is described by fuctio tht is ot eige fuctio of the opertor (yet well behved, d cceptble wve fuctio)? The the eergy ssocited with tht wve fuctio would ot yield precise vlue (like eigevlue) every time the property is mesured. Wht we would be ble to clculte i such cse is the verge vlue of the multiple sigle vlues tht eperimet would produce; d is lso clled the epecttio vlue. The sigle determitios re the eige vlues tht the opertor would produce. Cosider ormlized wve fuctio costructed by ddig two wve fuctios of prticle i bo, =1 d = sttes s follows; c 1 si d si c'si d 'si This is cceptble wve fuctio, well behved first d secod derivtives, BCs stisfied. 11
12 /16/17 d md H d md c d H 'si 'si c'si d 'si c 4d E E 'si m 'si ' #!?! But is ot eigefuctio of the eergy opertor. d d d d c si d si c cos d cos c si 4d si Normlized fuctio however; c si d d 1 si si si si si c cd d d c d 1 Normliztio =1 = =1 u-ormlized. Note the coditio: c d 1 c 1 d d 1 c d d re iterpreted s the probbility tht the stte emultes eigesttes =1 d = respectively Becuse wve fuctio is rel d ormlized * ( ) ( d ) ( ) ( d ) 1 I geerl E * ( ) H ( ) d 1
13 /16/17 Clcultig mometum: Solutios of 1D bo: is eigefuctio of the Hmiltoi ot eigefuctio of mometum opertor. The QM opertor for mometum p is; d p i d So we clculte the verge, <p> usig postulte 4. p ( ) p( ) d d p si i si d d p ( ) p ( ) d 13
14 /16/17 Clcultig positio: Solutios of 1D bo: Agi it is eigefuctio of the Hmiltoi ot eigefuctio of positio opertor. Note: Clculted <p> = ; despite KE = p /m!! Epli. The QM opertor for positio is; So we clculte <>; me usig postulte 4. ( ) ( d ) ( ) ( d ) ( ) ( ) d si si d Clculted <> = /; the verge vlue of the positio is hlf wy betwee the eds if 1D bo. Motio of prticles i cofied boudries leds to qutiztio (discrete vlues for properties) d the qutized sttes re described by stdig wve like wvefuctios. Motio of prticles i ifiite spce, i.e. ucofied spce (free prticles) leds to rge of vlues for the properties. The wvefuctio i such situtios resemble propgtig wve. 14
Lecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More informationPostulates of quantum mechanics
Postultes of qutum mechics P1. The stte of qutum mechicl sstem is completel specified b wvefuctio Ψ(q,t) P. For ever mesurble propert of the sstem, there eists correspodig opertor i QM (mesuremet i the
More informationEigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):
Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )
More informationis completely general whenever you have waves from two sources interfering. 2
MAKNG SENSE OF THE EQUATON SHEET terferece & Diffrctio NTERFERENCE r1 r d si. Equtio for pth legth differece. r1 r is completely geerl. Use si oly whe the two sources re fr wy from the observtio poit.
More informationChem 253A. Crystal Structure. Chem 253B. Electronic Structure
Chem 53, UC, Bereley Chem 53A Crystl Structure Chem 53B Electroic Structure Chem 53, UC, Bereley 1 Chem 53, UC, Bereley Electroic Structures of Solid Refereces Ashcroft/Mermi: Chpter 1-3, 8-10 Kittel:
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More information=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property
Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(
More informationChem 253B. Crystal Structure. Chem 253C. Electronic Structure
Chem 5, UC, Berele Chem 5B Crstl Structure Chem 5C Electroic Structure Chem 5, UC, Berele 1 Chem 5, UC, Berele Electroic Structures of Solid Refereces Ashcroft/Mermi: Chpter 1-, 8-10 Kittel: chpter 6-9
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationsin m a d F m d F m h F dy a dy a D h m h m, D a D a c1cosh c3cos 0
Q1. The free vibrtio of the plte is give by By ssumig h w w D t, si cos w W x y t B t Substitutig the deflectio ito the goverig equtio yields For the plte give, the mode shpe W hs the form h D W W W si
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More information[Q. Booklet Number]
6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More informationAbel Resummation, Regularization, Renormalization & Infinite Series
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 68 Article Jose
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationBC Calculus Path to a Five Problems
BC Clculus Pth to Five Problems # Topic Completed U -Substitutio Rule Itegrtio by Prts 3 Prtil Frctios 4 Improper Itegrls 5 Arc Legth 6 Euler s Method 7 Logistic Growth 8 Vectors & Prmetrics 9 Polr Grphig
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationDefinite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More information334 MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION
MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION TEST SAMPLE TEST III - P APER Questio Distributio INSTRUCTIONS:. Attempt ALL questios.. Uless otherwise specified, ll worig must be
More informationNotes 17 Sturm-Liouville Theory
ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (
More informationYOUR FINAL IS THURSDAY, MAY 24 th from 10:30 to 12:15
Algebr /Trig Fil Em Study Guide (Sprig Semester) Mrs. Duphy YOUR FINAL IS THURSDAY, MAY 4 th from 10:30 to 1:15 Iformtio About the Fil Em The fil em is cumultive for secod semester, coverig Chpters, 3,
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationThe Exponential Function
The Epoetil Fuctio Defiitio: A epoetil fuctio with bse is defied s P for some costt P where 0 d. The most frequetly used bse for epoetil fuctio is the fmous umber e.788... E.: It hs bee foud tht oyge cosumptio
More informationSurds, Indices, and Logarithms Radical
MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio
More informationCALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS
CALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS Abstrct Mirce I Cîru We preset elemetry proofs to some formuls ive without proofs by K A West d J McClell i 99 B
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationChapter 25 Sturm-Liouville problem (II)
Chpter 5 Sturm-Liouville problem (II Speer: Lug-Sheg Chie Reerece: [] Veerle Ledou, Study o Specil Algorithms or solvig Sturm-Liouville d Schrodiger Equtios. [] 王信華教授, chpter 8, lecture ote o Ordiry Dieretil
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationTopic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
More information1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.
Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b =
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationis continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a
. Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively?
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More informationIndices and Logarithms
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More informationDETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1
NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit
More informationCHAPTER 2: Boundary-Value Problems in Electrostatics: I. Applications of Green s theorem
CHAPTER : Boudr-Vlue Problems i Electrosttics: I Applictios of Gree s theorem .6 Gree Fuctio for the Sphere; Geerl Solutio for the Potetil The geerl electrosttic problem (upper figure): ( ) ( ) with b.c.
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationClosed Newton-Cotes Integration
Closed Newto-Cotes Itegrtio Jmes Keeslig This documet will discuss Newto-Cotes Itegrtio. Other methods of umericl itegrtio will be discussed i other posts. The other methods will iclude the Trpezoidl Rule,
More informationDiscrete Mathematics I Tutorial 12
Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationy udv uv y v du 7.1 INTEGRATION BY PARTS
7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product
More informationWaves in dielectric media. Waveguiding: χ (r ) Wave equation in linear non-dispersive homogenous and isotropic media
Wves i dieletri medi d wveguides Setio 5. I this leture, we will osider the properties of wves whose propgtio is govered by both the diffrtio d ofiemet proesses. The wveguides re result of the ble betwee
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES
ABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EHU (Uiversity of Bsque coutry) I Solid Stte Physics Addres: Prctictes Ad
More informationLesson 4 Linear Algebra
Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
8.1 Arc Legth Wht is the legth of curve? How c we pproximte it? We could do it followig the ptter we ve used efore Use sequece of icresigly short segmets to pproximte the curve: As the segmets get smller
More informationCHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER-0 WAVEFUNCTIONS, OBSERVABLES OPERATORS 0. Represettios i the sptil mometum spces 0..A Represettio of the wvefuctio i the
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Lecture 17
CS 70 Discrete Mthemtics d Proility Theory Sprig 206 Ro d Wlrd Lecture 7 Vrice We hve see i the previous ote tht if we toss coi times with is p, the the expected umer of heds is p. Wht this mes is tht
More informationINTEGRATION IN THEORY
CHATER 5 INTEGRATION IN THEORY 5.1 AREA AROXIMATION 5.1.1 SUMMATION NOTATION Fibocci Sequece First, exmple of fmous sequece of umbers. This is commoly ttributed to the mthemtici Fibocci of is, lthough
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More information2. Infinite Series 3. Power Series 4. Taylor Series 5. Review Questions and Exercises 6. Sequences and Series with Maple
Chpter II CALCULUS II.4 Sequeces d Series II.4 SEQUENCES AND SERIES Objectives: After the completio of this sectio the studet - should recll the defiitios of the covergece of sequece, d some limits; -
More informationStatistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006
Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite
More informationALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions
Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +
More information5.74 TIME-DEPENDENT QUANTUM MECHANICS
p. 1 MIT Deprtet of Cheistry 5.74, Sprig 005: Itroductory Qutu Mechics II Istructor: Professor Adrei Tooff 5.74 TIME-DEPENDENT QUANTUM MECHANICS The tie evolutio of the stte of syste is described by the
More informationStudent Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More information2017/2018 SEMESTER 1 COMMON TEST
07/08 SEMESTER COMMON TEST Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Electroic Systems Diplom i Telemtics & Medi Techology Diplom i Electricl Egieerig with Eco-Desig Diplom
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationClassical Electrodynamics
A First Look t Qutum Phsics Clssicl Electrodmics Chpter Boudr-Vlue Prolems i Electrosttics: I 11 Clssicl Electrodmics Prof. Y. F. Che Cotets A First Look t Qutum Phsics.1 Poit Chrge i the Presece of Grouded
More informationChapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Dr. Hamid R. Rabiee Fall 2013
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Dr. Hmid R. Rbiee Fll 03 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Outlie Itroductio to the -Trsform Properties
More information2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r
Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationInterpolation. 1. What is interpolation?
Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the
More informationSOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method.
SOLUTION OF SYSTEM OF LINEAR EQUATIONS Lecture 4: () Jcobi's method. method (geerl). (b) Guss Seidel method. Jcobi s Method: Crl Gustv Jcob Jcobi (804-85) gve idirect method for fidig the solutio of system
More informationLEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More informationLecture 2: Matrix Algebra
Lecture 2: Mtrix lgebr Geerl. mtrix, for our purpose, is rectgulr rry of objects or elemets. We will tke these elemets s beig rel umbers d idicte elemet by its row d colum positio. mtrix is the ordered
More informationREVISION SHEET FP1 (AQA) ALGEBRA. E.g., if 2x
The mi ides re: The reltioships betwee roots d coefficiets i polyomil (qudrtic) equtios Fidig polyomil equtios with roots relted to tht of give oe the Further Mthemtics etwork wwwfmetworkorguk V 7 REVISION
More informationError-free compression
Error-free compressio Useful i pplictio where o loss of iformtio is tolerble. This mybe due to ccurcy requiremets, legl requiremets, or less th perfect qulity of origil imge. Compressio c be chieved by
More informationQuadrature Methods for Numerical Integration
Qudrture Methods for Numericl Itegrtio Toy Sd Istitute for Cle d Secure Eergy Uiversity of Uth April 11, 2011 1 The Need for Numericl Itegrtio Nuemricl itegrtio ims t pproximtig defiite itegrls usig umericl
More informationEXERCISE a a a 5. + a 15 NEETIIT.COM
- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()
More information