HW #3. 1. Spin Matrices. HW3-soln-improved.nb 1. We use the spin operators represented in the bases where S z is diagonal:

Size: px
Start display at page:

Download "HW #3. 1. Spin Matrices. HW3-soln-improved.nb 1. We use the spin operators represented in the bases where S z is diagonal:"

Transcription

1 HW3-oln-mproved.nb HW #3. Spn Matrce We ue the pn operator repreented n the bae where S dagonal: S x = 880, <, 8, 0<<; S y = 880, -I<, 8I, 0<<; S = S x êê MatrxForm 0 ÅÅÅ ÅÅÅ 0 y S y êê MatrxForm 0 - Â Â 0 y S êê MatrxForm ÅÅÅ ÅÅÅ y 88, 0<, 80, -<<; (a) Obvouly two matrce commute when they are the ame: =. Alo, t obvou S D ant-ymmetrc n õ S D = -@S, S D. Therefore, t only reaman to verfy S x.s y - S y.s x - I S 880, 0<, 80, 0<< S y.s - S.S y - I S x 880, 0<, 80, 0<< S.S x - S x.s - I S y 880, 0<, 80, 0<< (b) We defne n Ø = Hn cof, n nf, col n x = Sn@D Co@fD; n y = Sn@D Sn@fD; n = Co@D Co@D S n = Smplfy@n x S x + n y S y + n S D 99 Co@D, Sn@D HCo@fD - Â Sn@fDL=, 9 Sn@D HCo@fD + Â Sn@fDL, - Co@D==

2 HW3-oln-mproved.nb n D è!!!!! 99- è!!!!!, I-è!!!!! =, 99 ÅÅ + Co@DM Cc@D ÅÅ, =, 9 ÅÅ Iè!!!!! + Co@DM Cc@D Å HCo@fD + Â Sn@fDL HCo@fD + Â Sn@fDL, === PowerExpand@%D 99-, H- + Co@DL Cc@D H + Co@DL Cc@D =, 99 ÅÅÅ ÅÅ, =, 9 ÅÅÅ Å HCo@fD + Â Sn@fDL HCo@fD + Â Sn@fDL, === Smplfy@%D 99-, =, 99H-Co@fD + Â Sn@fDL TanA E, =, 9CotA E HCo@fD - Â Sn@fDL, === Therefore, one can tae the the normaled egentate to be co y n wth egenvalue + ef and -n e-f y co wth egenvalue -. The tate wth pn along the nø drecton the former, and t probablty to have the potve S when meaured mply gven by» XS = +» S n = + \» = H, 0L co y ƒ n = co ef. ƒ (c) Between Ø n = Hn cof, n nf, coland Ø n ' = Hn' cof', n' nf', co 'L, the probablty» XS n' = +» S n = + \» = Hco ' ' Å, n e-f' L co y ƒ n =» co ' ef co ' + n Å e-f' n ef» = ƒ co Å ' co TrgExpandA CoA E ' + n Å n CoA E ' + co + SnA E co ' n Å SnA E n cohf - f'l + CoA + CoA E CoA E - CoA E SnA E + E CoA E SnA E SnA E Co@f - f DE CoA E CoA E Co@f D Co@f D SnA E SnA E - CoA E SnA E + SnA E SnA E + CoA E CoA E SnA E SnA E Sn@f D Sn@f D Smplfy@%D H + Co@ D Co@ D + Co@f D Co@f D Sn@ D Sn@ D + Sn@ D Sn@ D Sn@f D Sn@f DL Th nothng but I + nø ÿ Ø n 'M = h H + cohl = co, where h the angle between two vector, a expected from the rotatonal nvarance.. Sloppy Hydrogen Atom Accordng to the problem,

