# Modern Physics: Quantum Physics & Relativity

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Modern Physics:Quantum Physics & Relativity

You can’t get to Modern Physicswithout doing Classical Physics!The fundamental laws andprinciples of Classical Physicsare the basis Modern Physics.

Isaac Newton(1642 -1727)In Principia (1687 )Newton Invented Calculus 3 Laws of Motion Universal Law of GravityThe force of gravity is Universal: The same force that makesan apple fall to Earth, causes the moon to fall around theEarth and the planets to orbit the Sun.

James Clerk Maxwell 1860sLight is wave. The medium is the Ether.The Speed is constant, c.Maxwell’s Equationsc 1 0 o 3.0 x108 m / s

Michelson-Morely Experiment1887Measure the Speed of the Ether WindLight should travel faster with the windand slower against it during the year.

Rotate arms to produce interference fringes and finddifferent speeds of light caused by the Ether Wind, due toGalilean Relativity: light should travel slower against theEther Wind. From that you can find the speed of the wind.https://www.youtube.com/watch?v 7qJoRNseyLQ

Michelson-MorelyExperiment1887The speed of light is independent of the motion andis always c. The speed of the Ether wind is zero.OR: Lorentz ContractionIf there is an ether but we can’tdetect it because the apparatusshrinks by a factor :1 v2 / c2

A Problem with ElectrodynamicsThe force on a moving charge depends on the Frame.Charge Rest FrameWire Rest Frame(moving with charge)(At rest with wire)F 0F qvB sin

The Laws ofPhysics must be thesame in allreference frames!!

Einstein doesn’t like it:The Force Depends on Your PerspectiveCharge Rest Frame(moving with charge)F 0Wire Rest Frame(moving with wire)F qvB sin Einstein realized this inconsistency and could have chosen either: Keep Maxwell's Laws of Electromagnetism, and abandon Galileo'sSpacetime or, keep Galileo's Space-time, and abandon the Maxwell Laws.

Lorentz Contraction in WireMoving charges in the wire cause a Lorentz contraction in the distancebetween the moving charges in the wire so that from the rest frame ofthe charge outside the wire (or at rest in the wire) the moving chargesbunch up and thus give a net charge to the wire.

Eisntein Saves Maxwell!The force on a moving charge does NOT depend on the Frame.Charge Rest FrameWire Rest Frame(moving with charge)(moving with wire)F qvB sin F kq1q2 / r2From the rest frame of the charge, the wire is charged and it feels aCoulomb force. The forces in each frame are equal though they aredue to different causes!

Magnetic Fields are SR effects ofElectric Fields!

On the Electrodynamics of Moving Bodies1905

Einstein’s Principle of Relativity Maxwell’s equations are true in all inertial referenceframes. Maxwell’s equations predict that electromagneticwaves, including light, travel at speed c 3.00 108 m/s 300,000km/s 300m/µs . Therefore, light travels at speed c in all inertialreference frames.Every experiment has found that light travels at 3.00 108m/s in every inertial reference frame, regardless of how thereference frames are moving with respect to each other.

Special Theory:Inertial Frames:Frames do not acceleraterelative to eachother; FlatSpacetime – No GravityThey are moving on inertiaalone.General Theory:Noninertial Frames:Frames accelerate: CurvedSpacetime, Gravity &acceleration.

Albert Einstein1916The General Theory of Relativity

Postulates of SpecialRelativity19051. The laws of physics are the same in all inertial reference frames.2. The speed of light in a vacuum is constant in all inertialreference frames, independent of the relative motion of sourceand observer. We assume a vacuum for this class.3. Months later E mc2c 3.0 x10 m / s 300m / s8

Slower than Light?Light slows down in a medium like glass as a function ofwavelength. Who slows down more, red or blue in glass?

Faster than Light?Cherenkov radiation is electromagnetic radiation emitted when acharged particle passes through an insulator at a speed greater thanthe speed of light in that medium. The characteristic "blue glow" ofnuclear reactors is due to Cherenkov radiation.

