Particle Physics Formula Sheet

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "Particle Physics Formula Sheet"

Transcript

1 Particle Physics Formula Sheet Special Relativity Spacetime Coordinates: x µ = (c t, x) x µ = (c t, x) x = (x, x, x ) = (x, y, z) 4-Momentum: p µ = (E/c, p) p µ = (E/c, p) p = (p, p, p ) = (p x, p y, p z ) Derivatives: µ = ( x µ = c ) t, µ = ( c ) t, = ( x, x, ) ( x = x, y, ) z Metric: η µν = η µν = η µ ν = δ µ ν = Raising and Lowering Indices: a µ = η µν a ν a µ = η µν a ν Einstein Summation Convention: a µ b µ = a b + a b + a b + a b = a b a b Lorentz Transformations: a µ = Λ µ ν a ν η µν = Λ µ αλ ν β η αβ a µ = Λ µ ν a ν η µν = Λ µ α Λ ν β η αβ Properties of Lorentz Transformations:. η µν = Λ µ αλ ν β η αβ. det Λ µ ν =. (Λ ) µ α (Λ ) α ν = (Λ ) µ ν 4. Λ Boost Along the z-axis: γ β γ Λ µ ν = β γ γ with β = v c γ = x µ = Λ µ ν x ν = ( γ (c t β z), x, y, γ (z β c t) ) β p µ = Λ µ ν p ν = ( γ (E/c β p z ), p x, p y, γ (p z β E/c) )

2 Rotation About the y-axis: Λ µ cos θ sin θ ν = sin θ cos θ x µ = Λ µ ν x ν = ( c t, x cos θ z sin θ, y, z cos θ + x sin θ ) p µ = Λ µ ν p ν = ( E/c, p x cos θ p z sin θ, p y, p z cos θ + p x sin θ ) Relativistic Kinematics Conservation of 4-Momentum in Any Process: i=initial p µ i = j=final p µ j Massive Particles: Massless Particles: p µ p µ = E c p p = m c p µ p µ = p µ = p (, ε ) with ε ε = Important Matrices Matrices are written out explicitly, and 4 4 matrices are written out in terms of blocks. The identity matrix is = Pauli Matrices: σ = σ i = i σ = σ i σ j = δ ij + i ɛ ijk σ k [ σ i, σ j ] = i ɛ ijk σ k { σ i, σ j } = δ ij Dirac Matrices: γ = γ i σ i = σ i γ 5 = i γ γ γ γ = { γ µ, γ ν } = η µν /a = a µ γ µ = a γ a γ a γ a γ

3 Relativistic Wave Equations Klein-Gordon Equation (Spin- Particle): ( µ µ + m ) φ = µ µ = c t Dirac Equation: (i γ µ µ m) ψ = Maxwell s Equations (Massless Spin- Particle): µ F µν = with F µν = µ A ν ν A µ µ F µν = µ F νλ + ν F λµ + λ F µν = { E = B E t = { B = E + B t = Solutions of the Dirac Equation Some terms in these equations include implicit factors of the or 4 4 identity matrices. Plane-wave solutions: ψ(x) = { a e i pµ xµ u(p) (fermions) a e i pµ xµ v(p) (anti-fermions) Dirac Spinors: Conjugate Spinors: Conjugate Matrices: (/p m c) u (s) (p) = (/p + m c) v (s) (p) = ū (s) (p) (/p m c) = v (s) (p) (/p + m c) = ū = u γ v = v γ ψ = ψ γ Ā = γ A γ for any 4 4 matrix A. Complete Set of Plane-wave Solutions: u () (p) = N c p z E+m c c (p x+i p y) E+m c c (p x i p y) E+m c v () c p z (p) = N E+m c u () (p) = N v () (p) = N c (p x i p y) E+m c c p z E+m c c p z E+m c c (p x+i p y) E+m c E + m c with N = c

4 Fermi s Golden Rule Decays in the Rest Frame of the Decaying Particle: Γ = S p 8π m c M Scattering in the CM Frame: d σ = d Ω CM c S M p f 8π (E + E ) p i with p i = p = p and p f = p = p 4 Elastic Scattering in the Lab Frame (m = m, m 4 = m ): d σ = d Ω Lab p S M 8π m p (E + m c ) p p E cos θ Particle Data Mass in MeV/c, liftime in seconds, charge in units of the proton charge. Leptons (Spin /) Generation Flavor Charge Mass Lifetime First e ν e Second µ ν µ Third τ ν τ 4