3 HW3-oln-mproved.nb 3 Energy = ÅÅ m - e Z ÅÅÅ + Å d d m y d - Z e ÅÅÅ d Solve@D@Energy, dd ã 0, dd 99d Ø Å e m Z == Smplfy@Energy ê. %@@DDD - e4 m Z Th actually agree wth the exact reult. (One hould be cautoned, however, that the agreement wth the exact reult a concdence for th partcular example.) 3. Clacal Uncertanty Prncple (a) The Maxwell' euaton n vacuum are gven by Ø ÿ E Ø = 0 Ø ÿ B Ø = 0 Ø µ E Ø + t B Ø = 0 c Ø µ B Ø - t E Ø = 0 In th problem, there are only x and t dependence, and the only non-vanhng component are E y and B. Then the Maxwell' euaton reduce to x E y + t B = 0 -c x B - t E y = 0 Puttng them together, they reduce to a mple one-dmenonal euaton, c x E y - t E y = 0. Any functon of the combnaton c t - x atfe th euaton, namely Hc x - t L f Hc t - xl = 0. Becaue the form of E y gven n the problem a functon of c t - x only, t olve the Maxwell' euaton automatcally. The form can be etched a To mplfy the problem, defne g = 4 p n ê c. Then the electrc feld : Ey@x_, t_d = E0 * SnA c è!!! è!!! g g Å t - xe E -Hx-c tl^êh L E0 SnA c t è!!! g - x è!!! g E H-c t+xl - Å

4 HW3-oln-mproved.nb 4 Plot@Ey@x, td ê. 8t Ø 0, g Ø 400 p, E0 Ø, Ø 0<, 8x, -30, 30<, PlotRange Ø 8-, <D Ü Graphc Ü It ocllate ut le the plane wave, but localed. The "uncertanty" defned ung the formula analogou to the uantum mechancal wave functon. Frt the "norm," norm = Integrate@Ey@x, 0D^, 8x, -, <, Aumpton Ø 8g > 0, > 0<D General::pell : Poble pellng error: new ymbol name "norm" mlar to extng ymbol "Norm". More -g H- + g L E0 è!!! p We could et the overall normalaton E 0 = throughout nce t wll drop out after tang the norm correctly nto account. But we can alo leave t n a a chec that everythng worng correctly. Next the expectaton value Integrate@x * Ey@x, 0D^, 8x, -, <, Aumpton Ø 8g > 0, > 0<D 0 OK, th vanhe. Fnally the varance, temp = Integrate@x^ * Ey@x, 0D^, 8x, -, <, Aumpton Ø 8g > 0, > 0<D 4 -g E0 è!!! p H- + g + gl 3 varance = temp ê norm General::pell : Poble pellng error: new ymbol name "varance " mlar to extng ymbol "Varance ". More H- + g + gl ÅÅ H- + g L Note that the E0 dependence dd n fact drop out. FullSmplfy@varanceD J + g - + N g

5 HW3-oln-mproved.nb 5 One can wrte t a HD xl = I + Å g g - M, where g = 4 p n ê c. It epecally mple when g p, when HD xl =. (b) The Fourer tranform to the freuency doman a a functon of the varable "f" calculated below. We mght a well pc a pecfc poton le "x=0" to evaluate the Fourer tranform. fttemp = Integrate@Ey@0, td * E I p f t, 8t, -, <, Aumpton Ø 8g > 0, > 0, c > 0<D Integrate ::gener : Unable to chec convergence. More Ic è!!!! g + f p M E0 IfAf œ Real,  - Å c J- + 4 f p è!!!! g Å c N "##### ÅÅÅ p ÅÅ ÅÅÅ, IntegrateA c  f p t- c t Å SnA c t è!!! g E, 8t, -, <, Aumpton Ø c > 0 && g > 0 && > 0 && f RealEE ft@f_d = fttemp@@dd * fttemp@@, DD è!!!! g + f p M  - Ic Å c J- + 4 f p è!!!! c g Å N E0 "##### ÅÅÅ p ÅÅ Å c Agan tartng wth the norm (notng that ft@ f D * ft@ f D = -ft@ f D^ ): norm = Integrate@-ft@fD^, 8f, 0, <, Aumpton Ø 8g > 0, > 0, c > 0<D -g H- + g L E0 è!!! p ÅÅÅ Å 4 c Integrate@-ft@fD^, 8f, -, <, Aumpton Ø 8g > 0, > 0, c > 0<D -g H- + g L E0 è!!! p ÅÅÅ Å c Next, the average freuency, exptf = Integrate@f * -ft@fd^, 8f, 0, <, Aumpton Ø 8g > 0, > 0, c > 0<D ê norm c g è!!! g Erf@ è!!! g D ÅÅ H- + g L p exptf ê. Ø c è!!!! g ë H p nl g n Erf@ è!!! g D - + g LmtAErfA è!!! g E, g Ø E One can wrte t a n Erf@gê D, where g = 4 p n ê c. It epecally mple when g p, when t reduce to nothng but n. -E -g Fnally the varance n the freuency (remember to normale!)