Newton’s Principia in 1687.Galilean RelativityVelocities add:V U V'V’ 25m/sV 45m/sU 20 m/s

Relative Velocity Two observers moving relative to each other generally donot agree on the outcome of an experiment For example, observers A and B below see different pathsfor the ball and measure different velocities:Velocity of ballrelative toobserver BvvvvbB vbA vABVelocity of ballrelative toobserver AVelocity of Arelative toobserver B

What if instead of a ball, it is alight wave?Do velocities add according toGalilean Relativity?

Light travels in a vacuum at c in allinertial reference frames, regardlessof the relative motion.

Clocks slow downand rulers shrinkin order to keep thespeed of light thesame for allobservers!

Time is Relative!Space is Relative!Only the SPEEDOF LIGHT isAbsolute!

Time DilationSpaceship FrameEarth Frame:

Proper Time t t '1 vc22 t 'The smallest time interval t0 between two events ismeasured in the reference frame in which both events occur atthe same place (space ship) and is call the proper time. The“stretching out” of the time interval is called time dilation. Inanother frame moving at speed v with respect to the first, themeasured time interval is increased by a factor of gamma(Earth frame). The ‘event’ is the light bouncing in thespaceship.Space Ship Frame: Proper Time:Earth Lab Frame: Dilated Time: t ' t t '

v , 1c 11 vc22, 1

Time Dilation – Generalization If a clock is moving with respect to you, the timeinterval between ticks of the moving clock isobserved to be longer that the time intervalbetween ticks of an identical clock in yourreference frame All physical processes are measured to slow downwhen these processes occur in a frame movingwith respect to the observer– These processes can be chemical and biological as wellas physicalThe faster you move, the longer you live relative to people atrest! Relative to yourself, you age at a normal rate!

Time Dilation –Verification Time dilation is a veryreal phenomenon thathas been verified byvarious experiments These experimentsinclude:– Airplane flights– Muon decay

Time Dilation Verification –Muon Decays Muons are unstable particles that have thesame charge as an electron, but a mass207 times more than an electron Muons have a half-life of Δtp 2.2 µswhen measured in a reference frame atrest with respect to them (a) Relative to an observer on the Earth,muons should have a lifetime of Δtp (b) A CERN experiment measured lifetimesin agreement with the predictions ofrelativity

Sample ProblemSuperwoman can travel at 0.75c in her glass space ship.She has to fly to Beta Diva, 25.0 light years away as measured from Earth in theEarth frame, to battle alien evil guys.a)b)What is the total time for the trip for Superwoman?What is the total time you measure on Earth?What is the event? Which frame measures the proper time?Beta Diva t t '1 vc22 tPc yrly c yr 1 ly

As speed approach c, lengthscontract.

Length Contraction: DistancesL LP / LPv1 2c2Proper Length of ShipRest (Proper) Frame DistanceEarth Frame measures L0Moving Frame DistanceRocket Frame measures L

Length ContractionThe proper length of an object is longest in the referenceframe in which it is at rest. In another frame moving parallel tothe object, its length is shortened by a factor of gamma.L' LP / LPv1 2c2LPRuler (Proper) FrameMeasures length LPEarth FrameMeasures length L

Sample ProblemSuperwoman can travel at 0.75c in her glass space ship.She has to fly to Beta Diva, 25.0 light years away as measured from Earth in theEarth frame, to battle alien evil guys.a)b)c)d)What is the total time for the trip for Superwoman?What is the total time you measure on Earth?How far is Beta Diva as measured by Superwoman?As Superwoman leaves Earth, you measure her length as she flies overhead at0.75c. What is her length you measure? At rest she measures 2.75 m tall.Beta Diva

ExBinomial Expansion Trick for low speeds.Shrinking School Bus The binomial approximation is useful when we need tocalculate a relativistic expression for a nonrelativisticvelocity v c.Slide 36-95

Example

Example 36.9 The Shrinking School BusSlide 36-98

2-D Relative Motion Book ProblemA moving rod is observed to have a length of 2.00 mand to be oriented at an angle of 30.0 with respectto the direction of motion, as shown. The rod has aspeed of 0.995c. (a) What is the proper length of therod? (b) What is the orientation angle in the properframe?Conceptual Check:Is the proper length going to beshorter? Longer? The same?Is the proper angle going to besmaller? Larger? The same?

