Postulates of quantum mechanics
|
|
- Sara Clarke
- 5 years ago
- Views:
Transcription
1 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 lb opertig o the wvefuctio, ÂΨ) P3. Ψ(q,t) is obtied b solvig t-dep Schrodiger equtio, P4. I sigle mesuremet of the observble tht correspods to the opertor A, the ol vlues tht will ever be mesured re the eigevlues of tht opertor P5. If the sstem is i stte described b Ψ(q,t), the verge vlue of ll the mesuremets for the observble A * ψ Âψdτ A * ψψ d τ 1
2 Schrodiger equtio: m 3. Prticle-i-bo models: ˆ HΨ i Ψ, Hˆ ψ Eψ t ˆ + V q,, H ( ), where q:, (, z, ), ( r, θφ, ) or ( rr ) 1 d (,, ) z + + V z 4. Rigid-rotor d gulr mometum eigesttes 1 1 si 1 d,, θ + V ( r θφ) Re siθ θ θ si θ φ 5. moleculr vibrtios d t-dep perturbtio theor 1 Jˆ 1 r d V ( r, θφ, ) kr r r r 1 ˆ J Ze r d V r r r 4πε r 6. the hdroge tom:
3 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) rsiθcosφ rsiθcosφ z rcosθ r + + z cosθ tφ z r 3
4 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) z r r r siθ θ θ si θ φ r siθ ˆ 1 1 J siθ + siθ θ θ si θ φ ˆ 1 J r r r r dτ dddz ( )( si ) ( r si ) dτ dr rdθ r θdφ θ drdθdφ 4
5 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) rsiθcosφ rsiθcosφ z rcosθ r + + z cosθ tφ z r r θ φ + + r θ φ z, z, θφ, z, r, φ z, r, θ r θ φ + + r θ φ z, z, θφ, z, r, φ z, r, θ r θ φ + + z z r z θ z φ,, θφ,, r, φ, r, θ 5
6 Sphericl polr coordite r, θφ, ( ) cf, Crtesi coordite ( z,, ) r + + z r r r siθcosφ z, r r r siθcos φ siθsi φ cosθ z z,, z, r + + z cosθ tφ, z r z z cosθ z + + z r + + z, z ( ) 1 θ 3 siθ z ( + + z ) ( ) rcosθ r z, z,, tφ 1 φ rsiθsiφ ( cosφ ) z, ( r siθcosφ) z,,, 3 ( rsiθcosφ) θ cosθcosφ θ cosθsiφ θ siθ r r z r φ siφ φ cosφ φ rsiθ rsiθ z z 6
7 3.1 Prticle i 1D bo Coductio i metls Opticl properties i polmers Free electro moleculr orbitl theor ˆ d H m d + V V,, otherwise ψ π si h E where 1,,3 8m h E1 8m Zero-poit eerg: turl cosequece of the ucertit priciple 7
8 3.1 Prticle i 1D bo π h si, E where 1,,3 ψ 8m ψ hs wvelegth λ/ Ech ode of the -th stte is locted betwee odes of the (+1)-th stte holds for solutio to the 1D Schrodiger equtio mkig ψ orthogol More odes shorter wvelegth higher KE 8
9 3.1 Prticle i 1D bo π h si, E where 1,,3 ψ 8m 9
10 Outside the bo 3.1 Prticle i 1D bo d ψ ( E ) ψ ψ m d Iside the bo ˆ d H m d + V V,, otherwise d ψ Eψ ψ Asiα+ Bcosα m d ( ) where α BC.. ψ B me π ψ ( ) α, 1,, 3 π ψ si E h π me α 8m π ψ Asi * π ψψd A si d 1 π A 1 cos d A 1 A ˆ d ψ π Hψ ( ψ) Eψ m d m π h E m 8m 1
11 3.1. Epecttio vlues * ψ ψd * ψψd 3 1 π h si, E where 1,,3 ψ 8m π 1 π si d 1 cos d 1 π cos d 4 1 π π si si d 4 π π p p p ψ p ψd d * * ψψ h 4 p p p h h h h > π p π d π si si d i d π π π si cos d i π π π let t si, dt cos d t, t tdt i Prticle is movig with equl probbilit from right to left d from left to right h h p m H m 8m 4 11
12 3.1.3 Orthogolit π h si, E where 1,,3 ψ 8m π mπ d A si si d * ψ ψm * ( + ) π ( ) 1 m m π cos cos d + i) m 1 ( + m) π ( m) π si + si ( + m) π ( m) π ii) m i( A± B) ia ± ib 1 π cos 1 d + 1 ( ) 1 ψ ψ m d δ m e e e ( A± B) + i ( A± B) ( A+ i A)( B± i B) A B A B i( A B A B) A A B B+ A A B B ( A+ B) ( A B) cos si cos si cos si cos cos si si + si cos ± cos si cos cos si si cos cos cos si si cos 1 si Asi B cos A+ B + cos A B ( ( ) ( )) 1