6 HW3-oln-mproved.nb 6 exptf = Integrate@f * -ft@fd^, 8f, 0, <, Aumpton Ø 8g > 0, > 0, c > 0<D ê norm c H- + g H + gll ÅÅÅ 8 H- + g L p FullSmplfy@%D c H- + g H + gll ÅÅÅ 8 H- + g L p % ê. Ø c è!!!! g ë H p nl êê Smplfy H- + g H + gll n ÅÅÅ H- + g L g Lmt@%, g Ø D n Then the uare of the dperon n the freuency : dperon = exptf - exptf^ êê Smplfy - c I-H- + g L H- + g H + gll + g g Erf@ è!!! g D M ÅÅ ÅÅ ÅÅÅ ÅÅ 8 H- + g L p dperon ê. Ø c è!!!! g ë H p nl êê Smplfy - n I-H- + g L H- + g H + gll + g g Erf@ è!!! g D M ÅÅ ÅÅ ÅÅÅ ÅÅ H- + g L g Lmt@dperon, g Ø D % ê. Ø c è!!!! g ë H p nl c 8 p n ÅÅ g Namely, HD f L = n IH-e -g L HH-e -g L+ gl- g Erf@g ê D M ÅÅ whch mplfe to g H-e -g L HD f L = n ÅÅ g = ÅÅÅ c when g p. Therefore, HD xl HD f L = Å c. 8 p 6 p Once nterpreted a a photon, HD f L = c HD pl ë h, and hence HD xl HD pl = 4, a expected.

HW #3. 1. Spin Matrices. HW3.nb 1. We use the spin operators represented in the bases where S z is diagonal:

HW #3. 1. Spin Matrices. HW3.nb 1. We use the spin operators represented in the bases where S z is diagonal: HW3.nb HW #3. Spn Matres We use the spn operators represented n the bases where S s dagonal: S = 88,

More information

Scattering of two identical particles in the center-of. of-mass frame. (b)

Scattering of two identical particles in the center-of. of-mass frame. (b) Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and

More information

Harmonic oscillator approximation

Harmonic oscillator approximation armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon

More information

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted

The multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted Appendx Proof of heorem he multvarate Gauan probablty denty functon for random vector X (X,,X ) px exp / / x x mean and varance equal to the th dagonal term of, denoted he margnal dtrbuton of X Gauan wth

More information

Statistical Properties of the OLS Coefficient Estimators. 1. Introduction

Statistical Properties of the OLS Coefficient Estimators. 1. Introduction ECOOMICS 35* -- OTE 4 ECO 35* -- OTE 4 Stattcal Properte of the OLS Coeffcent Etmator Introducton We derved n ote the OLS (Ordnary Leat Square etmator ˆβ j (j, of the regreon coeffcent βj (j, n the mple

More information

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear

More information

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1

C/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1 C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned

More information

Root Locus Techniques

Root Locus Techniques Root Locu Technque ELEC 32 Cloed-Loop Control The control nput u t ynthezed baed on the a pror knowledge of the ytem plant, the reference nput r t, and the error gnal, e t The control ytem meaure the output,

More information

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters

Chapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters Chapter 6 The Effect of the GPS Sytematc Error on Deformaton Parameter 6.. General Beutler et al., (988) dd the frt comprehenve tudy on the GPS ytematc error. Baed on a geometrc approach and aumng a unform

More information

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order: 68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse

More information

The Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.

The Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom. The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E

More information

Solutions to exam in SF1811 Optimization, Jan 14, 2015

Solutions to exam in SF1811 Optimization, Jan 14, 2015 Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable

More information

Additional File 1 - Detailed explanation of the expression level CPD

Additional File 1 - Detailed explanation of the expression level CPD Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor

More information

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011 Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected

More information

Improvements on Waring s Problem

Improvements on Waring s Problem Improvement on Warng Problem L An-Png Bejng, PR Chna apl@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th paper, we wll gve ome mprovement for Warng problem Keyword: Warng Problem,

More information

Chapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder

Chapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder S-. The Method of Steepet cent Chapter. Supplemental Text Materal The method of teepet acent can be derved a follow. Suppoe that we have ft a frtorder model y = β + β x and we wh to ue th model to determne

More information

Physics 443, Solutions to PS 7

Physics 443, Solutions to PS 7 Physcs 443, Solutons to PS 7. Grffths 4.50 The snglet confguraton state s χ ) χ + χ χ χ + ) where that second equaton defnes the abbrevated notaton χ + and χ. S a ) S ) b χ â S )ˆb S ) χ In sphercal coordnates

More information

763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.

763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1. 7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars

More information

Now that we have laws or better postulates we should explore what they imply

Now that we have laws or better postulates we should explore what they imply I-1 Theorems from Postulates: Now that we have laws or better postulates we should explore what they mply about workng q.m. problems -- Theorems (Levne 7.2, 7.4) Thm 1 -- egen values of Hermtan operators

More information

A how to guide to second quantization method.

A how to guide to second quantization method. Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle

More information

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference

Team. Outline. Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference Team Stattc and Art: Samplng, Repone Error, Mxed Model, Mng Data, and nference Ed Stanek Unverty of Maachuett- Amhert, USA 9/5/8 9/5/8 Outlne. Example: Doe-repone Model n Toxcology. ow to Predct Realzed

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Supplementary material: Margin based PU Learning. Matrix Concentration Inequalities

Supplementary material: Margin based PU Learning. Matrix Concentration Inequalities Supplementary materal: Margn based PU Learnng We gve the complete proofs of Theorem and n Secton We frst ntroduce the well-known concentraton nequalty, so the covarance estmator can be bounded Then we

More information

Predictors Using Partially Conditional 2 Stage Response Error Ed Stanek

Predictors Using Partially Conditional 2 Stage Response Error Ed Stanek Predctor ng Partally Condtonal Stage Repone Error Ed Stane TRODCTO We explore the predctor that wll relt n a mple random ample wth repone error when a dfferent model potlated The model we decrbe here cloely

More information

8 Waves in Uniform Magnetized Media

8 Waves in Uniform Magnetized Media 8 Wave n Unform Magnetzed Meda 81 Suceptblte The frt order current can be wrtten j = j = q d 3 p v f 1 ( r, p, t) = ɛ 0 χ E For Maxwellan dtrbuton Y n (λ) = f 0 (v, v ) = 1 πvth exp (v V ) v th 1 πv th

More information

Three views of mechanics

Three views of mechanics Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental

More information

Multi-dimensional Central Limit Theorem

Multi-dimensional Central Limit Theorem Mult-dmensonal Central Lmt heorem Outlne ( ( ( t as ( + ( + + ( ( ( Consder a sequence of ndependent random proceses t, t, dentcal to some ( t. Assume t = 0. Defne the sum process t t t t = ( t = (; t

More information

No! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Survey Results. Class 15. Is the following possible?

No! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Survey Results. Class 15. Is the following possible? Survey Reult Chapter 5-6 (where we are gong) % of Student 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Hour Spent on ChE 273 1-2 3-4 5-6 7-8 9-10 11+ Hour/Week 2008 2009 2010 2011 2012 2013 2014 2015 2017 F17

More information

MULTIPLE REGRESSION ANALYSIS For the Case of Two Regressors

MULTIPLE REGRESSION ANALYSIS For the Case of Two Regressors MULTIPLE REGRESSION ANALYSIS For the Cae of Two Regreor In the followng note, leat-quare etmaton developed for multple regreon problem wth two eplanator varable, here called regreor (uch a n the Fat Food

More information

Modelli Clamfim Equazione del Calore Lezione ottobre 2014

Modelli Clamfim Equazione del Calore Lezione ottobre 2014 CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g

More information

Pythagorean triples. Leen Noordzij.

