Poisson brackets and canonical transformations

Size: px
Start display at page:

Download "Poisson brackets and canonical transformations"

Transcription

1 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 to smplfy the notaton defne the osson bracket 1 1 g g 1 g g g g Usng ths notaton wth g1=f g= we fnd q p p q df f f t Constants of the moton df If f s a constant of the moton (we also say an ntegral of the moton ) then 0 An example s n a problem where the angular momentum of the system under consderaton s conserved A way of vsualzng the meanng of df s to move wth the partcle and check the 1

2 rof O B Wrght Autumn 007 value of f from tme to tme If t s constant then In contrast the quantty f t df 0 s the change n f when we are fxed n one place wth a fxed momentum (You mght thnk ths means that nothng should change but sometmes the externally appled potentals or constrants vary wth tme) From above the conon for f f t df 0 can also be wrtten as If n adon f s not an explct functon of tme that s f both df 0 and f 0 t then f 0 We can use ths equaton to fnd the conon that f should be a f constant of the moton n the case when 0 t Example: free partcle n 1D p m dp Is p a constant of the moton? That s s 0? Let us fnd p p p p q q p =0 dp 0 and p=constant Example: partcle n a potental n 1D p V ( q) (where q=x) m

3 rof O B Wrght Autumn 007 p dp p dv dv p q dq dq 0 In ths case p s not constant but s constant snce general (see last week p 3) and n ths case 0 t d t n Formulas for osson brackets: f g g f f C 0 (where C s a constant) f f g f g f g 1 1 f f g f f g f f g f g f g g f t t t d df dg f g g f Also f q k f p and f p k k f q k (These can be proved usng kl kl q q k l 1for k l and 0 for k l ) kl qk and 0 where p l Specal cases of these relatons are q q 0 p p 0 q p k k There s also an mportant relaton called Jacob s dentty: (where 1for k and 0 for k ) k k f g h g h f h f g 0 (Ths can be proved by wrtng all the terms) 3

4 rof O B Wrght Autumn 007 There s also a theorem called osson s theorem: If f and g are constants of the moton then f g s also a constant of the moton That s df dg d f g Useful for fndng new constants of the moton roof: d df dg f g g f d f g 0 g f 0 0 df dg from above But f 0 and 0 then Canoncal transformatons We are free to choose the generalzed coordnates q 1 q n Whatever we choose d Lagrange s equatons wll apply: ( q1 qn q1 qn t) and q q Consder choosng a new set of coordnates Q Q ( q t) where 1 n ( Q Q Q Q t) and So 1 n 1 n d Q Q If these two Lagrangans lead to the same equatons of moton then for example or df( q t) where F s an arbtrary functon (as proved n Lecture 11 p 4) (The Lagrangan moton Can you see why?) where s a constant also gves the same equatons of 4

5 rof O B Wrght Autumn 007 We can do somethng smlar wth amlton s equatons ( q q p p t) Now consder a new set of n generalzed coordnates 1 n 1 n Q Q ( qp t) ( qp t) If the new Q obey Q Q for the new amltonan ( Q t) then we call the change of coordnates from q p to Q a canoncal transformaton The mechancal moton of the system descrbed by s the same as that descrbed by It s also possble to show see end of ths lecture that for a canoncal transformaton Q k k (where 1for k and 0 for k ) k Ths equaton can be used to test for a canoncal transformaton k ow to do a canoncal transformaton Use the defnton of the amltonan pq The defnton of s Q where ( Q t) For the two amltonans to descrbe the same problem try the general form df( Qq t) ( Q Q t) ( q q t) (1) 5