13 3.1.5 Correspodece priciple: serve s useful guide i the developmet of pproimte qutum theories (e, semiclssicl theories) π h si, E 1,, 3 ψ 8m As, Іψ І is the sme for ll As m, h, or, ΔE As h or size become mcroscopic, the predictios of QM become those of CM. 13
14 (Recll) Seprtio of vribles A sstem of two oiterctig prticle, H(q 1, q ) I f Hˆ Hˆ q Hˆ ( ) + ( q ) ( q, q ) ψ ( q ) ψ ( q ) 1 1 Ψ E E 1 ( ) ψ ( ) ψ ( ) ( ) ψ ( ) ψ ( ) where Hˆ q q E q + E Hˆ q q E q Hˆ Hˆ 1( q1) + Hˆ ( q) ( Hˆ 1( q1) + Hˆ ( q) ) Ψ ( q1, q) EΨ( q1, q) let Ψ ( q1, q) ψ1( q1) ψ( q) ( Hˆ 1( q1) + Hˆ ( q) ) ψ1( q1) ψ( q) Eψ1( q1) ψ( q) ψ( q) Hˆ 1( q1) ψ1( q1) + ψ1( q1) Hˆ ( q) ψ( q) Eψ1( q1) ψ( q) Hˆ 1( q1) ψ1( q1) Hˆ ( q) ψ( q) + E E1+ E ψ1( q1) ψ( q) Hˆ 1( q1) ψ1( q1) E1ψ1( q1) Hˆ ( q ) ψ ( q ) Eψ ( q ) A sstem of oiterctig prticle, H(q 1, q, ) If ( ) + ( ) + + ( q ) ( q, q, q ) ψ ( q ) ψ ( q ) ψ ( q ) ˆ ˆ ˆ ˆ H H1 q1 H q H where Hˆ Ψ E E + E + E 1 ( q ) ψ ( q ) Eψ ( q ) i i i i i i i 14
15 3. Prticle i D bo ˆ H, iside + b m 4 π π ψ, (, ) si si b b E 8m b where d 1,,3 h, + + m m Hˆ Hˆ + Hˆ iside ( ) ( ) ( ) ψ ( ) ψ ( ) Hˆ E ( ) ψ E h 8m si π ( ) ψ ( ) ψ ( ) Hˆ E ψ E ( ) h 8mb si π b b π π ψ, (, ) ( ) ( ) si si ψ ψ b b h h E, where d 1,,3 E + E + 8m 8mb 15
16 3. Prticle i D bo 4 π π ψ, (, ) si si b b h E, where d 1,,3 + 8m b ψ1,1 ψ ψ ψ,1 1,, If the bo is squre ( b) h E, ( ) degeerc + 8m M of the sttes re degeerte! ψ ψ 1,,1 π π si si π π si si E(h /8m )
17 3.3 Prticle i 3D bo ˆ H,, iside + + b z c m z 8 π π zπ z ψ,, (,, ) si si si z z bc b c E 8m b c where, d 1,,3 h z,, + + z z b c ( b c) E3. (p47) If prticle is i cubic bo, E,, ( ) h z + + z 8m 3h E111 8m E E E E (tripl degeerte) ψ11 ψ11 ψ11 17
18 3.4 Free-electro moleculr orbitl model E h 8m CC CC A simple model of the π electroic sttes i cojugted orgic molecule: 1. π e s i cojugted molecule c be seprted from the σ e s. the σ e s frmework is froze 3. ll iter electroic iterctios re eglected. 4. the effective potetil ctig o ech π e s is give b prticle-i--bo potetil 5. the epressio for the eerg levels of the π e s is give b E 6. π e s occup E ccordce with the Puli priciple (two e s ech) HOMO: the highest occupied moleculr orbitl LUMO: the lowest uoccupied moleculr orbitl LUMO HOMO If -stte is HOMO, h E E+ 1 E + 8m hc h hν + mi λm 8m ( ( ) ) 1 ( 1) butdiee For lier polees, the bsorptio wvelegth gets loger s the legth of the chi is icresed 18
19 E 3.1 ormliztio coditio : ψ dτ 1 ψ d d wvefuctio ψ is ot ormlizble. ( ψ mes tht the probbilit tht the prticle is where is zero. ) E 3. h 1 D: E 8m D: E 3 D: E, b c h 8m + 8m b c h z,, + + z b h E,, z + + z z 8m lowest stte is o-degeerte. ( ) (,, 1,, 3, ) ( z) ( ) ( ) ( ) The first ecited stte,,,1,1, 1,,1, 1,1, is tripl degeerte. 6h E E E E 8m
20 E 3.3 h E + 8m ( 1) ( ) A 5.6 A 1. the secod highest occupied the lowest empt h 8h E E ( 3 1 ) 8 m 8 m hc λ E 8m hc 19m 8h 3. the highest occupied the secod lowest empt h E4 E ( 4 ) 8 m 8m hc λ 86.3m 1h 1h 8m