5 Quarks (Spin /) Generation Flavor Charge Mass Other Quantum Numbers First d -/ 7 u / Second s -/ Strangeness = - c / Charm = + Third b -/ 4 Bottomness = - t / 74 Topness = + (Note: Quark masses are approximate.) Baryons (Spin /) Baryon Quark Content Charge Mass Lifetime p uud 98.7 n udd Λ uds Σ + uus Σ uds Σ dds Ξ uss Ξ dss Λ + c udc Baryons (Spin /) Baryon Quark Content Charge Mass Lifetime uuu, uud, udd, ddd,,, Σ uus, uds, dds,, Ξ uss, dss, Ω sss Vector Mesons (Spin ) Meson Quark Content Charge Mass Lifetime ρ u d, (u ū d d )/, d ū,,, K u s, d s, d d, s ū,,, ω (u ū + d d )/ ψ c c 97 7 D c d, c ū, u c, d c,,, - 8 Υ b b 946 5

6 Pseudoscalar Mesons (Spin ) Meson Quark Content Charge Mass Lifetime π ± u d, d ū ± π (u ū d d )/ K ± u s, s ū ± K, K d s, s d KS KL : 8.95 : 5. 8 η (u ū + d d s s )/ η (u ū + d d + s s)/ D ± c d, d c ± D, D c ū, u c D s ± c s, s c ± B ± u b, b ū ± B, B d b, b d Baryons, Mesons, and the Eightfold Way Light Baryons (l = ) n p udd dud udu duu Σ Σ Λ Σ + dsd sdd usd sud+dsu sdu uds dus+usd sud dsu+sdu usu suu Ξ Ξ dss sds uss sus The Baryon Octet χ : Antisymmetric in and dud ddu uud udu udd ddu uud duu dds dsd dus dsu+uds usd uus usu dds sdd uds sdu+dus sud uus suu sud sdu+dus dsu uds+usd usd dsu+uds sdu dus+sud sds ssd sus ssu dss ssd uss ssu χ : Antisymmetric in and χ : Antisymmetric in and 6

7 + ++ ddd ddu+dud+udd uud+udu+duu uuu Σ Σ Σ + dds+dsd+sdd uds+usd+dus+dsu+sud+sdu 6 uus+usu+suu Ξ Ξ dss+sds+ssd uss+sus+ssu Ω The Baryon Decuplet sss χ S : Completely Symmetric uds usd+dsu dus+sud sdu 6 χ A : Completely Antisymmetric Light Mesons (l = ) K K + K K + π π π + ρ ρ ρ + η η φ ω K K K K Pseudoscalar Meson Nonet (Octet plus Singlet η ) Vector Meson Nonet (Octet plus Singlet ω) 7

8 QED Feynman Rules. Make sure that fermion are labelled with an arrow that indicates whether they are particles or anti-particles.. Associate a 4-momentum p, p,..., p n with each external line, and draw an arrow next to each line pointing forward in time. Associate a 4-momentum q, q,... q m with each internal line, and draw an arrow next to each line (the direction is arbitrary for internal line).. External lines contribute the following factors Incoming Outgoing Electron = u Electron = ū Positron = v Positron = v Photon = ɛ µ Photon = ɛ µ 4. Each vertex contributes a factor i g γ µ, where the dimensionless coupling g is given by g = e 4π/ c. 5. Internal lines (propagators) contribute the following factors } = i (γµ q µ + m c) = i η µν q m c q 6. For each vertex, include a factor of the form (π) 4 δ (4) (±k ± k ± k ) () where the k i are the 4-momenta entering the vertex. The sign of each term is plus for a momentum arrow pointing into the vertex and minus if the momentum arrow points out of the vertex. 7. For each internal line with momentum q include a factor and integrate. d 4 q (π) 4 () 8. After performing integrals and using the various delta functions, the result will still contain an overall factor Replace this with a factor of i to obtain the amplitude M. (π) 4 δ (4) (p + p +... p n ). () 9. If there are multiple diagrams, include appropriate antisymmetrization factors. Include an extra minus sign between diagrams that differ only by exchange of two incoming (or outgoing) fermions or antifermions, or by exchange of an incoming fermion and outgoing antifermion (or vice versa). 8