6 rof O B Wrght Autumn 007 where F s some functon of q Q and t (If there s a constant other than 1 multplyng we do not call t a canoncal transformaton) We can see that the new Lagrangan descrbes the same problem because on ntegratng t t F( q( t1) Q( t1) t1) F( q( t) Q ( t) t) t1 t1 So applyng amlton s prncple t d 0 d s the same as applyng t1 d d t t1 0 so the two Lagrangans gve the same moton df We can wrte Eq (1) as pq Q or df p dq dq ( ) F F F Therefore p q Q t ere F F( qq t) s called the generatng functon for the canoncal transformaton If F F( qq ) only wth no explct dependence on t then we see that That s ( q p) ( Q ) ossble types of problem: 1) We are gven and q q( Q t) p p( Q t) Fnd and F( q Q t ) ) We are gven and F( q Q t ) Fnd and q q( Q t) p p( Q t) 6

7 rof O B Wrght Autumn 007 Example of a canoncal transformaton Consder the smple case of a system wth one q and one p In general for ( q p) we know that q p p q Consder the transformaton to the new coordnates and Q where q=- p=q Check that ths s a canoncal transformaton and fnd the generatng functon F Method 1 to check f t s a canoncal transformaton If t s a canoncal transformaton then Q () Q The relatons between q p and Q do not depend on t so we know that F/ t 0 Therefore ( Q ) ( q p) But and p Q q q Q Q p So Eqs () are satsfed and so the transformaton s canoncal Method to check f t s a canoncal transformaton A transformaton s canoncal f Q 1 Q Q that s f 1 Evaluatng ths drectly we get q p p q Q Q 0 (1 1) 1 so the transformaton s canoncal q p p q 7

8 rof O B Wrght Autumn 007 Fnd the generatng functon F Use F p from p 6 where F F( q Q) But p=q so q F q Q Therefore F Qdq Qq f ( Q) F Also But =-q so Q F q so F qdq Qq g( q) Q For both these solutons for F to be consstent F Qq arbtrary constant because t does not change the moton) ( We do not need to add an Also we know that because F/ t 0 Example For the same canoncal transformaton q=- p=q let as n a sngle partcle n 1D n a potental V Q Then V ( ) m p V ( q) m Utlty of canoncal transformatons Sometmes a partcular choce of p q mght mean that nether p or q are constants But by convertng to Q coordnates we mght be able to make or Q constant Also canoncal transformatons are very useful n quantum mechancs roof that Q 1 mples that the transformaton s canoncal Let us use the notaton of Jacobans for general functons F and G: F F F G F G q p q p q p p q G G q p F G F G 8

9 rof O B Wrght Autumn 007 where means the determnant of the matrx Ths matrx s called the Jacoban A shorthand for the Jacoban s F F q p ( F G) G G ( pq ) q p There s a chan rule for Jacobans: multplcaton ( F G) ( F G) ( u v) where ths s matrx ( x y) ( u v) ( x y) In partcular f we set F G= x y n ths equaton we fnd that ( x y) ( u v) I where ( u v) ( x y) I From the general rule that AB s the product of the determnants) we know therefore that A B (the determnant of a product of two matrces ( x y) 1 ( uv ) ( uv ) ( x y) (3) Our job s to evaluate Q ( Q ) p q ( pq ) Q Q p q owever from Eq (3) we know that Q q p Q ere p q p Q q Q 1 1 ( pq ) ( Q ) q p Q 9

10 rof O B Wrght Autumn 007 Let us frst fnd q p Q and then use ths to fnd Q To go further we shall assume the exstence of the generatng functon F defned n p 6 q p Q p p Q q p q p q q Q Q Q F F Also F F( q Q t) F( q( Q ) Q t) And p q Q Let us frst calculate p p and Q so that we can fnd q p Q p F( q( Q ) Q t) F q q q where q q( Q ) p F( q( Q ) Q t) F F q Q Q q q Q q Q Ths allows us to fnd q p Q F Q Q q Q qq q Q q q p q p q F q F F q qq F F But because q qq Q q p 1 Therefore But Q Q 1 Therefore q p Q Q 1 as requred Note that the quantty F qq always needs to be non-zero for a canoncal transformaton 10

Canonical transformations

Canonical transformations Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,

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

12. The Hamilton-Jacobi Equation Michael Fowler

12. The Hamilton-Jacobi Equation Michael Fowler 1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and

More information

PHYS 705: Classical Mechanics. Canonical Transformation II

PHYS 705: Classical Mechanics. Canonical Transformation II 1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 705: Classical Mechanics. Calculus of Variations II 1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013

Lagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013 Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned

More information

10. Canonical Transformations Michael Fowler

10. Canonical Transformations Michael Fowler 10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst

More information

Lecture 20: Noether s Theorem

Lecture 20: Noether s Theorem Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external

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

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

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

coordinates. Then, the position vectors are described by

coordinates. Then, the position vectors are described by Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

PHYS 705: Classical Mechanics. Hamilton-Jacobi Equation

PHYS 705: Classical Mechanics. Hamilton-Jacobi Equation 1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve

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

Lecture 10: Euler s Equations for Multivariable

Lecture 10: Euler s Equations for Multivariable Lecture 0: Euler s Equatons for Multvarable Problems Let s say we re tryng to mnmze an ntegral of the form: {,,,,,, ; } J f y y y y y y d We can start by wrtng each of the y s as we dd before: y (, ) (

More information

THEOREMS OF QUANTUM MECHANICS

THEOREMS OF QUANTUM MECHANICS THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn

More information

Quantum Mechanics I Problem set No.1

Quantum Mechanics I Problem set No.1 Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t

More information

PHYS 705: Classical Mechanics. Newtonian Mechanics

PHYS 705: Classical Mechanics. Newtonian Mechanics 1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

Notes on Analytical Dynamics

Notes on Analytical Dynamics Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame

More information

LAGRANGIAN MECHANICS

LAGRANGIAN MECHANICS LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m

More information

where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt

where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.

More information

CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 14 GENERAL PERTURBATION THEORY CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves

More information

Analytical classical dynamics

Analytical classical dynamics Analytcal classcal ynamcs by Youun Hu Insttute of plasma physcs, Chnese Acaemy of Scences Emal: yhu@pp.cas.cn Abstract These notes were ntally wrtten when I rea tzpatrck s book[] an were later revse to

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

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

Lagrangian Field Theory

Lagrangian Field Theory Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,

More information

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582

NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy

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

Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations

Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton

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

Integrals and Invariants of Euler-Lagrange Equations

Integrals and Invariants of Euler-Lagrange Equations Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,

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

Physics 181. Particle Systems

Physics 181. Particle Systems Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system

More information

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics) CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O

More information

Physics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2

Physics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2 Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on

More information

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on

More information

1 What is a conservation law?

1 What is a conservation law? MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2016 2017, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,

More information

8.323 Relativistic Quantum Field Theory I

8.323 Relativistic Quantum Field Theory I MI OpenCourseWare http://ocw.mt.edu 8.323 Relatvstc Quantum Feld heory I Sprng 2008 For nformaton about ctng these materals or our erms of Use, vst: http://ocw.mt.edu/terms. MASSACHUSES INSIUE OF ECHNOLOGY

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

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers. David Miller Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons

More information

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS) Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998

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

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

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Density matrix. c α (t)φ α (q)

Density matrix. c α (t)φ α (q) Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n

More information

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t

Snce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t 8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes

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

Calculus of Variations Basics

Calculus of Variations Basics Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y

More information

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,

More information

Classical Field Theory

Classical Field Theory Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to

More information

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or

More information

Review of Newtonian Mechanics

Review of Newtonian Mechanics hapter 1 Revew of Newtonan Mechancs 1.1 Why Study lasscal Mechancs? Quantum lmt Relatvstc lmt General relatvty Mathematcal technques Frst approxmaton Intuton 1.2 Revew of Newtonan Mechancs Basc defntons

More information

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Integrals and Invariants of

Integrals and Invariants of Lecture 16 Integrals and Invarants of Euler Lagrange Equatons NPTEL Course Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng, Indan Insttute of Scence, Banagalore

More information

= z 20 z n. (k 20) + 4 z k = 4

= z 20 z n. (k 20) + 4 z k = 4 Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5