21 P 3.1 ( ) ( b) () c h E 8m 1 E m 1
22 P 3. ψ ψd ψψd π π si si d 박스가운데있는입자의 verge vlue 를나타내면 : π π si si d π si si π π si d d
23 P 3.3 ( ) ( ) ψ ( ) BC. ψ d ψ, ψ should be cotiuous d well-behved wvefuctio. ( b) d ψ Eψ ψ Asiα+ Bcosα m d BC. ( ) ( ) ψ ψ B ψ ψ Asiα α π π α ψ Asiα ( ) d Asiα E Asiα m d ( Aα siα) EAsiα m α ( ) ( ) me me α ( ) 3
24 P 3.3 () c ψ ( ) A si α ( ), d m d m α ( A siα ( ) ) ( E Vs ) A siα ( ) ( A α siα ( ) ) ( E Vs ) A siα ( ) ( ) ( ) m E V m E V s α ( d) s Asiα A siα A siα differetil Aαcos α A α cosα A α cosα siα siα 1 1 α cos α α cosα 1 1 tα tα α α 4
25 P 3.4 p h h E p, λ, m λ + 1 π ( + 1) 1 h E m ( + 1) ( + ) 1 π 1 m π m ode 의수가증가하면 λ는감소, E는증가한다. 각각의 eigefuctio 은 orthogolit 를유지하기위해다른 ode 값을가진다. 5
26 P 3.6 ( ) h h E E + E + + 8m b 8m L L L 3L L 3 L, L L E () c h + 8m 9L L ( π ) 1 + 8m 9L π + 18mL ( 9 ) ( 9 ) 6
27 P 3.6 ( b) 7
28 P 3.7 ˆ H + + m z π ψ z,, z bc b c 8m b c cubic bo b c 8 π zπ z h z,, ( z,, ) si si si, E π π zπ z h ψ,, (,, ) si si si, z E z 3,, z 8m 1 ψ ( ψ1,,3 ψ3,1, ) ( + + z ) 1 8 π π 3πz 8 3π π πz si si si si si si ( A B) π π 3πz 3π π πz si si si A, si si si B ˆ 1 8 π 3π π π 3π π Hψ A+ B 3 A+ B A+ B m π 3 m 14π ψ m 7π E m ψ is eigefuctio. Similrl, ψ ψ ψ 1,1,1,, ( A B) is ot eigefuctio. 8
29 P 3.8 E ( z,, ) ( ) ( ) ( z) ψ ψ ψ ψ z z z π π zπ z si si si b b c c h h + + 8m 8mb 8mc b c.6 A z h ( z) ( ) ( z) ( ) ( ) ( ) Whe,, 1,1,1, it hs the lowest eerg. Whe,, 1,1,, 1,,1,,1,1, it hs the lowest ecittio eerg. E h + + 8m ( ) 34 ( J s) 31 1 kg ( m) h E + + 8m ( 1 1 ) 34 ( J s) 31 1 kg ( m) E E E 34 ( J s) 31 1 kg ( m) 18 6 ( 6 3)
30 P 3.9 k 이중결합의수 hc E ELUMO EHOMO λ π (( k+ 1) k ) m π ( k + 1) m π + m 1.4 A ( k+ ) m 1.4 A hc ( k + ) λ π k + 1 ( k 1) 31 1 ( )( ) 34 ( J s) π ( ) kg m 34 8 k + ( J s)( 3 1 m/ s) k m 8 m ( k + ) ( k + ) k + 1 k + 1 3
31 λ가 6 m보다커야하므로, ( k + ) 65m 6m k + 1 ( k+ ) ( k+ ) ( ) 65 k + 4k+ 4 1k+ 6 k k ( 15 + ) ( ) ± k 65 k 14.81,.35 f( k) k 14.81, k.35 k값은양수 k 15 ( ) CH CH CH CH CH CH 13 λ 65m 63m
32 P 3.1 ( ) ( ) ψ ψ ψ, h E + 8m ( ), ( 3 ) ( 3 ) + k k k 6 3k A hc E E+ 1 E λ ( 1 ) h 8m A c λ 8m A 88m 3h 4 si π π si b b 3
Using Quantum Mechanics in Simple Systems Chapter 15
/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
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 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 informationLecture 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 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 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 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 informationChapter 3 The Schrödinger Equation and a Particle in a Box
Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics
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 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 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 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 informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10
University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin
