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Mechanics

Physics 151^ Lecture 24

Continuous Systems and Fields

(Chapter 13)

What We Did Last Time „^ Built Lagrangian formalism for continuous system

„^ Lagrangian

Æ

„^ Lagrange’s equation

Æ

„^ Derived simple wave equation

„^ Energy and momentum conservation given by theenergy-stress tensor

„^ Conservation lawstake the form of (time derivative) = (flux into volume) „^ Ran out of time here

Æ^ See Goldstein 13.3 if interested

„^ Today’s lecture doesn’t use it

L^

dxdydz =^ ∫∫∫

L

,

0

d dx^ μ

η^ μ

η ⎛^

⎞

∂^

∂−

=

⎜^

⎟

⎜^

⎟

∂^

∂

⎝^

L^ ⎠

L

, ,

T^ μν

μ

μν

η^ ν

δ

∂≡ η

−

L^ ∂

L^

0

dT^ μν dx ν

=

Hamiltonian Formalism „^ For a discrete system, we define conjugate momenta

„^ For a continuous system, „^ Momentum should be „^ Hamiltonian „^ Let’s see how this works…

i

L i p^

∂= ^ q ∂^

Then

i^ i H^

p q^

L

=^

−^ ( ,^ ,

, ,

) i d

i dx

L^

t x^

dxdydz η

η η

=^ ∫∫∫

L^



( ,^

) t xi

π^

∂^ L=^ η∂

H^

dxdydz =^ ∫∫∫

H

πη= −

H^

L

where

The Rod Again „^ Consider the 1-dim elastic rod again

„^ Lagrangian density

x

2

2

1 2

d^

d K dt^

dx

η

η

⎡^ μ

⎤

⎛^

⎞^

⎛^

⎞

=^

−

⎢^

⎥

⎜^

⎟^

⎜^

⎟

⎝^

⎠^

⎝^

⎠

⎢^

⎥

⎣^

⎦

L

π

μη

∂^ L^ = = η∂





2

2 2

1 2 2 2

d^ 2

d K dt^

dx

K^ η d dx

η

πη

μ π

η ⎡^ μ

⎤

⎛^

⎞^

⎛^

⎞

=^

−^

=^

+

⎢^

⎥

⎜^

⎟^

⎜^

⎟

⎝^

⎠^

⎝^

⎠

⎢^

⎥

⎣^

⎦

⎛^

⎞

=^

+^

⎜^

⎟

⎝^

⎠

H^

L

Wait! What am I going todo with this term?

Hamiltonian Formalism „^ Hamiltonian formalism treats time as special

„^ Because of the way momentum is defined

„^ Natural structure of classical field theory is symmetricbetween time and space

„^ At least in Lagrangian formalism „^ Hamiltonian is not so usefulas 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

,

0

d dx^ μ

ρ μ

ρ η

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 t η

0

0

( , )

( ) sin

( ) sin

n^

n^

n

n^

n nx

x t^

q^ t

q^ t

k x

π L

η^

∞^

∞

=^

=

=^

=

∑^

∑^

Assuming η(0) =^ η

( L ) = 0

0

0

( , )

( )^

( )^

n

nxi

ik x

L 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

2

1 2 1

sin^

sin^

cos^

cos

2

n^

n^

m^

m^

n^ n^

n^

m^ m^

m

n^

m^

n^

m

d^

dK dt^

dx q^

k x^

q^

k x^

K^

q k^

k x^

q k^

k x

η

η

⎡^ μ μ

⎤

⎛^

⎞^

⎛^

⎞

=^

− ⎢^

⎥

⎜^

⎟^

⎜^

⎟

⎝^

⎠^

⎝^

⎠

⎢^

⎥

⎣^

⎦

⎡^

⎤

=^

−

⎢^

⎥

⎣^

⎦

∑^

∑^

∑^

∑

L

^



sin 0

sin^

2

L

n^

m^

L nm

k 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^

Kk

L^

L

dx^

K^

q^

q

μ

⎡^ μ

⎤^

⎡^

⎤

=^

−^

=^

−

⎢^

⎥^

⎢^

⎥

⎣^

⎦^

⎣^

⎦

∑^

∑^

∑

L∫

^



0 ( , )

( ) sin n

n n x t^

q^ t^

k x

η^

∞ = ∑=

What does this look like?

