Mechanics Continuous Systems, Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Continuous Systems, Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Lecture 24 Continuous Systems and Fields (Chapter 13) What We Did Last Time Built Lagrangian formalism for continuous system Lagrangian Lagrange’s equation Derived simple wave equation Energy and momentum conservation given by the energy-stress tensor Conservation laws take the form of (time derivative) = (flux into volume) Ran out of time here See Goldstein 13.3 if interested Today’s lecture doesn’t use it L dxdydz= ∫∫∫L , 0d dxµ µη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L , , Tµν µ µν ν η δ η ∂ ≡ − ∂ L L 0 dT dx µν ν = Hamiltonian Formalism Hamiltonian formalism treats time as special Because of the way momentum is defined Natural structure of classical field theory is symmetric between time and space At least in Lagrangian formalism Hamiltonian is not so useful as in the case of discrete systems Quantum field theory is built primarily on Lagrangian c.f. Non-relativistic QM is almost all Hamiltonian π η= ∂ ∂L , 0d dxµ ρ µ ρη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L Fourier Transformation Consider an elastic rod with finite length L At a given moment t, we can Fourier transform η(x,t) Or, using the complex form, x L ( , )x tη 0 0 ( , ) ( )sin ( )sinn n n n n nxx t q t q t k x L πη ∞ ∞ = = = =∑ ∑ Assumingη(0) = η(L) = 0 0 0 ( , ) ( ) ( ) n nxi ik xL n n n n x t q t e q t e π η ∞ ∞ = = = =∑ ∑ Re() assumed qn(t) is a complex function Fourier Transformation What happens to the Lagrangian? Integrate with x and use etc. 2 21 2 1 sin sin cos cos 2 n n m m n n n m m mn m n m d dK dt dx q k x q k x K q k k x q k k x η ηµ µ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ⎡ ⎤ = −⎢ ⎥⎣ ⎦ ∑ ∑ ∑ ∑ L 0 sin sin 2 L n m nm Lk x k xdx δ=∫ 2 2 2 2 2 2 0 2 2 2 2 2 2 L n n n n n n n n n q k q KkL Ldx K q qµµ ⎡ ⎤ ⎡ ⎤ = − = −⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∑ ∑ ∑∫ L 0 ( , ) ( )sinn n n x t q t k xη ∞ = =∑ What does this look like? Other Examples? Linear fields Harmonic oscillators Particles We know an excellent example: Electromagnetic field Corresponding particle = photon Photoelectric effect tells us Is it possible that all particles are quantized field? For a particle of mass m, Make correspondence with a harmonic oscillator But first of all, the field must satisfy relativity E ω= 2 4 2 2E m c p c= + 2 2 2 4 2 2m c p cω = + 2 4 2 2 2 2 m c k cω = + Must satisfy this dispersion relation Relativistic Field Theory We had difficulty with relativity and multi-particles Each particle’s EoM looked like When combined, we didn’t know whose time to use With field like η(x,t), time is just another parameter Action integral and Lagrange’s equations look symmetric for time and space Can we just call x0 = ct and call it done? Almost… s s s dp K dτ = Proper time of particle s I dxdydzdt= ∫∫∫L , 0d dxµ ρ µ ρη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L Lagrangian Density Everything depends on the action integral It must be Lorentz invariant All the equations will follow Write it as The volume element dx0dx1dx2dx3 is Lorentz invariant Because det(Lµν) = 1 for any Lorentz tensor You must construct L using covariant quantities Your field may be scalar (η) or 4-vector (ηµ) or tensor… You combine them so that the product is a scalar 0 1 2 3I dx dx dx dx= ∫L Lagrangian density L must be a Lorentz scalar Klein-Gordon Equation Let’s do Fourier in space volume V Klein-Gordon equation is then For each mode k, Dispersion relation is Corresponds to a particle with a finite mass iq eφ ⋅=∑ k rk k 1 iq e dV V φ − ⋅= ∫ k rk k takes all the values that satisfy the boundary condition 2 2 2 2 2 2 2 0 0 02 2 2 1 0id d q k q q e dx dx c dt cν ν φ φµ φ φ µ φ µ ⋅⎧ ⎫+ = −∇ + = + + =⎨ ⎬ ⎩ ⎭ ∑ k rk k k k where 2 2 02 1 0q k q q c µ+ + =k k k Harmonic oscillator! 