9 Casimir s Trick For a process with initial spin states s, final spin states s [ū (s) (p A ) Γ u (s ) (p B )] [ū (s) (p A ) Γ u (s ) (p B )] = Tr [ Γ (/p A + m A c) Γ (/p B + m B c) ] s,s [ v (s) (p A ) Γ v (s ) (p B )] [ v (s) (p A ) Γ v (s ) (p B )] = Tr [ Γ (/p A m A c) Γ (/p B m B c) ] s,s [ū (s) (p A ) Γ v (s ) (p B )] [ū (s) (p A ) Γ v (s ) (p B )] = Tr [ Γ (/p A + m A c) Γ (/p B m B c) ] s,s [ v (s) (p A ) Γ u (s ) (p B )] [ v (s) (p A ) Γ u (s ) (p B )] = Tr [ Γ (/p A m A c) Γ (/p B + m B c) ] s,s where u (s) (p) and v (s) (p) are Dirac spinors, their conjugates are ū (s) (p) and v (s) (p), and Γ and Γ are any complex 4 4 matrices. Factors of m c appearing in the traces on the right-hand side of these relations have implicit factors of the 4 4 identity matrix. Trace Identities If A and B are matrices and α is a complex number Tr [ A + B ] = Tr [ A ] + Tr [ B ] Tr [ α A ] = α Tr [ A ] Cyclic property of the Trace Tr [ A A... A n A n ] = Tr [ A n A A... A n ] =... = Tr [ A... A n A n A ] Gamma Matrix Identities If a µ and b µ are 4-vectors, and /a = γ µ a µ, then {γ µ, γ ν } = η µν implies /a /b + /b /a = a b Other contractions of gamma matrices follow from the anti-commutation relations γ µ γ µ = 4 γ µ γ ν γ µ = γ ν The trace of an odd number of gamma matrices is always zero. Some other traces are Tr [ I 4 4 ] = 4 Tr [ γ µ γ ν ] = 4 η µν ( Tr [ γ µ γ ν γ λ γ σ ] = 4 η µν η λσ η µλ η νσ + η µσ η νλ). Factors of the 4 4 identity matrix are implicit in many of these identities. 9

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

Space-Time Symmetries

Space-Time Symmetries Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a

Διαβάστε περισσότερα

Quantum Electrodynamics

Quantum Electrodynamics Quantum Electrodynamics Ling-Fong Li Institute Slide_06 QED / 35 Quantum Electrodynamics Lagrangian density for QED, Equations of motion are Quantization Write L= L 0 + L int L = ψ x γ µ i µ ea µ ψ x mψ

Διαβάστε περισσότερα

Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.

Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ. Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +

Διαβάστε περισσότερα

Dirac Matrices and Lorentz Spinors

Dirac Matrices and Lorentz Spinors Dirac Matrices and Lorentz Spinors Background: In 3D, the spinor j = 1 representation of the Spin3) rotation group is constructed from the Pauli matrices σ x, σ y, and σ k, which obey both commutation

Διαβάστε περισσότερα

Dirac Trace Techniques

Dirac Trace Techniques Dirac Trace Techniques Consider a QED amplitude involving one incoming electron with momentum p and spin s, one outgoing electron with momentum p and spin s, and some photons. There may be several Feynman

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

PHY 396 K/L. Solutions for problem set #12. Problem 1: Note the correct muon decay amplitude. The complex conjugate of this amplitude

PHY 396 K/L. Solutions for problem set #12. Problem 1: Note the correct muon decay amplitude. The complex conjugate of this amplitude PHY 396 K/L. Solutions for problem set #. Problem : Note the correct muon decay amplitude M(µ e ν µ ν e = G F ū(νµ ( γ 5 γ α u(µ ū(e ( γ 5 γ α v( ν e. ( The complex conjugate of this amplitude M = G F

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ. Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

Introduction to Elementary Particles Instructor s Solution Manual. 29th August 2008

Introduction to Elementary Particles Instructor s Solution Manual. 29th August 2008 Introduction to Elementary Particles Instructor s Solution Manual 9th August 8 V Acknowledgments: I thank Robin Bjorkquist, who wrote and typeset many of the solutions in the first four chapters; Neelaksh