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 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 informationBITSAT MATHEMATICS PAPER. If log 0.0( ) log 0.( ) the elogs to the itervl (, ] () (, ] [,+ ). The poit of itersectio of the lie joiig the poits i j k d i+ j+ k with the ple through the poits i+ j k, i
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 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 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 informationQn Suggested Solution Marking Scheme 1 y. G1 Shape with at least 2 [2]
Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme y G Shpe with t lest [] fetures correct y = f'( ) G ll fetures correct SR: The mimum poit could be i the first or secod qudrt. -itercept
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 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 informationProblem Set 3 Solutions
Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,
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 informationPhysics 235 Final Examination December 4, 2006 Solutions
Physics 35 Fi Emitio Decembe, 6 Soutios.. Fist coside the two u quks. They e idetic spi ½ ptices, so the tot spi c be eithe o. The Pui Picipe equies tht the ove wvefuctio be echge tisymmetic. Sice the
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 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 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 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 informationChapter 4 Postulates of Quantum Mechanics
F 014 Chem 350: Itroductory Qutum Mechics Chpter 4 Postutes of Qutum Mechics... 48 MthChpter C... 48 The Postutes of Qutum Mechics... 50 Properties of Hermiti Opertors... 53 Discussio of Mesuremet... 55
More informationSet 1 Paper 2. 1 Pearson Education Asia Limited 2014
. C. A. C. B 5. C 6. A 7. D 8. B 9. C 0. C. D. B. A. B 5. C 6. C 7. C 8. B 9. C 0. A. A. C. B. A 5. C 6. C 7. B 8. D 9. B 0. C. B. B. D. D 5. D 6. C 7. B 8. B 9. A 0. D. D. B. A. C 5. C Set Pper Set Pper
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 informationCITY UNIVERSITY LONDON
CITY UNIVERSITY LONDON Eg (Hos) Degree i Civil Egieerig Eg (Hos) Degree i Civil Egieerig with Surveyig Eg (Hos) Degree i Civil Egieerig with Architecture PART EXAMINATION SOLUTIONS ENGINEERING MATHEMATICS
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 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 informationPre-Calculus - Chapter 3 Sections Notes
Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied
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 informationAssessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Assessmet Ceter Elemetr Alger Stud Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetr Alger test. Reviewig these smples will give
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 informationPhysics 2D Lecture Slides Lecture 25: Mar 2 nd
Cofirmed: D Fial Eam: Thursday 8 th March :3-:3 PM WH 5 Course Review 4 th March am WH 5 (TBC) Physics D ecture Slides ecture 5: Mar d Vivek Sharma UCSD Physics Simple Harmoic Oscillator: Quatum ad Classical
More informationQuantum Mechanics I. 21 April, x=0. , α = A + B = C. ik 1 A ik 1 B = αc.