Beth and Charles are atrest relative to eachother. Anjay runs past atvelocity v while holding along pole parallel to hismotion. Anjay, Beth, andCharles each measurethe length of the pole atthe instant Anjay passesBeth. Rank in order,from largest to smallest,the three lengths LA, LB,and LC.A.B.C.D.E.LA LB LCLB LC LALA LB LCLA LB LCLB LC LA

Beth and Charles are atrest relative to eachother. Anjay runs past atvelocity v while holding along pole parallel to hismotion. Anjay, Beth, andCharles each measurethe length of the pole atthe instant Anjay passesBeth. Rank in order,from largest to smallest,the three lengths LA, LB,and LC.A.B.C.D.E.LA LB LCLB LC LALA LB LCLA LB LCLB LC LA

Last time .Time Dilation t t '1 vc22 tPLength Contractionv1 2cL' LP / LP 11 vc222, 1v , 1c

Length Contraction Paradox?Strange but not a paradox. They are not both true at the sametime to each other. They are true relative to their own frame!

Twin Paradox?Strange but not a paradox. Helen undergoes accelerations andthe situation is not symmetric. Special Relativity applies onlyto inertial, non-accelerating motion!!! Helen is younger whenshe returns because she is the one who was traveling,stopping, turning around, returning.

Quiz Twin ProblemThe identical twins Speedo and Goslo join amigration from the Earth to Planet X. It is 20.0 lyaway in a reference frame in which both planets areat rest. The twins, of the same age, depart at thesame time on different spacecraft. Speedo’s crafttravels steadily at 0.950c, and Goslo’s at 0.750c.Calculate the time difference between the twins afterGoslo’s spacecraft lands on Planet X. Which twin isthe older?

The Lorentz VelocityTransformationsConsider two reference frames S and S'. An object moves atvelocity u along the x-axis as measured in S, and at velocityu' as measured in S' . Reference frame S' moves withvelocity v relative to S, also along the x-axis.The Lorentz velocity transformations are:NOTE: It is important to distinguish carefully between v,which is the relative velocity between two reference frames,and u and u' which are the velocities of an object asmeasured in the two different reference frames.

Addition of Velocitiesv u'.5c cu cvu '.5cc1 21 2ccIf v .5c

ProblemSuperwoman can travel at 0.75c in her glass space ship.She has to fly to Beta Diva, 25.0 light years away as measured from Earth in theEarth frame, to battle alien evil guys.a)b)c)d)e)What is the total time for the trip for Superwoman?What is the total time you measure on Earth?How far is Beta Diva as measured by Superwoman?As Superwoman leaves Earth, you measure her length as she flies overhead at0.75c. What is her length you measure? At rest she measures 2.75 m tall.If she launches a space pod to Beta Diva while en route at 0.4c, what is thespeed at which it approaches Beta Diva as measured by Beta Diva?Beta Divav u'u vu '1 2c

The Lorentz TransformationsConsider two reference frames S andS'. An event occurs at coordinates x, y,z, t as measured in S, and the sameevent occurs at x', y', z', t' as measuredin S' . Reference frame S' moves withvelocity v relative to S, along the xaxis.The Lorentz transformationsfor the coordinates of one event are:S : ( x, y, x, t )S ' : ( x ', y ', x ', t ')