Pythagorean triples. Leen Noordzij. Pythagorean trple. Leen Noordz Dr.l.noordz@leennoordz.nl www.leennoordz.me Content A Roadmap for generatng Pythagorean Trple.... Pythagorean Trple.... 3 Dcuon Concluon.... 5 A Roadmap for generatng Pythagorean

More information

1 Vectors over the complex numbers

1 Vectors over the complex numbers Vectors for quantum mechancs 1 D. E. Soper 2 Unversty of Oregon 5 October 2011 I offer here some background for Chapter 1 of J. J. Sakura, Modern Quantum Mechancs. 1 Vectors over the complex numbers What

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

Molecular structure: Diatomic molecules in the rigid rotor and harmonic oscillator approximations Notes on Quantum Mechanics

Molecular structure: Diatomic molecules in the rigid rotor and harmonic oscillator approximations Notes on Quantum Mechanics Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons Notes on Quantum Mechancs http://quantum.bu.edu/notes/quantummechancs/molecularstructuredatomc.pdf Last updated

More information

From Biot-Savart Law to Divergence of B (1)

From Biot-Savart Law to Divergence of B (1) From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to

More information

Introduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015

Introduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015 Introducton to Interfacal Segregaton Xaozhe Zhang 10/02/2015 Interfacal egregaton Segregaton n materal refer to the enrchment of a materal conttuent at a free urface or an nternal nterface of a materal.

More information

Variable Structure Control ~ Basics

Variable Structure Control ~ Basics Varable Structure Control ~ Bac Harry G. Kwatny Department of Mechancal Engneerng & Mechanc Drexel Unverty Outlne A prelmnary example VS ytem, ldng mode, reachng Bac of dcontnuou ytem Example: underea

More information

LA PRISE DE CALAIS. çoys, çoys, har - dis. çoys, dis. tons, mantz, tons, Gas. c est. à ce. C est à ce. coup, c est à ce

LA PRISE DE CALAIS. çoys, çoys, har - dis. çoys, dis. tons, mantz, tons, Gas. c est. à ce. C est à ce. coup, c est à ce > ƒ? @ Z [ \ _ ' µ `. l 1 2 3 z Æ Ñ 6 = Ð l sl (~131 1606) rn % & +, l r s s, r 7 nr ss r r s s s, r s, r! " # $ s s ( ) r * s, / 0 s, r 4 r r 9;: < 10 r mnz, rz, r ns, 1 s ; j;k ns, q r s { } ~ l r mnz,

More information

PHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions

PHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions PHYS 5C: Quantum Mechancs Sprng 07 Problem Set 3 Solutons Prof. Matthew Fsher Solutons prepared by: Chatanya Murthy and James Sully June 4, 07 Please let me know f you encounter any typos n the solutons.

More information

Homework Notes Week 7

Homework Notes Week 7 Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we

More information

LECTURE 9 CANONICAL CORRELATION ANALYSIS

LECTURE 9 CANONICAL CORRELATION ANALYSIS LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of

More information

Small signal analysis

Small signal analysis Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea

More information

Not at Steady State! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Yes! Class 15. Is the following possible?

Not at Steady State! Yes! Only if reactions occur! Yes! Ideal Gas, change in temperature or pressure. Yes! Class 15. Is the following possible? Chapter 5-6 (where we are gong) Ideal gae and lqud (today) Dente Partal preure Non-deal gae (next tme) Eqn. of tate Reduced preure and temperature Compreblty chart (z) Vapor-lqud ytem (Ch. 6) Vapor preure

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

Quantum Mechanics I - Session 4

Quantum Mechanics I - Session 4 Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )

More information

1 Definition of Rademacher Complexity

1 Definition of Rademacher Complexity COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the

More information

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems

More information

Differentiating Gaussian Processes

Differentiating Gaussian Processes Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the

More information

Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions

Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY

ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG-6, 07725-Bucharest, Romana E-mal: acatrne@theory.npne.ro. Receved March 6, 2008

More information

Two Approaches to Proving. Goldbach s Conjecture

Two Approaches to Proving. Goldbach s Conjecture Two Approache to Provng Goldbach Conecture By Bernard Farley Adved By Charle Parry May 3 rd 5 A Bref Introducton to Goldbach Conecture In 74 Goldbach made h mot famou contrbuton n mathematc wth the conecture

