Quantum (1920 s-) quantum (1927-) PHYS-3301 Lectue 2 Classical phsics Newtonian Mechanics, Themodnamics Statistical Mechanics, El.-Mag. (1905) Mawell s Equations of electomagnetism (1873) Aug. 31, 2017 Quantum (1920 s-) quantum (1927-) Quantum (1920 s-) quantum (1927-) Classical phsics Nonclassical Phsics, El.-Mag. (1905) Classical phsics How Small, El.-Mag. (1905) Stuctue of the couse Moden Phsics is Small? How Fast is Fast?
Small => e.g. atomic size Quantum (1920 s-) quantum (1927-) Quantum (1920 s-) quantum (1927-) Classical phsics, El.-Mag. (1905) Classical phsics, El.-Mag. (1905) Fast => ~c c= the elocit of light u Fist Topic Moden Phsics Two basic ideas Time and space ae not absolutes. Paticles behae like waes and waes behae like paticles. Two banches Special Relatiit Quantum Mechanics With an undestanding of these banches, we can then eploe aeas of moden phsics such as supeconductiit, moden optics, nuclea phsics, paticle phsics and cosmolog - along with a host of othe aeas of science. 7 Chapte 2 Special Relatiit 1. Basic Ideas 2. Consequences of Einstein s Postulates 3. The Loentz Tansfomation Equations 4. The Twin Paado 5. The Dopple Effects 6. Velocit Tansfomation 7. Momentum & Eneg 8. Geneal Relatiit & a 1 st Look at Cosmolog 9. The Light Baie 10. The 4 th Dimension
What is elatiit? The dependence of aious phsical phenomena on elatie motion of the obsee & the obseed objects Popeties: An obsee in motion and an obsee at est ma disagee on the sequence of eents. Howee, the do agee on the esults. The laws that goen the motion ae the same. We can tansfom fom one fame to anothe using a Galilean Tansfomation. Galilean-Newtonian Relatiit Galilean-Newtonian elatiit is known as a classical theo. Einstein s special theo of elatiit is known as a moden theo. 9 10 z Galilean Tansfomation z K K K is at est and K is moing with elocit Aes ae paallel Refeence fame: A sstem of coodinates defined fo an obsee. Conention is to hae the obsee at the oigin of the sstem & to define all distances elatie to that point. Each efeence fame can hae it s own oigin. 11 K and K ae said to be INERTIAL CRDINATE SYSTEMS (fames of efeence) Refeence fame K is moing at constant elocit w..t. to K. (Let s estict motion to -ais) z Galilean Tansfomation z K K Time is absolute Inetial Refeence fame (K ): A fame of efeence in which an object epeiencing zeo net foce moes at constant elocit. i.e. F = 0, so a fame is not acceleating. Galilean Tansfomation holds = t = z = z t = t?? 12
Galilean Tansfomation Galilean Tansfomation K K EVENT K K EVENT Refeence fame K is moing at constant elocit w..t. to K. (Restict motion to -ais) Tansfom fom K to K: t = t = t = t Note: we will use u and u to designate the elocit of the ball in espectie coodinate sstems (i.e. elocit of an object moing elatie to a fame) = Time is absolute z = z t = t 13 constant elocit z = z t = t 14 Galilean Tansfomation K K EVENT t Galilean Tansfomation K K Let s check that the 2 nd Newton s Law is inaiant unde Galilean Tansfomations: diffeentiate again : d!! dt = d! u u dt acceleation is the same in all IRFs!! a! = a!! F! = F 15 z z What does it mean fo a quantit o law to be INVARIANT? The quantit o law emains unchanged when afte undegoing a tansfomation to a diffeent coodinate sstem. Non-inaiant Quantities = & = Inaiant Quantities a = a, F = ma = ma = F The Laws of Mechanics ae unchanged unde a Galilean tansfomation 16
Galilean Tansfomation s E&M K z z K In 1873, Mawell fomulated Equations of Electomagnetism. Mawell s equations descibe e well all obseed E&M phenomena, but the ae not inaiant unde G.T.! E = ρ / ε0 B = 0 B Some odd things: E = t a chage in motion poduces a B-field, but a chage E at est does not. B = µ0 J + ε 0 µ0 the speed of light is the same in all IRFs, at odds t with Galilean elocit addition. Histoical Pespectie Light is a wae. Waes equie a medium though which to popagate. Poposed solution: Light waes tael in a medium called ethe (o aethe: fom Geek, meaning uppe ai). (Like sound waes in ai o wate waes in wate) The Laws of Mechanics ae unchanged unde a Galilean tansfomation ε 0 µ0 = The Laws of E&M ae changed unde a Galilean tansfomation The ethe becomes the absolute and unique fame of efeence whee Mawell s equations hold. 1 c2 Mawell s equations assume that light obes the Newtonian-Galilean tansfomation. This bought into being an idea of a unique stationa RF (ethe), with espect to which all elocities ae to be measued What ae the options? At least one of the following statements must be wong: (a) the pinciple of elatiit applies to 17 both mechanical and E&M phenomena; (b) Mawell eqs ae coect; (c) G. T. ae coect. Wae Popagation 18 The Michelson-Mole Epeiment Epeiment designed to measue small changes in the speed of light was pefomed b Michelson (1852 1931, Nobel ) and Mole (1838 1923). Used an optical instument called an intefeomete that Michelson inented. Deice was to detect the pesence of the ethe. Stat two light beams at the same point & time, then send them on two paths Paallel to Ethe: 19 one path paallel to the motion of the eath though the ethe one pependicula Measue the time diffeence it takes fo them to aie at the same point Pependicula to Ethe: 20