Harmonic Oscillators „^ The Lagrangian represents an infinite array ofindependent harmonic oscillators

„^ Angular frequencies are

„^ Vibration of continuous system can be decomposedinto a set of discrete oscillators

„^ True for any linear system

„^ Lagrangian density must be 2

nd^ order homogeneous

function of the field’s derivatives „ Small oscillation around equilibrium always OK

2 2

2

2

2

n 2 n^

n

n

Kk

L^

q^

q

⎡^ μ

⎤

−

⎢^

⎥

⎣^

⎦

∑^



(^2) n n^

n Kk^

vk

ω^

μ

=^

=^

Wave velocityWavenumber

Phonons „^ Harmonic oscillator in QM has discrete energy levels

„^ Possible values of

E^ are

„^ What does this mean for the continuous system? „^ η

( x , t

) is a superposition of sine waves with different

k

„^ Each mode is a harmonic oscillator „^ Vibration energy comes in small-but-finite pieces of „^ As if it’s a bunch of particles

„^ Vibration can be seen as particles

„^ Called phonons in the case of mechanical vibration

1 2 (^

)^

(^

0,1, 2,

)

E^

m^

m

ω

=^

+^

=^

∞

=^

… 0

( , )

( ) sin n

n n x t^

q^ t^

k x

η^

∞ = ∑=

n^ vkn

ω^ =

1 2 (^

)

n^

n^

n

E^

m

ω

=^

+^

=

ω=^ n

Other Examples? „^ Linear fields

Æ^

Harmonic oscillators

Æ^

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

2

E^

m c

p c

=^

+

2 2

2 4

2 2 m c

p c

ω^ =

+

=

2 4 2

2 2 m c^^2

k c

ω^ =

+

=

Must satisfythis dispersionrelation

Relativistic Field Theory „^ We had difficulty with relativity and multi-particles

„^ Each particle’s EoM looked like „^ When combined, we didn’t knowwhose time to use

„^ With field like

η( x

, t ), time is just another parameter

„^ Action integral and Lagrange’s equationslook symmetric for time and space

„^ Can we just call

0 x =

ct^ and call it done?

„^ Almost…

s

s dp^ K = d^ τ^ s

Proper time of particle

s

I^

dxdydzdt =^ ∫∫∫

, L

0

d dx^ μ

ρ μ

ρ η

η ⎛^

⎞ ∂^

∂− = ⎜^

⎟ ⎜^

⎟ ∂^

∂ ⎝^ L^ ⎠

L

Lagrangian Density „^ Everything depends on the action integral

„^ It must be Lorentz invariant

Æ^

All the equations will follow

„^ Write it as „^ The volume element

(^0) dx (^1) dxdx 23 dx

is Lorentz invariant

„^ Because det(

μ L ) = 1 for any Lorentz tensorν

„^ You must construct

L^ using covariant quantities

„^ Your field may be scalar (

η) or 4-vector (

η) or tensor…μ

„^ You combine them so that the product is a scalar

0 1

2

3

I^

dx dx dx dx =^ L∫

Lagrangian density

L^ must be a Lorentz scalar

Field Equation „^ We derived Lagrange’s equation from Hamilton’sprinciple for continuous field in the last lecture

„^ Derivation is unchanged

Æ^

Same equations hold

„^ Note „^ Ready to look at an easy example…

,

0

d^ μ^ dx

ρ μ

ρ

η

η

⎛^

⎞

∂^

∂−

=

⎜^

⎟

⎜^

⎟

∂^

∂

⎝^

L^ ⎠

L

0 1

2

3 0

I^

dx dx dx dx

δ^

δ=

=

L∫

0

,^

(^ ) (^ )^

d^

d

d ct^

dt

d^

d^

d

dx^

d ct

dt ρ

ρ

η

η

ηρ

⎛^

⎞

⎛^

⎞^

⎛^

⎞

∂^

∂^

∂

⎜^

⎟

=^

=

⎜^

⎟^

⎜^

⎟

⎜^

⎟

⎜^

⎟^

⎜^

⎟

∂^

∂^

∂⎝

⎠

⎝^

⎠^

⎝^

⎠

L^

L^

L^

Didn’tchangethis term

Scalar Field „^ The simplest field is a scalar field

φ

„^ Lagrangian density may be a function of „^ For free field,

L^ has no explicit dependence on

x^ μ

„^ Only a few scalar quantities can be formed „^ Try „^ What kind of field is this?

, ( ,^

,^ ) x μ μ

φ φL

d^ φ^ μ dx

2 2 ,^ ,^

0 λ λ φ φ

μ φ =^

−

L

(^

)^

2 ,^

0

,

2

2

0

d^

d

dx^

dx

ν

ν

ν

ν

φ^

μ φ

φ^

φ

⎛^

⎞

∂^

∂−

=^

+^

=

⎜^

⎟

⎜^

⎟

∂^

∂

⎝^

L^ ⎠

L^

2

2 0

0

d^ ν^ dx dx

φ^ ν

μ φ+

=

Known as Klein-Gordon equation

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

i q e

φ^

⋅ =^ ∑

k rk k

1

i

q^

e^

dV V

−^ φ

⋅ =^

∫^

k r

k

k^ takes all the valuesthat satisfy theboundary condition

2

2 2

2

2

2

2

0

0

0

2 2

1 2

0

i

d^

d^

q^

k q^

q^ e

dx dx

c dt

c

φ ν ν

φ

μ φ

φ^

μ φ

μ^

⋅

⎧^

⎫

+^

=^

− ∇^

+^

=^

+^

+^

=

⎨^

⎬

⎩^

⎭

∑^

k r

k^

k^

k

k



where

2

(^20)

1 2

0

q^

k q^

q

c

μ

+^

+^

=

k^

k^

k

^

Harmonicoscillator!

