Mechanics Physics 151

Transcription

1 Mechanic Phyic 5 Lecture 6 Special Relativity (Chapter 7) What We Did Lat Time Defined covariant form of phyical quantitie Collectively called tenor Scalar, 4-vector, -form, rank- tenor, Found how to Lorentz tranform them Ue Lorentz tenor & metric tenor Covariant form of Newton equation with EM force dp Equation of motion K dτ = EM potential 4-vector (φ/c, A) EM field Faraday tenor I gave you a wrong one Faraday Tenor E x ν A A ν K = e uν ef u x x ν F ν i derivative of the EM potential E and B field φ 0 A x A A = = c x t x x0 ν B Faraday Tenor A A A A 3 z y x = = + y z x x3 F ν 0 Ex c Ey c Ez c Ex c 0 Bz B y = Ey c Bz 0 Bx Ez c By Bx 0 What I I gave gave you you (and (and the the textbook) wa wa wrong by by a factor c

2 Multi-Particle Sytem Conider a ytem with particle =,, Total momentum P = p dp Equation of motion for each particle K dτ = Different time for each particle! dp EoM for the total momentum i dt Not a very clean equation dp = γ = γk dτ Trouble ahead Momentum Conervation dp dp = γ = dt γ K dτ Imagine a -particle ytem with no external force But there are internal force between the particle Law of action and reaction K = K dp + = γk + γk Thi i zero only if γ = γ dt To conerve the total momentum in all frame, the particle interacting with each other mut have the ame velocity I thi a weird retriction, or what? Local Interaction Suppoe the particle interact only when in contact Force exchanged only when they collide They hare the ret frame momentarily Same γ Total momentum of a multi-particle ytem i conerved if the interaction between the particle are local Furthermore, Law of action/reaction K = K cannot work over a ditance intantaneouly There mut be delay Conervation law cannot hold Relativity and non-local interaction don t mix

3 Particle Colliion Interaction between particle mut be local Force exchanged when they collide Free motion between colliion Conider the colliion a a black box? We don t know what happen in the box (not claical) Motion outide the box i eay Relativitic Kinematic How much can we learn without opening the box? Center-of-Momentum Frame Local interaction conerve total 4-momentum i.e., total energy and total 3-momentum are conerved n n n E p = p =, p E = = c E p= p = = We know how to Lorentz tranform it ν ν p = p = Lp = Lp i a uual ν i ν i i Define the center-of-momentum frame in which p = 0 It the frame in which the total 3-momentum i zero Or, the center of ma i at ret Often called center-of-ma frame a well CoM Energy and Boot n E p = p =, = c p E CoM frame p, = 0 c There are two particularly ueful quantitie CoM energy E Lorentz invariance p p = p p E i the mallet poible E E c E = p c Boot β of CoM frame relative to the lab. frame Lorentz tranformation ν γe γβe p Lν p c = =, β = p E γ = c c E E 3

4 Two-Particle Colliion Conider colliion of particle on particle at ret p = ( E c, p ) p = ( mc, 0) Total 4-momentum i p= ( E c+ mc, p) Total CoM energy 4 E = p p c = ( m + m ) c + Em c Boot of CoM frame i p mγ v β = = E c+ m c mγ c+ m c Fixed-target colliion E grow lowly with E Approache v /c for large E Creation Threhold Suppoe we are trying to create a new particle In the bet cenario, particle and merge to create a new heavy particle 3 Total 4-momentum would be imply Total CoM energy i E c = p3 p3 = m3c E = mc 3 How much energy E do we have to give particle? 4 4 E = ( m + m ) c + Em c = m c E 3 For large m 3, E grow with m 3 p= p+ p = p3 CoM energy mut match the ma of the new particle ( m m m ) c = m 3 Fixed-Target v. Collider Conider hitting a proton with a proton to make a Higg particle, which i X time heavier than a proton ( m3 m m) c X E = mc p m For X > 00, we d need a >5000 GeV accelerator Particle collider are more energy-efficient p = ( E, p) p = ( E, p) p = p Laboratory i CoM p= ( E+ E,0) Jut need E + E = m c = Xm c 50 GeV + 50 GeV 3 p 4

5 Elatic Scattering Particle hit particle and get elatically cattered p 3 p 3 y p ϑ p Θ p CoM x p 4 Cro ection i calculated in CoM frame By treating it a a central-force problem Experiment i done in the laboratory frame p 4 We need to learn how to tranlate between the CoM and the laboratory frame Elatic Scattering Firt, what the boot? Total momentum i p = p + p = ( E c+ m c, p ) p p c β = = p E + m c 0 Let get γ a well p p p = ( E c+ m c) = m c + m c + Em 0 p E+ mc γ = = p p m c + m c + Em c 4 4 Elatic Scattering Now we can boot p to CoM p ( E c, p,0,0) p = ( E c, p,0,0) = E = γ ( E β pc ) p = γ ( p β E c) p 3 i given by rotating p by Θ p = ( E c, p co Θ, p in Θ,0) 3 Boot thi back to get p 3 E3 = γ ( E + β pc co Θ) p p = p in Θ 3 = γ( p co Θ+ β E c) 3 Scattering angle in lab frame i Velocity of p3 in Θ in Θ in CoM tanϑ = = = p γ (co Θ+ βe ( pc )) γ(co Θ+ β β ) 3 5

6 Elatic Scattering What happen to the kinetic energy? E3 = γ ( β co Θ) E β( co Θ) pc At Θ = 0 E3 = γ ( β ) E = E Make ene With a little bit of work T3 ρ(+ E ) Kinetic = ( co Θ) E = T mc energie T ( + ρ) + ρe ρ = m m Wort cae i Θ = π ( T3) min ( ρ) = T ( + ρ) + ρe Sign wrong in textbook Elatic Scattering ( T ) ( ρ) 3 min= T ( + ρ) + ρe Non-relativitic limit ( T3) min ( ρ) = T ( + ρ) Ultra-relativitic limit ( T3) min ( ρ) ( m m) c = = T ρe m T If m << m, i.e., the target i heavy, almot no energy i lot in the colliion (T 3 ) min i independent of T A T increae, the energy lo become very large Particle Decay Some particle are untable and decay after a while i i Ma of the mother i known from the 4-momenta of the daughter by calculating the total CoM energy mc = E = p p Alo called the invariant ma of the ytem Thi i how particle phyicit find particle that do not live long enough to be directly een Example: Dicovery of J/ψ (November 974) p = p 6

7 Particle Decay A day at the BABAR experiment at SLAC Collide e + and e to generate a few 00,000 Υ(4S) particle each of which decay into two B 0 meon ome of which decay into a J/ψ and a K S J/ψ decay into e + and e, or + and K S decay into π + and π Meaure 3-momenta of the table particle Mae known Calculate 4-momenta Rebuild the decay chain backward and calculate invariant mae of them all Do they match the expected mae? J/ψ ma Combine e + e or + and to ee if they make a J/ψ B 0 ma Combine J/ψ and K S to make B 0 Event /.5 MeV/c BABAR Found 440 ignal out of 30 million Υ(4S) generated Beam-Energy Subtituted Ma (MeV/c ) 7

8 Summary Dicued multi-particle ytem Only local force are amiable with relativity Relativitic kinematic for particle phyic CoM frame, CoM energy, invariant ma and boot Particle creation and collider experiment Elatic cattering Particle decay Next lecture: Lagrangian formalim 8