2 2 2 2 0( )k c kω µ= + 0m c µ= Mechanics Physics 151 Lecture 24 Continuous Systems and Fields (Chapter 13) What We Did Last Time Built Lagrangian formalism for continuous system Lagrangian Lagrange’s equation Derived simple wave equation Energy and momentum conservation given by the energy-stress tensor Conservation laws take the form of (time derivative) = (flux into volume) Ran out of time here See Goldstein 13.3 if interested Today’s lecture doesn’t use it L dxdydz= ∫∫∫L , 0d dxµ µη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L , , Tµν µ µν ν η δ η ∂ ≡ − ∂ L L 0 dT dx µν ν = Hamiltonian Formalism Hamiltonian formalism treats time as special Because of the way momentum is defined Natural structure of classical field theory is symmetric between time and space At least in Lagrangian formalism Hamiltonian is not so useful as in the case of discrete systems Quantum field theory is built primarily on Lagrangian c.f. Non-relativistic QM is almost all Hamiltonian π η= ∂ ∂L , 0d dxµ ρ µ ρη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L Fourier Transformation Consider an elastic rod with finite length L At a given moment t, we can Fourier transform η(x,t) Or, using the complex form, x L ( , )x tη 0 0 ( , ) ( )sin ( )sinn n n n n nxx t q t q t k x L πη ∞ ∞ = = = =∑ ∑ Assumingη(0) = η(L) = 0 0 0 ( , ) ( ) ( ) n nxi ik xL n n n n x t q t e q t e π η ∞ ∞ = = = =∑ ∑ Re() assumed qn(t) is a complex function Fourier Transformation What happens to the Lagrangian? Integrate with x and use etc. 2 21 2 1 sin sin cos cos 2 n n m m n n n m m mn m n m d dK dt dx q k x q k x K q k k x q k k x η ηµ µ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ⎡ ⎤ = −⎢ ⎥⎣ ⎦ ∑ ∑ ∑ ∑ L 0 sin sin 2 L n m nm Lk x k xdx δ=∫ 2 2 2 2 2 2 0 2 2 2 2 2 2 L n n n n n n n n n q k q KkL Ldx K q qµµ ⎡ ⎤ ⎡ ⎤ = − = −⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∑ ∑ ∑∫ L 0 ( , ) ( )sinn n n x t q t k xη ∞ = =∑ What does this look like? Other Examples? Linear fields Harmonic oscillators Particles We know an excellent example: Electromagnetic field Corresponding particle = photon Photoelectric effect tells us Is it possible that all particles are quantized field? For a particle of mass m, Make correspondence with a harmonic oscillator But first of all, the field must satisfy relativity E ω= 2 4 2 2E m c p c= + 2 2 2 4 2 2m c p cω = + 2 4 2 2 2 2 m c k cω = + Must satisfy this dispersion relation Relativistic Field Theory We had difficulty with relativity and multi-particles Each particle’s EoM looked like When combined, we didn’t know whose time to use With field like η(x,t), time is just another parameter Action integral and Lagrange’s equations look symmetric for time and space Can we just call x0 = ct and call it done? Almost… s s s dp K dτ = Proper time of particle s I dxdydzdt= ∫∫∫L , 0d dxµ ρ µ ρη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L