Mechanics Physics 151

Transcription

1 Mechaics Physics 151 Lecture 4 Cotiuous Systems ad Fields (Chapter 13) What We Did Last Time Built Lagragia formalism for cotiuous system Lagragia L = L dxdydz d L L Lagrage s equatio = dx η, η Derived simple wave equatio Eergy ad mometum coservatio give by the eergy-stress tesor L dt T η, L δ = Coservatio laws η, dx take the form of (time derivative) = (flux ito volume) Ra out of time here See Goldstei 13.3 if iterested Today s lecture does t use it 1

2 Hamiltoia Formalism For a discrete system, we defie cojugate mometa L pi = The H = pq q i i L For a cotiuous system, i L Mometum should be π (, tx i ) = η L = L dη ( ηη,,, t, x ) dxdydz dxi i Hamiltoia H = H dxdydz where H= πη L Let s see how this works The Rod Agai Cosider the 1-dim elastic rod agai 1 dη dη Lagragia desity L = K dt dx L π = = η 1 dη dη H= πη L= K η + dt dx π K dη = + dx Wait! What am I goig to do with this term? x

3 Hamiltoia Formalism Hamiltoia formalism treats time as special Because of the way mometum is defied π = L η Natural structure of classical field theory is symmetric betwee time ad space d L L At least i Lagragia formalism = dx η ρ, η ρ Hamiltoia is ot so useful as i the case of discrete systems Quatum field theory is built primarily o Lagragia c.f. No-relativistic QM is almost all Hamiltoia Fourier Trasformatio Cosider a elastic rod with fiite legth L η( x, t) x L At a give momet t, we ca Fourier trasform η(x,t) π x η( x, t) = q( t)si = q( t)sikx = L = Or, usig the complex form, π x i L ik x Re() assumed = = η( x, t) = q ( t) e = q ( t) e q (t) is a complex fuctio Assumig η() = η(l) = 3

4 Fourier Trasformatio What happes to the Lagragia? η( x, t) = q( t)sikx 1 dη dη L = K dt dx 1 = qsi kx qmsi kmx K qkcos kx qmkmcos kmx m m L Itegrate with x ad use si si L kx etc. kxdx m = δ m L L q kq L Kk L dx = K = q q = What does this look like? Harmoic Oscillators L Kk q The Lagragia represets a ifiite array of idepedet harmoic oscillators Kk Agular frequecies are ω = = vk q Wave velocity Waveumber Vibratio of cotiuous system ca be decomposed ito a set of discrete oscillators True for ay liear system Lagragia desity must be d order homogeeous fuctio of the field s derivatives Small oscillatio aroud equilibrium always OK 4

5 Phoos Harmoic oscillator i QM has discrete eergy levels 1 Possible values of E are E = ( m+ ) ω ( m=,1,, ) What does this mea for the cotiuous system? η(x,t) is a superpositio of sie waves with differet k Each mode is a harmoic oscillator η(, ) = ( )si 1 ω = = vk E = ( m + ) ω Vibratio eergy comes i small-but-fiite pieces of ω As if it s a buch of particles Vibratio ca be see as particles Called phoos i the case of mechaical vibratio x t q t k x Other Examples? Liear fields Harmoic oscillators Particles We kow a excellet example: Electromagetic field Correspodig particle = photo Photoelectric effect tells us E = ω Is it possible that all particles are quatized field? 4 For a particle of mass m, E = m c + p c Make correspodece with a harmoic oscillator 4 4 mc ω = mc + pc ω = +kc But first of all, the field must satisfy relativity Must satisfy this dispersio relatio 5

6 Relativistic Field Theory We had difficulty with relativity ad multi-particles Each particle s EoM looked like Whe combied, we did t kow whose time to use With field like η(x,t), time is just aother parameter Actio itegral ad Lagrage s equatios d L L = look symmetric for time ad space dx η ρ, η ρ Ca we just call x = ct ad call it doe? Almost dp K s s dτ = s Proper time of particle s I = L dxdydzdt Lagragia Desity Everythig depeds o the actio itegral It must be Loretz ivariat All the equatios will follow Write it as I = L dx dx dx dx 1 3 The volume elemet dx dx 1 dx dx 3 is Loretz ivariat Because det(l ) = 1 for ay Loretz tesor Lagragia desity L must be a Loretz scalar You must costruct L usig covariat quatities Your field may be scalar (η) or 4-vector (η ) or tesor You combie them so that the product is a scalar 6

