PH-101:Relativity and Quantum Mechanics

Transcription

1 PH-101:Relativity and Quantum Mechanics Special Theory of Relativity (5 Lectures) Text Book:1. An Introduction to Mechanics Author: Danieal Kleppner & Robert Kolenkow 2. Introduction to Special Relativity Author: Robert Resnick This Week: Thursday(Today), Friday(Tomorrow) Next Week: Monday, Thursday Quantum Mechanics (9 Lectures) Group-II, IV

2 E=m c 2

3 Some facts Einsteinain World Macroscopic world of our ordinary experiences, the speed u of any object is alwasy much less than c. Example: artificial satelite: u/c = , C= m/s Sound speed: u/c = Newtonian World Microscopic world: Electron moving in 10 MeV potential difference: u/c = Electron moving in 40 MeV potential difference: Newtonian u/c = Experimental u/c =

4 January 4, 1643, March 31, 1727 Newtonian World? m v=f ; I ω=n Newton's laws of Mechanics in Macroscopic world You know the Law and applied it in various physical situations

5 Foundation of Newtonian Mechanic was so robust that nobody has dared to say that this is wrong or has to be modified. Euler, D'Alembert, Lagrange, Hamilton, Laplace Generalization of mechanics, Mathemtical Structure Farady, Coulomb, ohm, Maxwell Electromagnetism Long time has passed

6 March 14, 1879, 230 year after April 18, 1955, Is Newton's world valid for any velocity? Certainly Not!! was bothered about more on the Principles base on which those Laws are defined. Symmetry Principles Try to convince you: It is Einsteinian world!! 100 year after September, 2016 Lot of open problems Enough time has passed, IT COULD BE YOUR WORLD!! Year 1905: Einstein, motivated by a desire to gain deeper insight into the nature of electromagnetism, push forward the idea of special theory of relativity

7 Where is the velocity U Symmetries Any physical law should be invariant under some special set of transformations. 1) Rotation and translation of the given corodiante system 2) Transformation from one observer (S) to other(s') who is moving with constant velocity with respect to S. v Linear Transformations

8 Rotation and Translation Rotation in x-y plane Translation along x F Tutorial problem x '=x cosθ+ y sin θ y '= x sin θ+ y cosθ z '=z x '=x+a y '= y z'=z r r ' We do not touch t in Galilean transformation Newton's Law is invariant under space rotaion m d 2 r dt 2 = F (r ) d 2 r d t = d2 r ' 2 d t 2 m d 2 r ' dt 2 = F ' (r ' ) a Linear transformation: Rotation Length is invariant under rotation and translation: ds 2 =dx 2 +dy 2 +dz 2 =dx ' 2 +dy ' 2 +dz ' 2 =ds ' 2

9 Galilean Transformation(GT) r '= r v t x'=x vt, y'=y, z'=z Space and Time Motion is along x-direction v F We do not touch t in Galilean transformation Newton's Law is invariant under GT m d 2 r dt 2 = F ( r ) Velocity addtion: d 2 r r ' d t 2=d2 d t 2 u' = u - v Not talking about: Non-linear transformation: r '= r v t 1 2 f t 2 m d 2 r ' dt 2 = F ' ( r ' ) r m d 2 r ' dt 2 = F ' ( r ' )+ f r ' Inertial frames Linear transformation: Rotation Depending on f: law changes

10 No mechanical experiments carried out entirely in one intertial frame can tell the observer what the motion of that frame is with respect to any other Inertial frame There is no prefered reference frame We can only speak about relative velocity: Galilean relativity Electromagnetism and Galilean relativity!!

