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Elementary Particles” Lecture 3 “Elementary Particles” Lecture 3 Niels Tuning Harry van der Graaf Niels Tuning (1) Plan Theory Detection and sensor techn. Quantum Quantum Forces Mechanics Field Theory Interactions Light Scintillators with Matter PM Tipsy Accelerators Bethe Bloch Medical Imag. Cyclotron Photo effect X-ray Compton, pair p. Proton therapy Bremstrahlung Cherenkov Experiments Fundamental Astrophysics Charged Particles Physics Cosmics Particles ATLAS Grav Waves Km3Net Si Neutrinos Virgo Gaseous Lisa Pixel … Special General Gravity Optics Relativity Relativity Laser Niels Tuning (2) Plan Theory Today Detection and sensor techn. Niels 2) Niels 2) Niels 7) + 10) Quantum Quantum Forces Mechanics Field Theory 5) + 8) 4) Harry Particles 3) Harry Light RelativisticIn teractions 6) + 9) with Matter Ernst-Jan 1) Harry 11) +12) Martin Fundamental 6) Ernst-Jan Accelerators Martin Physics 13) + 14) Astrophysics Charged Excursions Particles Experiments 1) Niels 9) Ernst-Jan 9) Ernst-Jan 9) Ernst-Jan Special General Gravity Optics Relativity Relativity Niels Tuning (3) Schedule 1) 11 Feb: Accelerators (Harry vd Graaf) + Special relativity (Niels Tuning) 2) 18 Feb: Quantum Mechanics (Niels Tuning) 3) 25 Feb: Interactions with Matter (Harry vd Graaf) 4) 3 Mar: Light detection (Harry vd Graaf) 5) 10 Mar: Particles and cosmics (Niels Tuning) 6) 17 Mar: Astrophysics and Dark Matter (Ernst-Jan Buis) 7) 24 Mar: Forces (Niels Tuning) break 8) 21 Apr: e+e- and ep scattering (Niels Tuning) 9) 28 Apr: Gravitational Waves (Ernst-Jan Buis) 10) 12 May: Higgs and big picture (Niels Tuning) 11) 19 May: Charged particle detection (Martin Franse) 12) 26 May: Applications: experiments and medical (Martin Franse) 13) 2 Jun: Nikhef excursie 14) 8 Jun: CERN excursie Niels Tuning (4) Plan 1) Intro: Standard Model & Relativity 11 Feb 1900-1940 2) Basis 18 Feb 1) Atom model, strong and weak force 2) Scattering theory 1945-1965 3) Hadrons 10 Mar 1) Isospin, strangeness 2) Quark model, GIM 1965-1975 4) Standard Model 24 Mar 1) QED 2) Parity, neutrinos, weak inteaction 3) QCD 1975-2000 5) e+e- and DIS 21 Apr 2000-2015 6) Higgs and CKM 12 May Niels Tuning (5) Thanks • Ik ben schatplichtig aan: – Dr. Ivo van Vulpen (UvA) – Prof. dr. ir. Bob van Eijk (UT) – Prof. dr. M. Merk (VU) Niels Tuning (6) Exercises Lecture 2: QM and Scattering Niels Tuning (7) Exercises Lecture 2: QM and Scattering Niels Tuning (8) Answers Homework Theory 2 1 Celebrating Bohr a) v = (↵qeqp)/(mr) L = mvr = p↵qeqpmr ) 2 b) L =pp↵qeqpmr = ~ r = ~ /(↵qeqpm). ) 9 2 2 19 2 (I tried SI units: ↵qq = keqq =(9 10 Nm /C ) (1.6 10− C) ) 2 34 2 ⇥9 ⇥19 2 ⇥ 30 10 r = ~ /(↵qeqpm)=(10− ) /(9 10 (1.6 10− ) 10− )=0.4 10− m ⇥ ⇥ ⇥ ⇥ ⇥ ↵qeqp ↵qeqp ↵qeqp Etot = Ekin + Epot = 2r r = 2r 2 m(↵qeqp) − Etot = 2 . − 2~ I tried natural units here: E = ↵2m/2) = (1/137)2 0.5 MeV / 2 = -13.3 eV Exercisestot − Lecture− 2: ⇥QM and Scattering 3 Spinors 2 Yukawa’s massive force carrier We saw that the requirement of a relativistically correct, but linear equation led to the Dirac equation, (iγµ@ m) = 0, with being a four component spinor. 