“Elementary Particles” Lecture 2 Niels Tuning Harry van der Graaf Ernst-Jan Buis Martin Fransen Niels Tuning (1)
Plan Theory Detection and sensor techn. Quantum Quantum Forces Mechanics Field Theory Light Interactions Scintillators with Matter PM Accelerators Tipsy Bethe Bloch Medical Imag. Cyclotron Photo effect X-ray Compton, pair p. Proton therapy Bremstrahlung Experiments Cherenkov Fundamental Astrophysics Charged Particles Physics Cosmics ATLAS Particles Km3Net Grav Waves Si Neutrinos Virgo Gaseous Lisa Pixel … Special General Optics Gravity Relativity Relativity Laser Niels Tuning (2)
Plan Today Theory 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 Martin 11) +12) Fundamental Accelerators 6) Ernst-Jan Martin Physics 13) + 14) Astrophysics Charged Excursions Particles Experiments 9) Ernst-Jan 1) Niels 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)
Thanks • Ik ben schatplichtig aan: – Dr. Ivo van Vulpen (UvA) – Prof. dr. ir. Bob van Eijk (UT) – Prof. dr. M. Merk (VU) Niels Tuning (5)
Plan 1) Intro: Standard Model & Relativity 11 Feb 2) Basis 1900-1940 18 Feb 1) Atom model, strong and weak force 2) Scattering theory 3) Hadrons 1945-1965 10 Mar 1) Isospin, strangeness 2) Quark model, GIM 4) Standard Model 1965-1975 24 Mar 1) QED 2) Parity, neutrinos, weak inteaction 3) QCD 5) e + e - and DIS 1975-2000 21 Apr 6) Higgs and CKM 2000-2015 12 May Niels Tuning (6)
Exercises Lecture 1: Special Relativity x’ = ct’ Niels Tuning (7)
Exercises Lecture 1: Special Relativity 2 Niels Tuning (8)
Exercises Lecture 1: Special Relativity Niels Tuning (9)
Exercises Lecture 1: Special Relativity 2 Relativistic momentum Given 4-vector calculus, we know that p µ p µ = E 2 /c 2 − ⃗ p 2 = m 2 0 c 2 . a) Show that you get in trouble when you use E = mc 2 and ⃗ p = m ⃗ v . b) Show that E = γ m 0 c 2 and ⃗ p 2 = m 2 v obey E 2 /c 2 − ⃗ 0 c 2 . p = γ m 0 ⃗ a) Using E = mc 2 and ⃗ p = m ⃗ v , one finds: p 2 = m 2 c 2 − m 2 v 2 = m 2 ( c 2 − v 2 ) ̸ = m 2 c 2 E 2 /c 2 − ⃗ b) Using E = γ m 0 c 2 and ⃗ p = γ m 0 ⃗ v , one finds: p 2 = γ 2 ( m 2 ( c 2 − v 2 )) = m 2 c 2 1 − v 2 /c 2 E 2 /c 2 − ⃗ 1 − v 2 /c 2 = m 2 c 2 . Niels Tuning (10)
Exercises Lecture 1: Special Relativity 3 Center-of-mass energy Niels Tuning (11)
Exercises Lecture 1: Special Relativity Niels Tuning (12)
Lecture 1: Standard Model & Relativity • Standard Model Lagrangian • Standard Model Particles Niels Tuning (13)
Lecture 1: Standard Model & Relativity • Theory of relativity – Lorentz transformations (“boost”) – Calculate energy in colissions • 4-vector calculus • High energies needed to make (new) particles Niels Tuning (14)
Outline for today • Quantum mechanics: equations of motions of wave functions – Schrodinger, Klein Gordon, Dirac • Forces – Strong force, pion exchange – Weak nuclear force, decay • Scattering Theory – Rutherford (classic) and QM – “Cross section” – Coulomb potential – Yukawa potential – Resonances Niels Tuning (15)
D. Griffiths “Introduction to Elementary Particles” • Lecture 1: – ch.3 Relativistic kinematics • Lecture 2: ch.5.1 Schrodinger equation – – ch.7.1 Dirac equation – ch.6.5 Scattering • Lecture 3: – ch.1.7 Quarkmodel – ch.4 Symmetry/spin • Lecture 4: – ch.7.4 QED – ch 11.3 Gauge theories • Lecture 5: – ch.8.2 e+e- – ch.8.5 e+p • Lecture 6: – ch.11.8 Higgs mechanism Niels Tuning (16)
Lecture 2: QM, Dirac and Scattering • Introduce “matter particles” – spinor ψ from Dirac equation • Introduce “force particles” • Introduce basic concepts of scattering processes Niels Tuning (17)
Quantum mechanics
From classic to quantum Why does the black body Why does the electron not spectrum look like it does? fall onto the nucleus? � Finite number of wavelengths ( E=h ν ) � Finite number of nuclear orbits • The wavefunction ψ describes a system (eg. particle) • Physical quantities are given by operators Niels Tuning (19)