11 Another Piller: Maxwell's equation of electromagnetism June 13, November 5, 1879, v c: velocity of light E= 0, B=0 B= 1 c 2 E t E= B t E :Electric field B :Magnetic field Relative velocity of light: c'= c-v With respect to whom c is defined? Michelson-Morley Experiment: could not prove the existence of any frame Unfortunately these set of equtions are not invariant under GT Why are we so obsessed with such an invariance property under GT? Laws should not be dependent upon who is observing! WHO? Class of observes: moving with constant velocity relative to each other Inertial observers

12 There is a problem!! 1) Electromagnetism? 2) Different relativity principle for Electromagnetism 2) Newtonian Mechanics? Einstein did series of 1. thought experiments : based on light and measurement of time! 2. measurement of time

13 Einstein was a Genius worried about basic principle based on which laws are defined 1. Newton's law does not contain velocity explicity 2. Maxwell's equatioin contains velocity of light c explicitly Since velocity of light, C, involves no reference to a medium, C, must be universal constant!! This simple but bold statement changes the very notion of time

14 How and why? Follow Einstein's path: Do some thought experiments 1. Velocity of light is finite 2. Usual Galileon relativity 3. Every observer has his own clock to measure time 4. Every observer has his own light detector Newton's law does not talk about how do we measurement the time

15 Galileo: Time is sacred, it does not transfrom... Einstein: It is more subtle than what you think... t1 t2 How do we measure time? Relativity of simultaneity t1=t2 Finite velocity of light plays crucial role in defining observer's Time Time is observer dependent!! This can be very well explained in space and time diagram

16 'Space and Time' diagram Inertial observer: x=vt x= ( v c ) ct Galilean relativity does not say anything about the magnitude of v ct v=c v<c v=c v>c x

17 ct Events are at two different points Space and Time diagram O0 Talking about a class of observers: moving with constant velocity relative to each other Inertial Observers x=vt, x= ( v c ) ct t x'=f (x,t), t '=g(t, x) event 1 event 2 must be true x Real Experiment

18 ct Events are at two different points Space and Time diagram O1 Talking about a class of observers: moving with constant velocity relative to each other Inertial Observers t1' x=vt, x= ( v c) ct t t2' x'=f (x,t), t '=g(t, x) event 1 event 2 must be true x Thought Experiment

19 ct Events are at two different points Space and Time diagram O2 O0 O1 Talking about a class of observers: moving with constant velocity relative to each other Inertial Observers x=vt, x= ( v c ) ct t x'=f (x,t), t '=g(t, x) event 1 event 2 must be true x Thought Experiment

20 Events are at the same point ct Death t3 v<c Sequence of events: t1' < t2' < t3' t2 Playing v=c x=vt x= ( v c ) ct t1 Birth Sequence of events: t1 < t2 < t3 Thought Experiment x

21 ct v>c t3' t2' t1' Going into the past t3 Sequence of events: t3' < t2' < t1' Death t2 Playing v=c x=vt x= ( v c ) ct t1 Birth Sequence of events: t1 < t2 < t3 Thought Experiment x

22 What we have obsered so far 1. Measurement of time does depened on the position of an event 2. There should exist a limiting velocity which is the velocity of light 3. Galiloe's idea of observer independence should be valid, However, Galilean tranformation has to be modified

23 Einstein's postulates: 1) The laws of physics are the same in all intertial frame. 2) The speed of ligh in free space is same in all inerial frame. What we have learnt: (x, t) & (x', t'): We must find out a linear relation among them keeping velocity of light constant. Assuming: x '=A (x v t),t '=B(t f (x)) y '= y, z '=z x 2 + y 2 + z 2 =c 2 t 2 x ' 2 + y ' 2 +z ' 2 =c 2 t ' 2 A 2 (x 2 2 x v t +v 2 t 2 )+ y 2 +z 2 =B 2 c 2 (t 2 2 f (x)t +f (x) 2 ) ( A 2 x 2 B 2 c 2 f (x) 2 )+2t ( A 2 x v+b 2 c 2 f (x))+ y 2 +z 2 =(B 2 A 2 v 2 c 2 )c2 t 2 =x 2 =0 f (x)= A2 v x B 2 c 2 Consistent solution exists for: A = B = 1 A 2( 2) 1 v2 =1 c

24 Lorentz Transformations x '=A (x v t),t '=B(t v x c ) 2 y '= y, z '=z A=B= x '=γ(x v t),t '=γ(t v x c 2 ) y '= y, z '=z 1 1 v2 c 2 γ Newtonian Limit(non-relativistic) v c 1 x '=(x v t),t '=t y '= y, z'=z Since all ovservers are equivalent, the inverse transformation woule be: x=γ(x '+v t '),t=γ(t '+ v x ' c 2 ) y= y ', z=z ' Time and space get inter-connected Space and Time Space-time