15 µ 15 8 24 a) r 10− m t =−r/c =10− /(3 10 )=3 10− s. ⇠ ) ⇥ ⇥ 2 2 2 a) H =(~↵ p~ + βm) 22 gives E = p + m24 if the matrices anticommute, ↵i, ↵j = b) E ~/t =6·.6 10− MeVs/(3 10− s) = 200 MeV. { } ⇠↵i↵j + ↵j↵i =⇥ 0. Usually we use⇥ the γ matrices, γ =(β, ~↵). Show that indeed γ1γ2 = γ2γ1, using the Pauli-Dirac representation, - 0 0 ~ β = ; ~↵ = . 3 Spinors 0 ~ 0 ✓ − ◆ ✓ ◆ ipx µ a)Withγ1γ2 == u(γp2)γe−1 : and pµ i@µ we get (γ pµ m)u(p) = 0. Looking for the eigenvectors, it is easier− to go back to the! original form; Hu−=(~↵ p~ + βm)u = Eu, which leads to 10 0 0 0001· 0001 m ~ p~ u u Hu01= 0· 0 A0010= E A 0010 γ1 = ↵1 = 0 ~ p~ m 1 0uB uB1 = 0 1 00✓ · 10− ◆✓ 0100◆ ✓ ◆ 0 100 − − with ~ the Pauli matrices,B 00 0 1 C B 1000C B 1000C B − C B C B − C @ A @ A @ A 01 0 i 10 10 σ1 = ; 10σ2 = 0− 0 ; σ3000= i; = 000. i 10 i 0 0 1 − 01 − ✓ ◆ 01✓ 0 0◆ 00✓ −i ◆0 ✓ 00◆ i 0 γ2 = ↵2 = 0 1 0 1 = 0 1 And thus: 00 10 0 i 00 0 i 00 − − B 00 0 1 C B i 000C B i 00 0C B (~ p~)−uB C=(BE m)uA C B − (1) C @ · A @ − A @ A (~ p~)u =(E + m)u , (2) i 000· A B i 000 − where uA and uB are two-component0 i 00 objects. Let’s inspect this two-fold0 i degeneracy,00 and γ1γ2 = 0 1 γ2γ1 = 0 − 1 find the observable that distinguishes00 thei 0 two components. 00i 0 − B 000i C B 000 i C B C B − C Niels Tuning (9) b) Consider an electron@ with the momentumA in the z-direction,@ p~ =(0, 0,p). WhatA do you find for ~ p~ ? · 1 c) What is the eigenvalue of 1~ pˆ for the eigenfunction 2 · 0 χ = 1 ✓ ◆ withp ˆ = p/~ p~ the vector in the direction of p~ with unit length. What does this value correspond| | to, you think? 1 d) Supposep ˆ can point in any direction, what is then the meaning of 2~ pˆ? What are the possible eigenvalues? · 2 3 Spinors We saw that the requirement of a relativistically correct, but linear equation led to the Dirac equation, (iγµ@ m) = 0, with being a four component spinor. µ − a) H =(~↵ p~ + βm) gives E2 = p2 + m2 if the matrices anticommute, ↵ , ↵ = · { i j} ↵i↵j + ↵j↵i = 0. Usually we use the γ matrices, γ =(β, ~↵). Show that indeed γ1γ2 = γ2γ1, using the Pauli-Dirac representation, 0 0 ~ β = ; ~↵ = . 0 ~ 0 ✓ − ◆ ✓ ◆ ipx µ With = u(p)e− and pµ i@µ we get (γ pµ m)u(p) = 0. Looking for the eigenvectors, it is easier to go back to the! original form; Hu−=(~↵ p~ + βm)u = Eu, which leads to · m ~ p~ u u Hu = A = E A ~ p~ m· u u ✓ · − ◆✓ B ◆ ✓ B ◆ with ~ the Pauli matrices, 01 0 i 10 10 σ = ; σ = ; σ = ; = . 