More information

Dynamics of a Superconducting Qubit Coupled to an LC Resonator

Dynamics of a Superconducting Qubit Coupled to an LC Resonator Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of

More information

The generating function of a canonical transformation

The generating function of a canonical transformation ENSEÑANZA Revsta Mexcana de Físca E 57 158 163 DICIEMBRE 2011 The generatng functon of a canoncal transformaton G.F. Torres del Castllo Departamento de Físca Matemátca Insttuto de Cencas Unversdad Autónoma

More information

Random Walks on Digraphs

Random Walks on Digraphs Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

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

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography

CSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve

More information

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,

Example: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41, The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson

More information

Thermodynamics and statistical mechanics in materials modelling II

Thermodynamics and statistical mechanics in materials modelling II Course MP3 Lecture 8/11/006 (JAE) Course MP3 Lecture 8/11/006 Thermodynamcs and statstcal mechancs n materals modellng II A bref résumé of the physcal concepts used n materals modellng Dr James Ellott.1

More information

Mathematical Preparations

Mathematical Preparations 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More 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

Lecture 10 Support Vector Machines II

Lecture 10 Support Vector Machines II Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed

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

Classical Mechanics ( Particles and Biparticles )

Classical Mechanics ( Particles and Biparticles ) Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Hidden Markov Models & The Multivariate Gaussian (10/26/04)

Hidden Markov Models & The Multivariate Gaussian (10/26/04) CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models

More information

5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR

5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR 5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 00-5000 cm - spectral regon. The absorpton of a photon

More information

6. Hamilton s Equations

6. Hamilton s Equations 6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

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

Classical Mechanics Virtual Work & d Alembert s Principle

Classical Mechanics Virtual Work & d Alembert s Principle Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often

More information

Determinants Containing Powers of Generalized Fibonacci Numbers

Determinants Containing Powers of Generalized Fibonacci Numbers 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal

More information

a b a In case b 0, a being divisible by b is the same as to say that

a b a In case b 0, a being divisible by b is the same as to say that Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

Fixed points of IA-endomorphisms of a free metabelian Lie algebra

Fixed points of IA-endomorphisms of a free metabelian Lie algebra Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ

More information

Affine and Riemannian Connections

Affine and Riemannian Connections Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Polynomials. 1 More properties of polynomials

Polynomials. 1 More properties of polynomials Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a

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

Feature Selection: Part 1

Feature Selection: Part 1 CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there

More information

The Second Anti-Mathima on Game Theory

The Second Anti-Mathima on Game Theory The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player

More information

Radar Trackers. Study Guide. All chapters, problems, examples and page numbers refer to Applied Optimal Estimation, A. Gelb, Ed.

Radar Trackers. Study Guide. All chapters, problems, examples and page numbers refer to Applied Optimal Estimation, A. Gelb, Ed. Radar rackers Study Gude All chapters, problems, examples and page numbers refer to Appled Optmal Estmaton, A. Gelb, Ed. Chapter Example.0- Problem Statement wo sensors Each has a sngle nose measurement

More information

Rate of Absorption and Stimulated Emission

Rate of Absorption and Stimulated Emission MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld

More information

Ph 219a/CS 219a. Exercises Due: Wednesday 12 November 2008

Ph 219a/CS 219a. Exercises Due: Wednesday 12 November 2008 1 Ph 19a/CS 19a Exercses Due: Wednesday 1 November 008.1 Whch state dd Alce make? Consder a game n whch Alce prepares one of two possble states: ether ρ 1 wth a pror probablty p 1, or ρ wth a pror probablty

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system. Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

find (x): given element x, return the canonical element of the set containing x;

find (x): given element x, return the canonical element of the set containing x; COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:

More information

Lecture 10 Support Vector Machines. Oct

Lecture 10 Support Vector Machines. Oct Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs hyscs 151 Lecture Canoncal Transformatons (Chater 9) What We Dd Last Tme Drect Condtons Q j Q j = = j, Q, j, Q, Necessary and suffcent j j for Canoncal Transf. = = j Q, Q, j Q, Q, Infntesmal CT

More information