25 ct {x,y,z} {ct} {x,y,x,ct} Space and Time Space-Time We can access only this part Future light cone x Past light cone Maximum velocity is the velocity of light c = constant(vacuum)

26 Offsprings Time dilation (Moving clock runs slow!!) Length Constraction (Running cat is safe for being shorter!!) Relativistic Doppler effect

27 Time Dilation Two events: emission and absorption of light at the same point t a =γ(t ' a + v x ' c 2 ) t b =γ(t ' b + v x ' c 2 ) S' S t b t a =γ(t ' b t ' a ) T =γt 0 γ= 1 1 v2 c 2 > 1 T 0 = 2 L 0 c T 0 T=γT 0 T = 2 L 0 T 2 v 2 =4 ( ( ct 2 ) 2 L 0 2) = T 0 T >T 0 c 1 v2 v 2 Moving clock runs slow 1 v2 v 2

28 Length contraction Before talking any such counter intutive things, let us talk abou what do mean by lenght measurement Length of a stick is the distance between its ends measured at the same instant of time t1 t2

29 Length contraction Condition: Length of a stick is the distance between its ends at the same instant of time x ' a =γ(x a v t ) x ' b =γ(x b v t ) y y' v x ' b x ' a =γ(x b x a ) L 0 =γ L a L 0 b L= L 0 γ γ>1 x,x' Therefore: L<L 0

30 S'-frame Two events: emission and absorption of light at the same point T 0 = 2 L 0 c T 0 =t ' 2 t ' 1 S-frame Δ t 1 = L c +Δ t 1v c ; Δ t 2 = L c Δ t 2v c T =t 2 t 1 T =Δ t 1 +Δ t 2 = L c v + L c+v = 2 L c (c 2 v 2 ) We learnt: T = T 0 v 2 1 c 2 2 L c (c 2 v 2 ) = 2 L 0 c v2 1 c 2 L=( 1 v2 c 2) L 0

31 The Doppler effect L L'=vT frequency= velocity wavelenthg ν 0 = 1 T 0 ; for u=0 ν d = u L L' = u u T v T = 1 T u u v = 1 T 0 u u v (Newotonian)Non-relativistic: T =T 0 For light signal: u = c ν d = 1 T u u v = 1 γ T 0 0 c c v =ν c+(+v) c+( v) Relativistic: T =γt 0

32 The Doppler effect L ν d = L'=uT u L L' = u u T v T cosθ = 1 T u u v cosθ ν 0 = 1 T 0 ; for u=0 Non-relativistic: T =T 0 ν d = 1 γ T 0 c c v cosθ Relativistic: T =γ T 0 Navigation, GPS

33 Velocity addition theorem Assuming: u' is the velocity with respect to the moving frame dx dt x'=γ (x v t ), t '=γ(t v x c 2 ) d (x ' +v t ' ) =u=γ =γ dt u=γ 2 (u' + v) ( 1 v u d (x '+v t ') dt ' ) c 2 dt ' dt y y' v u' x,x' u'= u v 1 u v c 2 u max =c

34 Lorentz Transformation: Invariance property What we have learnt: (x, t) & (x', t'): We must find out a linear relation among them keeping velocity of light constant. Assuming: x '=A (x v t),t '=B(t v x c 2 ) y '= y, z '=z x 2 + y 2 + z 2 =c 2 t 2 x' 2 + y ' 2 + z ' 2 =c 2 t ' 2 A 2( 2) 1 v2 =1 c If you think little deep, actually we demand: Let us understand how do we derive A S ' 2 = c 2 t ' 2 +x ' 2 + y ' 2 + z ' 2 = c 2 t 2 +x 2 + y 2 +z 2 =S 2 For light, S =S'=0