2

2 2

(^20) (^

)

k^

c^ k ω

μ =^

+

0 m^

=^ μ= c

The Field – What is It? „^

gives particles with mass

„^ OK, but what is the field

φ^ itself?

„^ Vibration of elastic material

Æ^ Phonons

„^ Vibration of electromagnetic field

Æ^ 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 λ φ φ

μ φ =^

−

L^

0 m^

=^ μ= c

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

Æ^

Particles with spin = 1

„^ Electromagnetic field is an obvious example

„^ Corresponding particle is photon, with spin 1 „^ Recall

is a 4-vector

„^ Connectionwith

E^ and

(^ B

,^ )

A^

c μ

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^

A

F^

x^

x ν

μ

μν

μ

ν

−^

−^

− −

−

− ⎡^

⎤

⎢^

⎥

∂^

∂^

⎢^

⎥

=^

−^

= ⎢^

⎥

∂^

∂^

⎢^

⎥

⎢^

⎥

⎣^

⎦

Electromagnetic Field „^ EM field interacts with charge

„^ In terms of

μυ F

,

„^ Defining 4-current as

,

ρ ε^0

∇^ ⋅^

= E

0 1 2 t c

μ

∂

∇ ×^

−^

E = ∂

B^

j

(^ , )

j^

c μ^

ρ=

j

0 dF^

j μν dx

μ ν

μ= −

0

0

F x^

c^

c

ν ν

ρ ε

∂^

∇ ⋅

= −^

= −

∂

E^

(^

)^

0

1 2

i^

i i^

i

F^

E^

j

x^

c^

t

ν ν

μ

∂^

∂

= − ∇ ×

+^

= −

∂^

∂

B

Maxwell’sequations

Let’s pick a unitin which

μ= 1^0

0

dF^

j μν dx

μ +ν

=

What’s the Lagrangian?

Electromagnetic Field „^ To build

L, we can use

μ A ,

μν F

and

μ j

„^ Easy to find

L^ that works:

„^ Field equation is

,^

, 2

2

0

F

d^

d^ F

j

dx^

A^

A^

dx^

A

d^

F^

F^

j

dx dF

j λρ dx

λρ^

μ

ν

ν

μ ν

μ

μ ν μν νμ

μ

ν μν

μ ν

⎛^

⎞^

∂

∂^

∂−

=^

−

⎜^

⎟

⎜^

⎟

∂^

∂^

∂

⎝^

⎠

−^

+

=^

−

= −^

−^

=

L^

L

F^ F^^4

j A λρ^ λρ

λ^ λ

=^

+

L

,^

, A^

A

ρ^ λ^

λ ρ =^

− What wewanted

Free EM Field „^ Does it satisfy the usual wave equation?

„^ For free field (

μ^ j = 0), the field equation reduces to

„^ This doesn’t give you the usual plane waves etc. „^ Problem: Given

E^ and

B ,^ A

μ^ is not uniquely defined

„^ Extra condition to fix this ambiguity „^ Impose Lorentz gauge condition

F^ F^^4

j A

λρ^ λρ

λ^ λ

=^

+

L

0

μν dF ν dx

=

2

2

0

d^

A^

A^

A^

A

dx^

x^

x^

x^ x

x^ x

ν

μ

ν

μ

ν

ν

ν

μ

ν

μ

ν

⎛^

⎞

∂^

∂^

∂^

∂

−^

=^

−^

=

⎜^

⎟

⎜^

⎟

∂^

∂^

∂^ ∂

∂^ ∂

⎝^

⎠

0

μ A ∂ =μ x ∂

2

2

2

2

2 1

0

A^

d A

A

x^ x

c^

dt μ

μ^

μ

ν ν ∂^

=^

− ∇^

=

∂^ ∂

EM waveswith

v^ =^

c

Gauge Conditions „^ We may add a gradient of any function

Λ^ to

μ A

„^ A

μ^ 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

∂Λ μ ′^ =

+ ∂

2

2

A^

A

F^

F^

F

x^

x^

x^ x

x^ x

ν

μ

μν

μν

μν

μ

ν

μ^ ν

ν^ μ

′^

′

∂^

∂^

∂ Λ

∂ Λ

′^ =

−^

=^

+^

−^

=

∂^

∂^

∂^ ∂

∂^ ∂

0

∇^ ⋅^

= A

0

μ A ∂ =μ

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

Æ^ Harmonic oscillators

„^ Quantum field theory has enjoyed great success indescribing elementary particles and their interactions

Summary „^ We’ve come a long way

„^ Covered all the essentials in Goldstein

„^ Lagrangian, conservation laws, special relativity,Hamiltonian, canonical transformations „^ Central force, rigid body, oscillation „^ Also talked about a lot of frivolous but intriguing topics

„^ I don’t expect you to keep everything in your brain

„^ Hopefully, it will come back and help you when you see it inthe more advanced courses of physics

„^ At least you’ll know which book to look up