Quatum Mechaics I 1 April, 14 Assigmet 5: Solutio 1 For a particle icidet o a potetial step with E < V, show that the magitudes of the amplitudes of the icidet ad reflected waves fuctios are the same Fid
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 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 informationAutar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates
Autr Kw Bejmi Rigsby http://m.mthforcollege.com Trsformig Numericl Methods Eductio for STEM Udergrdutes http://m.mthforcollege.com . solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the
More informationStudents must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTutor.com PhysicsAdMthsTutor.com Jue 009 4. Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer
More information(200 terms) equals Let f (x) = 1 + x + x 2 + +x 100 = x101 1
SECTION 5. PGE 78.. DMS: CLCULUS.... 5. 6. CHPTE 5. Sectio 5. pge 78 i + + + INTEGTION Sums d Sigm Nottio j j + + + + + i + + + + i j i i + + + j j + 5 + + j + + 9 + + 7. 5 + 6 + 7 + 8 + 9 9 i i5 8. +
More informationPhysics 2D Lecture Slides Lecture 22: Feb 22nd 2005
Physics D Lecture Slides Lecture : Feb d 005 Vivek Sharma UCSD Physics Itroducig the Schrodiger Equatio! (, t) (, t) #! " + U ( ) "(, t) = i!!" m!! t U() = characteristic Potetial of the system Differet
More informationMathematical Notations and Symbols xi. Contents: 1. Symbols. 2. Functions. 3. Set Notations. 4. Vectors and Matrices. 5. Constants and Numbers
Mthemticl Nottios d Symbols i MATHEMATICAL NOTATIONS AND SYMBOLS Cotets:. Symbols. Fuctios 3. Set Nottios 4. Vectors d Mtrices 5. Costts d Numbers ii Mthemticl Nottios d Symbols SYMBOLS = {,,3,... } set
More informationELG4156 Design of State Variable Feedback Systems
ELG456 Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for
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 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 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 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 informationImportant Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
More informationdx x x = 1 and + dx α x x α x = + dx α ˆx x x α = α ˆx α as required, in the last equality we used completeness relation +
Physics 5 Assignment #5 Solutions Due My 5, 009. -Dim Wvefunctions Wvefunctions ψ α nd φp p α re the wvefunctions of some stte α in position-spce nd momentum-spce, or position representtion nd momentum
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 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 informationDifferentiation Formulas
AP CALCULUS BC Fil Notes Trigoometric Formuls si θ + cos θ = + t θ = sec θ 3 + cot θ = csc θ 4 si( θ ) = siθ 5 cos( θ ) = cosθ 6 t( θ ) = tθ 7 si( A + B) = si Acos B + si B cos A 8 si( A B) = si Acos B
More informationChapter 11 Design of State Variable Feedback Systems
Chpter Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for
More informationSharjah Institute of Technology
For commets, correctios, etc Plese cotct Ahf Abbs: hf@mthrds.com Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet ALGERA Lws of Idices:. m m + m m. ( ).. 4. m m 5. Defiitio of logrithm:
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 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 informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationHandout #2. Introduction to Matrix: Matrix operations & Geometric meaning
Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly
More informationWestchester Community College Elementary Algebra Study Guide for the ACCUPLACER
Westchester Commuity College Elemetry Alger Study Guide for the ACCUPLACER Courtesy of Aims Commuity College The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry
More informationSOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM
Jourl of Applied Mthemtics d Computtiol Mechics () 57-6 SOUION O DIERENIA EQUAION OR HE EUER-ERNOUI EAM Izbel Zmorsk Istitute of Mthemtics Czestochow Uiversit of echolog Częstochow Pold izbel.zmorsk@im.pcz.pl
More informationNational Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/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 workig.