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님

상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님 상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님 Motivation Bremsstrahlung is a major rocess losing energies while jet articles get through the medium. BUT it should be quite different from low energy

Διαβάστε περισσότερα

Srednicki Chapter 55

Srednicki Chapter 55 Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

Διαβάστε περισσότερα

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

Reminders: linear functions

Reminders: linear functions Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

Section 9.2 Polar Equations and Graphs

Section 9.2 Polar Equations and Graphs 180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)

MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3) 1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations

Διαβάστε περισσότερα

Tutorial problem set 6,

Tutorial problem set 6, GENERAL RELATIVITY Tutorial problem set 6, 01.11.2013. SOLUTIONS PROBLEM 1 Killing vectors. a Show that the commutator of two Killing vectors is a Killing vector. Show that a linear combination with constant

Διαβάστε περισσότερα

Second Order RLC Filters

Second Order RLC Filters ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Partial Differential Equations in Biology The boundary element method. March 26, 2013 The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

Διαβάστε περισσότερα

= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ.

= {{D α, D α }, D α }. = [D α, 4iσ µ α α D α µ ] = 4iσ µ α α [Dα, D α ] µ. PHY 396 T: SUSY Solutions for problem set #1. Problem 2(a): First of all, [D α, D 2 D α D α ] = {D α, D α }D α D α {D α, D α } = {D α, D α }D α + D α {D α, D α } (S.1) = {{D α, D α }, D α }. Second, {D

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

D Alembert s Solution to the Wave Equation

D Alembert s Solution to the Wave Equation D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique

Διαβάστε περισσότερα

Phys624 Quantization of Scalar Fields II Homework 3. Homework 3 Solutions. 3.1: U(1) symmetry for complex scalar

Phys624 Quantization of Scalar Fields II Homework 3. Homework 3 Solutions. 3.1: U(1) symmetry for complex scalar Homework 3 Solutions 3.1: U(1) symmetry for complex scalar 1 3.: Two complex scalars The Lagrangian for two complex scalar fields is given by, L µ φ 1 µ φ 1 m φ 1φ 1 + µ φ µ φ m φ φ (1) This can be written

Διαβάστε περισσότερα

Section 9: Quantum Electrodynamics

Section 9: Quantum Electrodynamics Physics 8.33 Section 9: Quantum Electrodynamics May c W. Taylor 8.33 Section 9: QED / 6 9. Feynman rules for QED Field content: A µ(x) gauge field, ψ(x) Dirac spinor Action Z» S = d x ψ(iγ µ D µ m)ψ Z

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

4 Dirac Equation. and α k, β are N N matrices. Using the matrix notation, we can write the equations as imc

4 Dirac Equation. and α k, β are N N matrices. Using the matrix notation, we can write the equations as imc 4 Dirac Equation To solve the negative probability density problem of the Klein-Gordon equation, people were looking for an equation which is first order in / t. Such an equation is found by Dirac. It

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2

If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

Now, suppose the electron field Ψ(x) satisfies the covariant Dirac equation (i D m)ψ = 0.

Now, suppose the electron field Ψ(x) satisfies the covariant Dirac equation (i D m)ψ = 0. PHY 396 K. Solutions for homework set #7. Problem 1a: γ α γ α 1 {γα, γ β }g αβ g αβ g αβ 4; S.1 γ α γ ν γ α γ α γ ν g να γ ν γ α γ α γ ν γ ν γ α γ α 4 γ ν ; S. γ α γ µ γ ν γ α γ α γ µ g µα γ µ γ α γ ν

Διαβάστε περισσότερα

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

Questions on Particle Physics

Questions on Particle Physics Questions on Particle Physics 1. The following strong interaction has been observed. K + p n + X The K is a strange meson of quark composition u s. The u quark has a charge of +2/3. The d quark has a charge

Διαβάστε περισσότερα

Trigonometric Formula Sheet

Trigonometric Formula Sheet Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ

Διαβάστε περισσότερα

If we restrict the domain of y = sin x to [ π 2, π 2

If we restrict the domain of y = sin x to [ π 2, π 2 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

Derivation of Optical-Bloch Equations

Derivation of Optical-Bloch Equations Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be

Διαβάστε περισσότερα

Higher Derivative Gravity Theories

Higher Derivative Gravity Theories Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)

Διαβάστε περισσότερα

Solutions to Exercise Sheet 5

Solutions to Exercise Sheet 5 Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: HW# 1 solutions Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds! MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.