Wavefunction Each particle may be described by a wave function ψ ( x,y,z,t ), real or complex, having a single value for a given position ( x,y,z ) and time t • In QM a particle is not localized • Probability of finding a particle somewhere in a volume V of space: 2 P ( r , t ) dV ( r , t ) dV = Ψ • Probability to find particle 2 ( r , t ) dV 1 anywhere in space = 1 ∫ Ψ = Ø condition of normalization: all space Niels Tuning (20)
Operator Any physical quantity is associated with an operator § An operator O: the “ recipe ” to transform ψ into ψ ’ – We write: O ψ = ψ ’ • If O ψ = o ψ then – ψ is an eigenfunction of O and – o is the eigenvalue. We have “ solved ” the wave equation O ψ = o ψ by finding simultaneously ψ and o that satisfy the equation. Ø o is the measure of O for the particle in the state described by ψ Niels Tuning (21)
Correspondence? • What operator belongs to which physical quantity? Classical quantity QM operator E kin Niels Tuning (22)
Example Let ’ s try operating: • Wavefunction: ikx t ( ) ( x , t ) A cos[ kx t ] i sin[ kx t ] Ae − ω Ψ = − ω + − ω = • Momentum operator : � � ∂ i ( kx t ) i ( kx t ) i ( kx t ) ˆ p ( x , t ) Ae ikAe � kAe � k ( x , t ) − ω − ω − ω Ψ = = = = Ψ x i x i ∂ • Or energy operator: ∂ ˆ i ( kx t ) i ( kx t ) i ( kx t ) ( ) � � � ( ) E x , t i Ae − ω i ( i ) Ae − ω Ae − ω E x , t Ψ = = − ω = ω = Ψ t ∂ Ø Ψ is indeed eigenfunction ( ђ k and ђω are the eigenvalues for ˆ p and ˆ E) Niels Tuning (23)
Expectation value Average value of physical quantity: expectation value Think of the Staatsloterij: x : prize i p ( x ) : probabilit y to win that prize i E ( X ) x p ( x ) 0 . 697 13 . 50 9 . 41 EUR = ∑ = × = + ∞ i i [ ] ˆ * i W ( x , t ) W ( x , t ) dx ∫ = Ψ Ψ − ∞ + ∞ + ∞ [ ] [ * ] 2 ikx ikx ikx ( x ) Ae with ( x ) dx Ae Ae dx 1 , ψ = ∫ ψ = ∫ = Example: − ∞ − ∞ where A 0 as limits of integratio n → → ∞ + ∞ � ∂ ⎡ ⎤ [ ] * ikx ikx p Ae Ae dx = ∫ ⎢ ⎥ i x ∂ ⎣ ⎦ − ∞ + ∞ � + ∞ [ ] * [ ] [ ] [ * ] ikx ikx ikx ikx p Ae ik Ae dx � k Ae Ae dx � k p = ∫ = ∫ = = i − ∞ − ∞ ≡ 1 Niels Tuning (24)
Heisenberg How to describe a particle that is “ localized ” somewhere, but which is also “ wave-like ” ? ikx Ae ψ = Ø k can be any value: Ø Fourier decomposition of many frequencies Ø The more frequencies you add, the more it gets localized Ø The worse you know p, the better you know x ! Niels Tuning (25)
Heisenberg How to describe a particle that is “ localized ” somewhere, but which is also “ wave-like ” ? Ø Fourier decomposition of many frequencies Ø The more frequencies you add, the more it gets localized Ø The worse you know p, the better you know x ! Niels Tuning (26)
Twitter Niels Tuning (27)
Heisenberg • Uncertainty relation à Commutation relation Ø A wave function cannot be simultaneously an eigenstate of position and momentum • Suppose it was , what then?? Ø Then the operators would commute: ( ) ( ) ( ) P X XP P x Xp x p x p 0 − Ψ = − Ψ = − Ψ = x x x 0 0 0 0 0 0 Niels Tuning (28)
Schr ö dinger Classic relation between E and p: Quantum mechanical substitution: (operator acting on wave function ψ ) Schrodinger equation: Solution: (show it is a solution) Niels Tuning (29)
Intermezzo: “radial Schrödinger equation” • Polar coordinates • Separate variables • Three differential equations for R, φ , θ : • Radial Schrödinger equation – Potential is augmented by “centrifugal barrier” : (apparent centrifugal force) Linear momentum Potential energy Angular momentum Niels Tuning (30)
Klein-Gordon Relativistic relation between E and p: Quantum mechanical substitution: (operator acting on wave function ψ ) Klein-Gordon equation: or : Solution: with eigenvalues: But! Negative energy solution? Niels Tuning (31)
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