35 Lorentz Transformation: Invariance property x '=γ(x v t),t '=γ(t v x c 2 ) y '= y, z '=z Tranformation is linear in ( t, x, y, z) dx '=γ(dx v dt),dt '=γ(dt v dx c 2 ) dy '=dy,dz '=dz Invariant quantities: ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 =ds ' 2 d τ 2 = dt 2 dx2 + dy 2 + dz 2 c 2 S 2 = c 2 t 2 + x 2 + y 2 + z 2 S ' 2 = c 2 t 2 + x 2 + y 2 +z 2 S 2 =S ' 2 Invariant =dt 2[ 1 1 c 2 (( dx dt ) 2 + ( dy dt ) 2 + ( dz dt ) 2)] Invariant time: proper-time =dt 2 ( u 2 1 2) c

36 Newtonian Rotation+Galileon dx '=dx cosθ+dy sin θ dy '= dx sin θ+dy cosθ dz '=dz dx '=dx v dt, dy '=dy, dz '=dz dt '=dt ds 2 =dx 2 +dy 2 +dz 2 =ds ' 2 Distance: measured at same instant of time Time t: Invariant under all observers dt With respect to which we measure physical variation Einsteinian Rotation+ Lorentz x '=x cosθ+ y sin θ y '= x sin θ+ y cosθ z '=z dx '=γ(dx v dt), dt '=γ(dt v dx c 2 ) dy '=dy, dz '=dz ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 =ds ' 2 τ Proper time : Invariant under all observers d τ

37 Concept of ovserver independent time (proper time) Why do need to define such a quantity? Newtonina dynamics: we use time (t) as a parameter(observer independent): measure all the dynamical 'changes of state of a system (position, momentum,...) Now you know: Newtonin time is not invariant any more!! Define invariant time(proper time) d τ 2 = dt 2 dx2 + dy 2 + dz 2 c 2 2[ 2)] =dt 1 1 (( dx ) 2 + ( dy ) 2 + ( dz ) c 2 dt dt dt =dt 2 ( u 2 1 2) c d τ=dt 1 u2 c 2=dt ' 2 u' 1 c =d τ ' 2 u: velocity of the particle under study

38 Newtonian world 3-dimensionl Eucledian space S 2 =x 2 + y 2 +z 2 ds 2 =dx 2 +dy 2 +dz 2 Definitio: three-position r i (x, y, z), r 1 =x, r 2 = y, r 3 =z dr i (dx,dy,dz), dr 1 =dx, dr 2 =dy, dr 3 =dz Einstein's world 4-dimensionl Einsteinina space-time S 2 = c 2 t 2 + x 2 + y 2 +z 2 ds 2 = c 2 dt 2 +dx 2 +dy 2 +dz 2 Definitio: Four-position R i (ict, x, y, z), R 1 =ict, R 2 =x, R 2 = y, R 3 =z d R i (i c dt, dx,dy,dz), d R 1 =ic dt, d R 1 =dx, d R 2 =dy, d R 3 =dz r 2 =r r=r 2 1 +r r 3 R 2 =R R=R 2 1 +R 2 2 +R R 4 =x 2 + y 2 +z 2 = c 2 t 2 + x 2 + y 2 +z 2

39 Definitions: Position, Momentum, Acceleration Newtonian Relativistic Time (t) as a parameter with respect to which we define all the changes. Proper time as a parameter with respect to which we define all the changes. Definition: 3-vectors 3-Position: r i ( x,y,z ),r 1 =x... 3-momentum: p i =m 0 dr i dt m 0( dx dt, dy dt, dz dt ) 3-acceleration: a i = d 2 ri dt 2 ( d2 x dt 2, d2 y dt 2, d2 z dt 2 ) d p i dt =F i Definition: 4-vectors 4-Position: R μ (ict, x, y, z ), R 1 =ict... 4-momentum: d R μ μ =m 0 dτ m 0( dt ic dτ, dx dτ, dy dτ, dz ) dτ 4-acceleration: a μ = d 2 R μ dτ 2 ( ic d2 t dτ 2, d 2 x dτ 2, d2 y dτ 2, d2 z dτ 2 ) d μ d τ =f μ