Διαβάστε περισσότερα

Homework 4 Solutions Weyl or Chiral representation for γ-matrices. Phys624 Dirac Equation Homework 4

Homework 4 Solutions Weyl or Chiral representation for γ-matrices. Phys624 Dirac Equation Homework 4 Homework 4 Solutions 4.1 - Weyl or Chiral representation for γ-matrices 4.1.1: Anti-commutation relations We can write out the γ µ matrices as where ( ) 0 σ γ µ µ = σ µ 0 σ µ = (1, σ), σ µ = (1 2, σ) The

Διαβάστε περισσότερα

Orbital angular momentum and the spherical harmonics

Orbital angular momentum and the spherical harmonics Orbital angular momentum and the spherical harmonics March 8, 03 Orbital angular momentum We compare our result on representations of rotations with our previous experience of angular momentum, defined

Διαβάστε περισσότερα

Spinors and σ-matrices in 10 dimensions It is well known that in D-dimensional space-time Dirac spinor has 2 D 2

Spinors and σ-matrices in 10 dimensions It is well known that in D-dimensional space-time Dirac spinor has 2 D 2 PiTP Study Guide to Spinors in D and 0D S.J.Gates Jr. John S. Toll Professor of Physics Director Center for String and Particle Theory University of Maryland Tel: 30-405-6025 Physics Department Fax: 30-34-9525

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

8.324 Relativistic Quantum Field Theory II

8.324 Relativistic Quantum Field Theory II 8.324 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 200 Lecture 2 3: GENERAL ASPECTS OF QUANTUM ELECTRODYNAMICS 3.: RENORMALIZED LAGRANGIAN Consider the Lagrangian

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

Geodesic Equations for the Wormhole Metric

Geodesic Equations for the Wormhole Metric Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

1 String with massive end-points

1 String with massive end-points 1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε

Διαβάστε περισσότερα

PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL NOTES for 6.1 Trigonometric Identities PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot

Διαβάστε περισσότερα

DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C

DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response

Διαβάστε περισσότερα

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0. DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec

Διαβάστε περισσότερα

Muon & Tau Lifetime. A. George January 18, 2012

Muon & Tau Lifetime. A. George January 18, 2012 Muon & Tau Lifetime A. George January 18, 01 Abstract We derive the expected lifetime of the muon, assuming only the Feynman Rules, Fermi s Golden Rule, the Completeness Relations for Dirac Spinors), and

Διαβάστε περισσότερα

Lecture 21: Scattering and FGR

Lecture 21: Scattering and FGR ECE-656: Fall 009 Lecture : Scattering and FGR Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA Review: characteristic times τ ( p), (, ) == S p p

Διαβάστε περισσότερα

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω 0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +

Διαβάστε περισσότερα

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We

Διαβάστε περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch: HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

Section 7.6 Double and Half Angle Formulas

Section 7.6 Double and Half Angle Formulas 09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)

Διαβάστε περισσότερα

Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α

Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α Α Ρ Χ Α Ι Α Ι Σ Τ Ο Ρ Ι Α Π Ο Λ Ι Τ Ι Κ Α Κ Α Ι Σ Τ Ρ Α Τ Ι Ω Τ Ι Κ Α Γ Ε Γ Ο Ν Ο Τ Α Σ η µ ε ί ω σ η : σ υ ν ά δ ε λ φ ο ι, ν α µ ο υ σ υ γ χ ω ρ ή σ ε τ ε τ ο γ ρ ή γ ο ρ ο κ α ι α τ η µ έ λ η τ ο ύ

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011 Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι

Διαβάστε περισσότερα

Derivation of Quadratic Response Time Dependent Hartree-Fock (TDHF) Equations

Derivation of Quadratic Response Time Dependent Hartree-Fock (TDHF) Equations Derivation of Quadratic Response Time Dependent Hartree-Fock (TDHF) Equations Joshua Goings, Feizhi Ding, Xiaosong Li Department of Chemistry, University of Washington Nember 21, 2014 1 Derivation of Equations

Διαβάστε περισσότερα

Relativistic particle dynamics and deformed symmetry

Relativistic particle dynamics and deformed symmetry Relativistic particle dynamics and deformed Poincare symmetry Department for Theoretical Physics, Ivan Franko Lviv National University XXXIII Max Born Symposium, Wroclaw Outline Lorentz-covariant deformed