1 10 2 i −0 3 0 1 01 ✓ ◆ ✓ ◆ ✓ − ◆ ✓ ◆ And thus: (~ p~)u =(E m)u (1) · B − A (~ p~)u =(E + m)u , (2) Exercises Lecture 2: QM· andA ScatteringB where uA and uB are two-component objects. Let’s inspect this two-fold degeneracy, and find the observable that distinguishes the two components. b) Consider an electron with the momentum in the z-direction, p~ =(0, 0,p). What do you find for ~ p~ ? · c) What is the eigenvalue of 1~ pˆ for the eigenfunction 2 · 0 χ = 1 ✓ ◆ withp ˆ = p/~ p~ the vector in the direction of p~ with unit length. What does this value correspond| | to, you think? 1 d) Supposep ˆ can point in any direction, what is then the meaning of 2~ pˆ? What are the possible eigenvalues? · b) 2 p 0 ~ p~ = σ p = · 3 0 p ✓ − ◆ c) 1 1 0 1 0 ( ~ pˆ)χ = σ = 2 · 2 3 1 −2 1 ✓ ◆ ✓ ◆ The eigenvalue -1/2 is the z-component of the spin. d) 1~ pˆ is the spin component in the direction of motion. 2 · Possible eigenvalue: 1/2. Niels Tuning (10) ± e) Positive and negative helicity: 1000 1 (1) ~ (1) σ3 0 uA 0 10 0 0 (1) (⌃ pˆ)u = (1) = 0 − 1 0 1 =+u · 0 σ3 u 0010 pz/(E + m) ✓ ◆ B ! B 000 1 C B 0 C B − C B C @ A @ A (⌃~ pˆ)u(2) = u(2) · − 4 Rutherford scattering and Cross section e) dσ Z Z ↵ 2 1 1 2 = 1 2 =(2mZ Z ↵)2 2 4 ✓ 1 2 2 2 2 ✓ d⌦ mv0 4 sin 4m v0 sin ⇣ ⌘ 2 ⇣ 2 ⌘ 0 2 2 0 2 2 2 2 2 2 2 ✓ q = 0 1 =(mv0) (sin ✓+(1 cos ✓) )=2(mv0) (1 cos ✓)=4m v0 sin − − mv0 sin ✓ − − 2 − B mv (1 cos ✓) C B 0 − C @ A 2 Exercises Lecture 2: QM and Scattering e) Let’s consider the operator ~ pˆ 0 ⌃~ pˆ · , · ⌘ 0 ~ pˆ ✓ · ◆ b) What are its eigenvalues for p 0 ~ (1)p~ = σ3p = (2) (1) u· A (2) 0 uAp u = (1) ,u✓= −(2) ◆ uB ! uB ! c) where: 1 1 0 1 0 ( ~ pˆ)χ = σ = 3 1 1 1 2 · 12 −20 0 u(1) = ,u(1) = ~ p/~ (E+m) ✓ u◆(2) = ✓ ,u◆ (2) = ~ p/~ (E+m) TheA eigenvalue0 -1/2B is· the z-component0 of theA spin. 1 B · 1 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ d) (Hint:1~ pˆ is rotate the spin your component frame such thatin thep~ points direction along of motion.the z-axis, such that you only 2 · needPossible to worry eigenvalue: about p3.)1/2. ± e) Positive and negative helicity: 4 Rutherford scattering 1000 1 (1) ~ (1) σ3 0 uA 0 10 0 0 (1) We calculate(⌃ pˆ)u the= distribution of scattering(1) = angles0 for− charged particles1 0 on a charged1 tar-=+u · 0 σ3 u 0010 pz/(E + m) get, like alpha particles✓ scattering◆ o↵B gold! nuclei as done by Ernest Rutherford in 1913. B 000 1 C B 0 C B − C B C @ A @ A (⌃~ pˆ)u(2) = u(2) · − Niels Tuning (11) 4 Rutherford scattering and Cross section e) dσ Z Z ↵ 2 1 1 2 = 1 2 =(2mZ Z ↵)2 2 4 ✓ 1 2 2 2 2 ✓ d⌦ mv0 4 sin 4m v0 sin ⇣ ⌘ 2 ⇣ 2 ⌘ 0 2 2 0 2 2 2 2 2 2 2 ✓ q = 0 1 =(mv0) (sin ✓+(1 cos ✓) )=2(mv0) (1 cos ✓)=4m v0 sin − − mv0 sin ✓ − − 2 a) The incoming particle− arrives with an impact parameter b, and initial velocity v0.
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