40 dot product of vectors What you know In a same way Definitio: three-vectors 3-Position: 3 r 2 =r r= i=1 Any 3-vector (say 3-momentum): dr i p i =m 0 dt m 0( dx dt, dy dt, dz ) dt 3 p 2 = p p= i=1 r i (x, y, z) ; r 1 =x... r i r i =r 1 2 +r 2 2 +r 3 2 =x 2 + y 2 + z 2 p i p i = p p p 3 2 =m 0 2 (ẋ 2 + ẏ 2 + ż 2 ) Definitio: four-vectors 4-Position: 4 R 2 =R R= μ=1 R μ (i ct, x, y, z), r 1 =i ct... Any 4-vecto (say 4-momentum): R μ R μ =R 1 2 +R 2 2 +R 3 2 +R 4 2 = c 2 t 2 + x 2 + y 2 + z 2 d R μ =m μ 0 dτ m 0( dt ic dτ, dx dτ, dy dτ, dz ) dτ 4 2 = = i i = i=1 =m 2 0 ( c 2 t ' 2 +x ' 2 + y ' 2 + z ' 2 ) A A i ( A 1, A 2, A 3 ) A μ (i A 1, A 2, A 3, A 4 )

41 Let us look at term by term of 4-momentum Momentum 4-vector (say 4-momentum): dr μ =m μ 0 dτ m 0( dt ic dτ, dx dτ, dy dτ, dz ) dτ v2 d τ=dt 1 c =d τ ' 2 1 =i m 0 c dt d τ =i m 0 c v 2 1 c 2 = ( i c ) m c2 =i p t dx 2 =m 0 d τ = m = p y ; 4 = p z v 2 c 2 dx dt =m v x= p x m= m 0 v 2 1 c 2 4 μ =(i p t, p x, p y, p z ) ; 2 = i=1 Relativistic energy i i = m 2 c 2 + p x 2 + p y 2 + p z 2

42 Let us closely look at the first component of momentum Momentum 4-vector (say 4-momentum): dr μ μ =m 0 dτ m 0( dt ic dτ, dx dτ, dy dτ, dz ) dτ v2 d τ=dt 1 c =d τ ' 2 1 =i m 0 c dt d τ =i m 0 c v 2 1 c 2 = ( i c ) m c2 = i c (m 0 c m 0 v 2...)(for v c) KE : Kinetic energy δ 1 = 1 (v) 1 (0)= ( i c ) 1 2 m 0 v 2 = ( i c ) (KE) Rest mas energy Einstein's Total relativistic energy = E =(m 0 c m 0 v 2...)=mc 2 μ =(i E 4 c, p x, p y, p z ) ; 2 = i=1 i i = ( E2 c 2 ) + p 2 x+ p 2 2 y + p z

43 μ =(i E c, p x, p y, p z ) (i E c, p) Energy Mass v2 d τ=dt 1 c =d τ ' 2 Relativistic energy Relativistic Mass E=mc 2 m= m 0 1 v 2 c = i=1 i i = ( E2 c 2 ) + p 2 x+ p 2 y + p 2 z = m 2 c 2 + p 2 x + p 2 2 y + p z

44 Relativistic energy of a particle of momentum p, and mass m 0 4-momentum Massive particle: 2 = ( E2 c 2 ) 2 = m 0 2 c 2 + p 2 x+ p 2 y + p 2 z = m 2 c 2 + p 2 x + p 2 2 y + p z = =m 2 ( c 2 +v x 2 +v y 2 +v z 2 ) m v2 c 2 = m 0 2 c 2 (c 2 v x 2 v y 2 v z 2 ) E 2 =p 2 c 2 +m 0 2 c 4 Relativistic energy of a massless particle: Photon m 0 =0 2 =0 E= p c μ = (i E c, p)=(i p, p)=(i p, p) NO Newtonian analog

45 Conservation Law d μ d τ =f μ=0 If there is no external force 1 3 μ 1 + μ 2 = μ 3 + μ E 1 +E 2 =E 3 +E 4 m 1 u 1 +m 2 u 2 =m 3 u 3 +m 4 u 4