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS

CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3

Διαβάστε περισσότερα

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013 Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering

Διαβάστε περισσότερα

General 2 2 PT -Symmetric Matrices and Jordan Blocks 1

General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 General 2 2 PT -Symmetric Matrices and Jordan Blocks 1 Qing-hai Wang National University of Singapore Quantum Physics with Non-Hermitian Operators Max-Planck-Institut für Physik komplexer Systeme Dresden,

Διαβάστε περισσότερα

Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain

Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain PHY 396 K. Solutions for homework set #5. Problem 1a: Starting with eq. 3 proved in class and applying the Leibniz rule, we obtain [ γ κ γ λ, S µν] γ κ [ γ λ, S µν] + [ γ κ, S µν] γ λ γ κ ig λµ γ ν ig

Διαβάστε περισσότερα

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits. EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.

Διαβάστε περισσότερα

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

Διαβάστε περισσότερα

ECE Spring Prof. David R. Jackson ECE Dept. Notes 2

ECE Spring Prof. David R. Jackson ECE Dept. Notes 2 ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =

Διαβάστε περισσότερα

SPECIAL FUNCTIONS and POLYNOMIALS

SPECIAL FUNCTIONS and POLYNOMIALS SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

( ) 2 and compare to M.

( ) 2 and compare to M. Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

Math 6 SL Probability Distributions Practice Test Mark Scheme

Math 6 SL Probability Distributions Practice Test Mark Scheme Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry

Διαβάστε περισσότερα

1 Lorentz transformation of the Maxwell equations

1 Lorentz transformation of the Maxwell equations 1 Lorentz transformation of the Maxwell equations 1.1 The transformations of the fields Now that we have written the Maxwell equations in covariant form, we know exactly how they transform under Lorentz

Διαβάστε περισσότερα

Numerical Analysis FMN011

Numerical Analysis FMN011 Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =

Διαβάστε περισσότερα

Parametrized Surfaces

Parametrized Surfaces Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some

Διαβάστε περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

Appendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3

Appendix A. Curvilinear coordinates. A.1 Lamé coefficients. Consider set of equations. ξ i = ξ i (x 1,x 2,x 3 ), i = 1,2,3 Appendix A Curvilinear coordinates A. Lamé coefficients Consider set of equations ξ i = ξ i x,x 2,x 3, i =,2,3 where ξ,ξ 2,ξ 3 independent, single-valued and continuous x,x 2,x 3 : coordinates of point

Διαβάστε περισσότερα

Dirac Matrices and Lorentz Spinors

Dirac Matrices and Lorentz Spinors Dirac Matrices and Lorentz Spinors Background: In 3D, the spinor j = 2 1 representation of the Spin3) rotation group is constructed from the Pauli matrices σ x, σ y, and σ z, which obey both commutation

Διαβάστε περισσότερα

g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King

g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i

Διαβάστε περισσότερα

Symmetric Stress-Energy Tensor

Symmetric Stress-Energy Tensor Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor

Διαβάστε περισσότερα

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3) Q1. (a) A fluorescent tube is filled with mercury vapour at low pressure. In order to emit electromagnetic radiation the mercury atoms must first be excited. (i) What is meant by an excited atom? (1) (ii)

Διαβάστε περισσότερα

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset

Διαβάστε περισσότερα

Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -

Διαβάστε περισσότερα

10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations

10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations //.: Measures of Angles and Rotations I. Vocabulary A A. Angle the union of two rays with a common endpoint B. BA and BC C. B is the vertex. B C D. You can think of BA as the rotation of (clockwise) with

Διαβάστε περισσότερα

Quantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids, superconductors

Quantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids, superconductors BYU PHYS 73 Statistical Mechanics Chapter 7: Sethna Professor Manuel Berrondo Quantum Statistical Mechanics (equilibrium) solid state, magnetism black body radiation neutron stars molecules lasers, superuids,

Διαβάστε περισσότερα

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

The Hartree-Fock Equations

The Hartree-Fock Equations he Hartree-Fock Equations Our goal is to construct the best single determinant wave function for a system of electrons. We write our trial function as a determinant of spin orbitals = A ψ,,... ϕ ϕ ϕ, where

Διαβάστε περισσότερα