More information

Electric and magnetic field sensor and integrator equations

Electric and magnetic field sensor and integrator equations Techncal Note - TN12 Electrc and magnetc feld enor and ntegrator uaton Bertrand Da, montena technology, 1728 oen, Swtzerland Table of content 1. Equaton of the derate electrc feld enor... 1 2. Integraton

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

APPLICATIONS: CHEMICAL AND PHASE EQUILIBRIA

APPLICATIONS: CHEMICAL AND PHASE EQUILIBRIA 5.60 Sprn 2007 Lecture #28 pae PPLICTIOS: CHMICL D PHS QUILIBRI pply tattcal mechanc to develop mcrocopc model for problem you ve treated o far wth macrocopc thermodynamc 0 Product Reactant Separated atom

More information

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac

More information

Assortment Optimization under MNL

Assortment Optimization under MNL Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.

More information

Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology

Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology Electromagnetc catterng Graduate Coure Electrcal Engneerng (Communcaton) 1 t Semeter, 1390-1391 Sharf Unverty of Technology Content of lecture Lecture : Bac catterng parameter Formulaton of the problem

More information

Strong Markov property: Same assertion holds for stopping times τ.

Strong Markov property: Same assertion holds for stopping times τ. Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up

More information

Advanced Quantum Mechanics

Advanced Quantum Mechanics Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton

More information

Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q

Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals

More information

T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a. A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r )

T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a. A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r ) v e r. E N G O u t l i n e T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r ) C o n t e n t s : T h i s w o

More information

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands

1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of

More information

! " # $! % & '! , ) ( + - (. ) ( ) * + / 0 1 2 3 0 / 4 5 / 6 0 ; 8 7 < = 7 > 8 7 8 9 : Œ Š ž P P h ˆ Š ˆ Œ ˆ Š ˆ Ž Ž Ý Ü Ý Ü Ý Ž Ý ê ç è ± ¹ ¼ ¹ ä ± ¹ w ç ¹ è ¼ è Œ ¹ ± ¹ è ¹ è ä ç w ¹ ã ¼ ¹ ä ¹ ¼ ¹ ±

More information

Economics 130. Lecture 4 Simple Linear Regression Continued

Economics 130. Lecture 4 Simple Linear Regression Continued Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do

More information

This appendix presents the derivations and proofs omitted from the main text.

This appendix presents the derivations and proofs omitted from the main text. Onlne Appendx A Appendx: Omtted Dervaton and Proof Th appendx preent the dervaton and proof omtted from the man text A Omtted dervaton n Secton Mot of the analy provded n the man text Here, we formally

More information

Chat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall, 1980

Chat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall, 1980 MT07: Multvarate Statstcal Methods Mke Tso: emal mke.tso@manchester.ac.uk Webpage for notes: http://www.maths.manchester.ac.uk/~mkt/new_teachng.htm. Introducton to multvarate data. Books Chat eld, C. and

More information

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016

U.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016 U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and

More information

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1 Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons

More information

MATH Sensitivity of Eigenvalue Problems

MATH Sensitivity of Eigenvalue Problems MATH 537- Senstvty of Egenvalue Problems Prelmnares Let A be an n n matrx, and let λ be an egenvalue of A, correspondngly there are vectors x and y such that Ax = λx and y H A = λy H Then x s called A

More information

CHAPTER X PHASE-CHANGE PROBLEMS

CHAPTER X PHASE-CHANGE PROBLEMS Chapter X Phae-Change Problem December 3, 18 917 CHAPER X PHASE-CHANGE PROBLEMS X.1 Introducton Clacal Stefan Problem Geometry of Phae Change Problem Interface Condton X. Analytcal Soluton for Soldfcaton

More information

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA

4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA 4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected

More information

Effects of Ignoring Correlations When Computing Sample Chi-Square. John W. Fowler February 26, 2012

Effects of Ignoring Correlations When Computing Sample Chi-Square. John W. Fowler February 26, 2012 Effects of Ignorng Correlatons When Computng Sample Ch-Square John W. Fowler February 6, 0 It can happen that ch-square must be computed for a sample whose elements are correlated to an unknown extent.