7 Field Equatio We derived Lagrage s equatio from Hamilto s priciple for cotiuous field i the last lecture Derivatio is uchaged Same equatios hold d 1 3 L L δi = δ L dx dx dx dx = = dx η ρ, ηρ d L d L d L Did t Note = = d d dx ηρ ηρ η ρ, d( ct) dt d( ct) dt Ready to look at a easy example chage this term Scalar Field The simplest field is a scalar field φ Lagragia desity may be a fuctio of (,, x ) L φ φ, For free field, L has o explicit depedece o x Oly a few scalar quatities ca be formed λ Try L = φ φ φ, λ, dφ dx d dx L L d = ( φ, ) + φ = φ, φ dx What kid of field is this? d φ + φ= dx dx Kow as Klei-Gordo equatio 7

8 Klei-Gordo Equatio Let s do Fourier i space volume V φ i = qe kr k k Klei-Gordo equatio is the d φ d φ 1 ikr + φ = φ + φ = q k q q e k + k + k = dx dx c dt k c where For each mode k, 1 ikr q e dv V φ k = 1 c q k q k k q k + + = Dispersio relatio is ω k = c ( k + ) Correspods to a particle with a fiite mass k takes all the values that satisfy the boudary coditio Harmoic oscillator! m = c The Field What is It? λ = φ φ φ gives particles with mass m = c L, λ, OK, but what is the field φ itself? Vibratio of elastic material Phoos Vibratio of electromagetic field Photos The field φ does t have to be physical It exists oly i the sese that quatized excitatio of φ are physical (particles) Mystical ether aybody? QM calls it wave fuctio, whose (amplitude) is iterpreted as the probability of a particle beig there Still a idirect defiitio of existece 8

9 Vector Field Field ca be more complicated tha scalar How about a 4-vector, for example? Such field represets particles with spis 4-vector field Particles with spi = 1 Electromagetic field is a obvious example Correspodig particle is photo, with spi 1 Recall A = ( φ c, A) is a 4-vector Coectio E c E c E c x y z with E ad B E c B B A A x z y F = = x x E c B B y z x E c B B z y x Electromagetic Field EM field iteracts with charge Maxwell s ρ 1 E E = B = j equatios ε c t I terms of F υ, F E ρ = = x c cε Defiig 4-curret as j = ( ρc, j), F i i i i = ( B) + = j x c t 1 E df dx = j Let s pick a uit i which = 1 df + j = dx What s the Lagragia? 9

10 Electromagetic Field To build L, we ca use A, F ad j λρ F Fλρ λ Easy to fid L that works: L = + j Aλ 4 Field equatio is = Aρ λ A d λρ L L d F Fλρ j = dx A, A dx A, d F + F = j dx df = j = dx, λ, ρ What we wated Free EM Field λρ F Fλρ λ L = + j Aλ 4 Does it satisfy the usual wave equatio? df For free field (j = ), the field equatio reduces to = dx d A A A A = = dx x x x x x x This does t give you the usual plae waves etc. Problem: Give E ad B, A is ot uiquely defied Extra coditio to fix this ambiguity Impose Loretz gauge coditio A A 1 d A = = A = x x x c dt EM waves with v = c 1

11 Gauge Coditios We may add a gradiet of ay fuctio Λ to A Λ A A Λ Λ A = A + F = = F + = F x x x x x x x A is ot fully specified without a gauge coditio You ve probably see Coulomb gauge i Physics 15b A = This is ot Loretz ivariat Natural relativistic extesio is the Loretz gauge All gauge coditios give you same physics Some are easier tha the others to solve A = Relativistic Field Theory Classical field theory ca be made relativistic Not very difficult although I omitted may subtleties Lagragia desity L must be a Loretz scalar Built usig covariat fields ad currets This limits the possible forms of L Guided physicists toward correct picture of Nature Quatizatio of the field produces particles Fourier trasformatio Harmoic oscillators Quatum field theory has ejoyed great success i describig elemetary particles ad their iteractios 11

12 Summary We ve come a log way Covered all the essetials i Goldstei Lagragia, coservatio laws, special relativity, Hamiltoia, caoical trasformatios Cetral force, rigid body, oscillatio Also talked about a lot of frivolous but itriguig topics I do t expect you to keep everythig i your brai Hopefully, it will come back ad help you whe you see it i the more advaced courses of physics At least you ll kow which book to look up 1