More informationFor students entering Honors Precalculus Summer Packet
Hoors PreClculus Summer Review For studets eterig Hoors Preclculus Summer Pcket The prolems i this pcket re desiged to help ou review topics from previous mthemtics courses tht re importt to our success
More informationDefinition Integral. over[ ab, ] the sum of the form. 2. Definite Integral
Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ
More informationAdd Maths Formulae List: Form 4 (Update 18/9/08)
Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Fuctios Asolute Vlue Fuctio f ( ) f( ), if f( ) 0 f( ), if f( ) < 0 Iverse Fuctio If y f( ), the Rememer: Oject the vlue of Imge the vlue of y or f() f()
More informationIndicate if the statement is True (T) or False (F) by circling the letter (1 pt each):
Indicate if the statement is (T) or False (F) by circling the letter (1 pt each): False 1. In order to ensure that all observables are real valued, the eigenfunctions for an operator must also be real
More informationOrthogonality, orthogonalization, least squares
ier Alger for Wireless Commuictios ecture: 3 Orthogolit, orthogoliztio, lest squres Ier products d Cosies he gle etee o-zero vectors d is cosθθ he l of Cosies: + cosθ If the gle etee to vectors is π/ (90º),
More informationRepeated Root and Common Root
Repeted Root d Commo Root 1 (Method 1) Let α, β, γ e the roots of p(x) x + x + 0 (1) The α + β + γ 0, αβ + βγ + γα, αβγ - () (α - β) (α + β) - αβ (α + β) [ (βγ + γα)] + [(α + β) + γ (α + β)] +γ (α + β)
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 informationTime: 2 hours IIT-JEE 2006-MA-1. Section A (Single Option Correct) + is (A) 0 (B) 1 (C) 1 (D) 2. lim (sin x) + x 0. = 1 (using L Hospital s rule).
IIT-JEE 6-MA- FIITJEE Solutios to IITJEE 6 Mthemtics Time: hours Note: Questio umber to crries (, -) mrks ech, to crries (5, -) mrks ech, to crries (5, -) mrks ech d to crries (6, ) mrks ech.. For >, lim
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 informationModule 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur
Module 4 Sigl Represettio d Bsed Processig Versio ECE IIT, Khrgpur Lesso 5 Orthogolity Versio ECE IIT, Khrgpur Ater redig this lesso, you will ler out Bsic cocept o orthogolity d orthoormlity; Strum -
More information2a a a 2a 4a. 3a/2 f(x) dx a/2 = 6i) Equation of plane OAB is r = λa + µb. Since C lies on the plane OAB, c can be expressed as c = λa +
-6-5 - - - - 5 6 - - - - - - / GCE A Level H Mths Nov Pper i) z + z 6 5 + z 9 From GC, poit of itersectio ( 8, 9 6, 5 ). z + z 6 5 9 From GC, there is o solutio. So p, q, r hve o commo poits of itersectio.
More informationLecture 7: Polar representation of complex numbers
Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with
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 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 informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009
University of Wshington Deprtment of Chemistry Chemistry Winter Qurter 9 Homework Assignment ; Due t pm on //9 6., 6., 6., 8., 8. 6. The wve function in question is: ψ u cu ( ψs ψsb * cu ( ψs ψsb cu (
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 informationSolutions to Problem Set 7
8.0 Clculus Jso Strr Due by :00pm shrp Fll 005 Fridy, Dec., 005 Solutios to Problem Set 7 Lte homework policy. Lte work will be ccepted oly with medicl ote or for other Istitute pproved reso. Coopertio
More informationDo the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
More informationSet 3 Paper 2. Set 3 Paper 2. 1 Pearson Education Asia Limited 2017
Set Pper Set Pper. D. A.. D. 6. 7. B 8. D 9. B 0. A. B. D. B.. B 6. B 7. D 8. A 9. B 0. A. D. B.. A. 6. A 7. 8. 9. B 0. D.. A. D. D. A 6. 7. A 8. B 9. D 0. D. A. B.. A. D Sectio A. D ( ) 6. A b b b ( b)
More informationOrthogonal functions - Function Approximation
Orthogol uctios - Fuctio Approimtio - he Problem - Fourier Series - Chebyshev Polyomils he Problem we re tryig to pproimte uctio by other uctio g which cosists o sum over orthogol uctios Φ weighted by
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 informationAlgebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents
Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit
More informationExponents and Radical
Expoets d Rdil Rule : If the root is eve d iside the rdil is egtive, the the swer is o rel umber, meig tht If is eve d is egtive, the Beuse rel umber multiplied eve times by itself will be lwys positive.
More informationLecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box
561 Fall 013 Lecture #5 page 1 Last time: Lecture #5: Begi Quatum Mechaics: Free Particle ad Particle i a 1D Box u 1 u 1-D Wave equatio = x v t * u(x,t): displacemets as fuctio of x,t * d -order: solutio
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 informationLimits and an Introduction to Calculus
Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit
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 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 informationLinford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
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 informationName: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!
Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe
More informationWe now turn to our first quantum mechanical problems that represent real, as
84 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like
More information