The Michelson-Mole Epeiment Paallel to Ethe: Pependicula to Ethe: Time fo each tip is gien b Poblem: The speed of Eath moing though the ethe is ~10,000 times smalle than the speed of light. Instumentation of the peiod was not sensitie enough to measue such small timing diffeences. Solution: bsee the intefeence between the two beams. 24
The Michelson-Mole Epeiment If the two beams ae moing at diffeent speeds, thee will be a fast am and a slow am. Deduce Δt fom obseed # finges. utcomes of Michelson-Mole Epeiment bseation: No finge. T=peiod of light wae The speed of light was shown to be independent of the motion of the obsee. Helped cast awa the ethe hpothesis E&M eplained the oigin of light, but did not equie a medium fo its wae popagation 1907 Nobel Pize awaded to Michelson (1st Ameican Phsicist) 25 Einstein s Postulates Big poblem at the tun of the centu: 1. Michelson-Mole showed that the Galilean tansfomation did not hold fo Mawell s equation. 2. Mawell s equations could not be wong. 3. Galilean tansfomation did hold fo the laws of. 4. Einstein poposed a solution 26 Einstein s Pinciple of Relatiit (1905) Einstein's Pinciple of Relatiit (the fist postulate of the Special Theo of Relatiit): The laws of phsics ae the same (coaiant) in all inetial efeence fames. 2 nd Postulate: Light moes with the same speed (c) elatie to all obsees The second postulate: The speed of light in acuum is the same to all obsees, egadless of thei motion elatie to the light souce. Thus, Mawell s Equations ae in line with Einstein s Pinciple of Relatiit. Conclusion: Galilean Tansfomations must go. Einstein needed to find a new tansfomation (we will see this toda) The idea of uniesal and absolute time is wong! ne has to come up with coect tansfomations that wok fo both mechanical and E&M phenomena (an speed up to ~c). Consequentl, the laws of hae to be modified to be coaiant unde new (coect) tansfomations. bsees in all inetial sstems measue the same alue fo the speed of light in a acuum. (c = 2.9979 10 8 m/s) Anna measues: Speed of light = c? Bob measues: Speed of light = c and not +c
The Ultimate Speed The speed of light has been defined to be eactl: c = 299 792 458 m/s Einstein s Postulates of Relatiit: Light taels at this ultimate speed, as do an massless paticles. No entit that caies eneg o infomation can eceed this speed limit. No paticle that does hae a mass, can actuall each c. Electons hae been acceleated to at least 0.999 999 999 95 times the speed of light still less than c. 29 Light Souce, Medium and Michelson- Mole Epeiment (Discussion sessions, see Appendi A) Definition of an Eent Consequences of Einstein s Postulates: 1. Relatie Simultaneit 2. Time Dilation 3. Length Contaction Consequence 1: Relatie Simultaneit, o The absence of absolute simultaneit Simultaneous Flash Simultaneit is the popet of two eents happening at the same time in a fame of efeence. An eent is a phsical occuence, independent of an efeence fame Accoding to Einstein's Theo of Relatiit, simultaneit is not an absolute popet between eents; what is simultaneous in one fame of efeence will not necessail be simultaneous in anothe. Simultaneous Aial => AN EVENT
Simultaneous Aial!The same EVENT Fo Bob as well! 2 nd Postulate 2 nd Postulate The speed of light in acuum is the same fo all inetial obsees, egadless of the motion of the souce. THEN: BUT THEN: Simultaneous Aial!AN EVENT Fo Bob as well! Simultaneous Emission is impossible fo Bob Simultaneous Aial!AN EVENT Fo Bob as well! Simultaneous Emission is impossible fo Bob
Both emission and aial ae simultaneous Aial is simultaneous But Emission is not simultaneous Consequence 2: Time Dilation, o The absence of absolute time Two eents occuing at the same location in one fame will be sepaated b a longe time inteal in a fame moing elatie to the fist. This is not an optical illusion. Space and time ae diffeent fo all obsees in elatie motion. The time fo the light to etun:!= 2"/c
The time fo the light to etun:! = 2"/c Longe Path (L) + 2 nd Postulate PRECISELY => T = 2 L/c L 2 = H 2 + ( T/2) 2 # = 2"/c The time fo the light to etun:! > 2"/c L The time fo the light to etun: # > 2"/c Longe Path + 2 nd Postulate T = # (1 (/c) 2 ) -1/2