Lorentz Transformations, Pairs of EventsThe Lorentz transformations can be written in a form suitablefor describing pairs of eventsFor S to S’For S’ to S x ' γ x v t x γ x ' v t ' v t ' γ t 2 x c v t γ t ' 2 x ' c

Relativity of Simultaneity Two events occurring simultaneously in onereference frame are not simultaneous in anyreference frame moving relative to it. Neitherreference frame is ‘correct’: There are nopreferred or special reference frames.It’s all relative baby! Any two observers at rest in the same frameMUST measure simultaneous events to besimultaneous if even if they don’t SEE it thatway. That means that they measure the timedifference between them to be zero. Use theLorentz transformationO’: front is struck before backO: sees strikes simultaneousSeeing is NOT the same as Being!The time when an event is SEEN is not ecessarilywhen the event actually happens!

SimultaneityFireworks go off at the same time according to Earth clocks intwo cities, Alum and Boron, that are 300 km apart. The peoplein a spaceship that is flying in a straight line from Alum toBoron at 0.8c also observe the fireworks when the spaceshipis directlty over Alum. Do they see the fireworks in the twocities simultaneously? If the people in the spaceship say thefireworks were not simultaneous in Alum and Boron, howlong before or after the fireworks flashed at Alum did thefireworks flash at Boron according to their calculations?Sketch the situation.

The Spacetime Interval in Invariant Consider two events that are separated in time by aninterval t, and are separated in space by an interval x. Let us define the spacetime interval s between the twoevents to be: The spacetime interval s has the same value in all inertialreference frames. That is, the spacetime interval between two events is aninvariant.Slide 36-80

Example 36.7 Using the Spacetime IntervalSlide 36-81

E2 mc

Invariant MassInvariant mass is independent of frame of reference and isthe mass as measured in the proper frame:Rest Energy:E0 m0cInvariant mass:m0 E0 / c22If a mass is moving, than the total energy increases bya gamma factor:Total Energy:E mc 2

2 m E/cMass is bound energy. c2 is the conversion factor – themagnitude is 90 quadrillion joules per kilogram. Even aspeck of matter with a mass of only 1 milligram has a restenergy of 90 billion Joules!!!c 9 x10 J / kg2161 GeV/c2 1.783 10-27 kgThe rest mass of a proton is 0.938 GeV/c2

Proof: E Pure energyconverted intoparticles: pairproduction2mcIrène and Frédéric Joliot-Curie1933

Proof of2E mchttps://www.youtube.com/watch?v hW7DW9NIO9M&index 42&list PL908547EAA7E4AE74

Relativistic EnergyThe total energy E of a particle isThis total energy consists of a rest energyand a relativistic expression for the kinetic energyThis expression for the kinetic energy is very nearly ½mu2when u c.

Kinetic EnergyThe kinetic energy is the total energy less the rest energy:KE E E0 (1 )m0c2

Relativistic Momenutm & MassImagine you are at rest and a bullet shoots past approaching c.What is the momentum of the particle relative to you?p muThe speed is bounded, is the momentum? NO!As the bullet’s speed increases, themomentum and mass increase:“Relativistic” Mass:rest energy: E0 m0 c 2total energy E m0 c 2m m0Note: We don’t use this notation!m m0

Total EnergyE mcE0 mc22p muu 2 upc muc m c ( mc 2 ) Eccu2u2 22 22E mc E (1 2 ) (mc ) E 2 E E02 E 2 2 E 2 E02ccu21 2c2mc 22E ( pc) E0222If m 0 (photon) E 2 ( pc)2 (m0c 2 )2 ( pc)2photon momentum:p E /c

If a photon has momentum, doesit have mass?