Lagrangian Density Everything depends on the action integral It must be Lorentz invariant All the equations will follow Write it as The volume element dx0dx1dx2dx3 is Lorentz invariant Because det(Lµν) = 1 for any Lorentz tensor You must construct L using covariant quantities Your field may be scalar (η) or 4-vector (ηµ) or tensor… You combine them so that the product is a scalar 0 1 2 3I dx dx dx dx= ∫L Lagrangian density L must be a Lorentz scalar Klein-Gordon Equation Let’s do Fourier in space volume V Klein-Gordon equation is then For each mode k, Dispersion relation is Corresponds to a particle with a finite mass iq eφ ⋅=∑ k rk k 1 iq e dV V φ − ⋅= ∫ k rk k takes all the values that satisfy the boundary condition 2 2 2 2 2 2 2 0 0 02 2 2 1 0id d q k q q e dx dx c dt cν ν φ φµ φ φ µ φ µ ⋅⎧ ⎫+ = −∇ + = + + =⎨ ⎬ ⎩ ⎭ ∑ k rk k k k where 2 2 02 1 0q k q q c µ+ + =k k k Harmonic oscillator! 2 2 2 2 0( )k c kω µ= + 0m c µ= The Field – What is It? gives particles with mass OK, but what is the field φ itself? Vibration of elastic material Phonons Vibration of electromagnetic field Photons The field φ doesn’t have to be “physical” It “exists” only in the sense that quantized excitation of φ are physical (particles) QM calls it wave function, whose (amplitude)2 is interpreted as the probability of a particle being there Still an indirect definition of “existence” 2 2 , , 0 λ λφ φ µ φ= −L 0m c µ= Mystical ether anybody? Vector Field Field can be more complicated than scalar How about a 4-vector, for example? Such field represents particles with spins 4-vector field Particles with spin = 1 Electromagnetic field is an obvious example Corresponding particle is photon, with spin 1 Recall is a 4-vector Connection with E and B ( , )A cµ φ= A 0 0 0 0 x y z x z y y z x z y x E c E c E c E c B B E c B B E c B B A AF x x ν µ µν µ ν − − − − − − ⎡ ⎤ ⎢ ⎥ ∂ ∂ ⎢ ⎥= − = ⎢ ⎥∂ ∂ ⎢ ⎥ ⎢ ⎥⎣ ⎦ Free EM Field Does it satisfy the usual wave equation? For free field (jµ = 0), the field equation reduces to This doesn’t give you the usual plane waves etc. Problem: Given E and B, Aµ is not uniquely defined Extra condition to fix this ambiguity Impose Lorentz gauge condition 4 F F j A λρ λρ λ λ= +L 0dF dx µν ν = 2 2 0d A A A A dx x x x x x x ν µ ν µ ν ν ν µ ν µ ν ⎛ ⎞∂ ∂ ∂ ∂ − = − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ 0A x µ µ ∂ = ∂ 2 2 2 2 2 1 0A d A A x x c dt µ µ µ ν ν ∂ = −∇ = ∂ ∂ EM waves with v = c Gauge Conditions We may add a gradient of any function Λ to Aµ Aµ is not fully specified without a gauge condition You’ve probably seen Coulomb gauge in Physics 15b This is not Lorentz invariant Natural relativistic extension is the Lorentz gauge All gauge conditions give you same physics Some are easier than the others to solve A A x µ µ µ ∂Λ′ = + ∂ 2 2A AF F F x x x x x x ν µ µν µν µν µ ν µ ν ν µ ′ ′∂ ∂ ∂ Λ ∂ Λ′ = − = + − = ∂ ∂ ∂ ∂ ∂ ∂ 0∇⋅ =A 0Aµµ∂ = Relativistic Field Theory Classical field theory can be made relativistic Not very difficult – although I omitted many subtleties… Lagrangian density L must be a Lorentz scalar Built using covariant fields and currents This limits the possible forms of L Guided physicists toward correct picture of Nature Quantization of the field produces particles Fourier transformation Harmonic oscillators Quantum field theory has enjoyed great success in describing elementary particles and their interactions