More information

Vapnik-Chervonenkis theory

Vapnik-Chervonenkis theory Vapnk-Chervonenks theory Rs Kondor June 13, 2008 For the purposes of ths lecture, we restrct ourselves to the bnary supervsed batch learnng settng. We assume that we have an nput space X, and an unknown

More information

One Dimension Again. Chapter Fourteen

One Dimension Again. Chapter Fourteen hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons

More information

), it produces a response (output function g (x)

), it produces a response (output function g (x) Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans

More information

Another converse of Jensen s inequality

Another converse of Jensen s inequality Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout

More information

SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:

SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table: SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be

More information

Eigenvalues of Random Graphs

Eigenvalues of Random Graphs Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the

More information

The Prncpal Component Transform The Prncpal Component Transform s also called Karhunen-Loeve Transform (KLT, Hotellng Transform, oregenvector Transfor

The Prncpal Component Transform The Prncpal Component Transform s also called Karhunen-Loeve Transform (KLT, Hotellng Transform, oregenvector Transfor Prncpal Component Transform Multvarate Random Sgnals A real tme sgnal x(t can be consdered as a random process and ts samples x m (m =0; ;N, 1 a random vector: The mean vector of X s X =[x0; ;x N,1] T

More information

Srednicki Chapter 14

Srednicki Chapter 14 Srednc Chapter 4 QFT Problems & Solutons A. George September, Srednc 4.. Derve a generalzaton of Feynman s formula, = Γ ( α ) A αa... A αn n Γ(α df xα n ) (n )! ( x A ) Hnt: start wth: Γ(α) A α = dt t

More information

REDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].

REDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1]. REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß

More information

Limited Dependent Variables

Limited Dependent Variables Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

More information

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department

More information

Lorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3

Lorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3 Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac

More information

On the U-WPF Acts over Monoids

On the U-WPF Acts over Monoids Journal of cence, Ilamc Republc of Iran 8(4): 33-38 (007) Unverty of Tehran, IN 06-04 http://jcence.ut.ac.r On the U-WPF ct over Monod. Golchn * and H. Mohammadzadeh Department of Mathematc, Unverty of

More information

Physics 111. CQ1: springs. con t. Aristocrat at a fixed angle. Wednesday, 8-9 pm in NSC 118/119 Sunday, 6:30-8 pm in CCLIR 468.

Physics 111. CQ1: springs. con t. Aristocrat at a fixed angle. Wednesday, 8-9 pm in NSC 118/119 Sunday, 6:30-8 pm in CCLIR 468. c Announcement day, ober 8, 004 Ch 8: Ch 10: Work done by orce at an angle Power Rotatonal Knematc angular dplacement angular velocty angular acceleraton Wedneday, 8-9 pm n NSC 118/119 Sunday, 6:30-8 pm

More information

Linear Approximation with Regularization and Moving Least Squares

Linear Approximation with Regularization and Moving Least Squares Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

Multi-dimensional Central Limit Argument

Multi-dimensional Central Limit Argument Mult-dmensonal Central Lmt Argument Outlne t as Consder d random proceses t, t,. Defne the sum process t t t t () t (); t () t are d to (), t () t 0 () t tme () t () t t t As, ( t) becomes a Gaussan random

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force. The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

More information

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several

More information

Math 702 Midterm Exam Solutions

Math 702 Midterm Exam Solutions Math 702 Mdterm xam Solutons The terms measurable, measure, ntegrable, and almost everywhere (a.e.) n a ucldean space always refer to Lebesgue measure m. Problem. [6 pts] In each case, prove the statement

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede Fall 0 Analyss of Expermental easurements B. Esensten/rev. S. Errede We now reformulate the lnear Least Squares ethod n more general terms, sutable for (eventually extendng to the non-lnear case, and also

More information

8.592J: Solutions for Assignment 7 Spring 2005

8.592J: Solutions for Assignment 7 Spring 2005 8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I

5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I 5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can

More information

Poisson brackets and canonical transformations

Poisson brackets and canonical transformations rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order

More information