You Try: Sample ProblemSuperwoman can travel at 0.75c in her glass space ship.She has to fly to Beta Diva, 25.0 light years away as measured from Earth in theEarth frame, to battle alien evil guys.f) If her rest mass is 65 kg, what is her rest Energy?g) What is her total relativistic Energy you measure? What does she measure? Is itdifferent?2h) What is her kinetic energy?00rest energy: E m ctotal energy E m0 c 2KE E E0 ( 1)m0cBeta Diva2

In problems with collisions or decays,Use conservation of Energy and momentum!Conservation of Energy:Ei E fEtotal mcConservation of momentum:2 p pp muif

43. An unstable particle at rest breaks into two fragmentsof unequal mass. The mass of the first fragment is2.50 10–28 kg, and that of the other is 1.67 10–27 kg.If the lighter fragment has a speed of 0.893c after thebreakup, what is the speed of the heavier fragment?

62. An unstable particle with amass of 3.34 10–27 kg is initiallyat rest. The particle decays intotwo fragments that fly off alongthe x axis with velocitycomponents 0.987c and –0.868c.Find the masses of thefragments.Conservation of Energy:Ei E fEtotal mcConservation of momentum:p mu2 p pif

Energy Released: The Mass DefectAtomic Decay: Parent atoms have more mass than product atoms.The difference is released in the form of Kinetic energy.E 2 mc m mparents mproducts

Compare ReactionsEnergy Released per kg of Fuel (J/kg)Chemical@ 700KC O2 - CO23.3 x 107Fission@ 1000Kn U-235 - Ba-143 Kr-91 2 nFusion@ 108KH-2 H-3 - He-4 n2.1 x 10123.4 x 1014

A rechargeable AA battery with a mass of 25.0 g cansupply a power of 1.20 W for 50.0 min. (a) What is thedifference in mass between a charged and an unchargedbattery? (b) What fraction of the total mass is this massdifference?a) E P t 1.20 J s 50 min 60 s min 3 600 J m Ec2 3 600 J 3 108ms 2 4.00 10 14 kg m 4.00 10 14 kg 12b) 1.60 10m25 10 3 kg

E 2 mcSolar Flux: 3.77 x1026Wdm 4.19 x109 kg / sdt

Creating Matter From Pure EnergyMatter-AntiMatterE 2 mc

Atomic Mass Units1u 1/12 mass of Carbon-121u 1.6605x10 27kg 931.5MeV / c238.0508u 234.0436u24.0026u m 238.0508 234.0436 4.0026 u 0.0046uE mc 0.0043 931.5MeV / c 4.3MeV22Most of this energy is Kinetic!

Electron Volts1 eV is the energy an electron gains across a 1 volt potential.E qV1eV (1.6 x10 19 C )(1J / C ) 191eV 1.6 x1019J OR 1J 10 eVThe Tevatron: 1 Trillion (Tera) Electron VoltsMass is given as GeV/c2

The Tevatron: Fermi Lab E 2 mc

LHC: Large Hadron ColliderE 2 mcHiggs Boson 3TeVThe protons will each have an energy of 7 TeV, giving a totalcollision energy of 14 TeV. It will take around 90 s for anindividual proton to travel once around the collider.

E 2 mcStanford Linear Accelerator (SLAC)The International Linear Collider is a proposed future international particle accelerator. Itwould create high-energy particle collisions between electrons and positrons, theirantimatter counterparts. The ILC would provide a tool for scientists to address many of themost compelling questions of the 21st century-questions about dark matter, dark energy,extra dimensions and the fundamental nature of matter, energy, space and time.http://www.slac.stanford.edu/

E 2 mcIn the ILC's design, two facing linear accelerators, each 20 kilometers long,hurl beams of electrons and positrons toward each other at 99.9999999998percent of the speed of light (with a few TeV energy). Each beam containsten billion electrons or positrons compressed down to a minuscule threenanometer thickness. As the particles hurtle down the collider,superconducting accelerating cavities operating at temperatures nearabsolute zero pump more and more energy in

Modern Physics: Quantum Physics & Relativity. You can’t get to Modern Physics without doing Classical Physics! The fundamental laws and principles of Classical Physics are the basis Modern Physics

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