2020년 9월 10일 목요일

(The Limits of Classical Physics-1) Classical Derivation of Rayleigh-Jeans Formula (Gasiorowicz chapter 1)

< Reference >
 I referred to Professor Joon-Gon Choi's lecture on quantum mechanics 1 at Korea University. 
 Robert Eisberg, Robert Resnick, “Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles”, 1.3. LASSICAL THEORY OF CAVITY RADIATION

  Theoretical research in the field of thermal radiation began in 1859 with the work of Kirchhoff, who showed that for a given λ, the ratio of the emissive power E to the absorptivity A, defined as the fraction of incident radiation of wavelength λ that is absorbed by the body, is the same for all bodies. Kirchhoff considered two emitting and absorbing parallel plates and showed from the equilibrium condition that the energy emitted was equal to the energy absorbed (for each λ), that the ratio E/A must be the same for the two plates. Soon thereafter, he observed that for a black body, defined as a surface that totally absorbs all radiation that falls on it, so that A=1, the function \(E\left( {\lambda ,T} \right)\) is a universal function.

The energy from the outside of the blackbody is measured in proportion to the amount of energy from the inside of the blackbody through the hole dA of the blackbody.

The total energy (\( \in \left( {{\rm{\lambda }},T} \right)\)) generated by the energy density inside the black body comes out through the hole dA

\(E\left( {{\rm{\lambda }},T} \right)\) : energy(\( \in \left( {{\rm{\lambda }},T} \right)\)) emitted per unit area(\(dA\)) per unit time (\({\rm{\Delta }}t\))

\(u\left( {{\rm{\lambda }},T} \right)\) : Energy density in blackbody

Eλ,T=(λ,T)dAΔt (λ,T)=0cΔt0π/2 02πuλ,Tr2sinθdrdθdϕdAcosθ4πr2 =0cΔtr2dr0π/2sinθcosθdθ 02πdϕ uλ,TdA14πr2=0cΔtdr0π/2sinθcosθdθ 02πdϕ uλ,TdA14π =cΔt122π uλ,TdA14π=cΔt4dAuλ,T Eλ,T=c4uλ,T


  Energy Density

uλ,T=uν,Tdλdν=uν,Tcν2 dλdν=-cν2     (λν=c, λ=cν)

  Classical Derivation of Rayleigh-Jeans Law

Boundary Condition

 Periodic boundary condition

fx,y,0=fx,y,L fx,0,z=fx,L,z f0,y,z=f(L,y,z)

The function that satisfies the periodic boundary condition is typically an oscillator.

 One dimension

1Leikx,  eikL=1, k=2πLn , n =integer 

 Three dimension

1L3/2ei(k1x+k2y+k3z), k1=2πLn1,k2=2πLn2,k3=2πLn3, kiL=2πni  (i=1,2,3)

Equipartition Theorem(Robert Eisberg, Robert Resnick)

 The prediction comes from classical kinetic theory, and it is called the law of equipartition of energy. This law states that for a system of gas molecules in thermal equilibrium at temperature T, the average kinetic energy of a molecule per degree of freedom

12kT , k= Boltzmann constant

 However, each oscillating standing wave has a total energy which is twice its average kinetic energy. This is a common property of physical systems which have a single degree of freedom that execute simple harmonic oscillations in time; familiar cases are a pendulum or a coil spring.

 Rayleigh assumed the classical law of equipartition energy. He said, “one dimensional waves always have 2 degrees of freedom, one for potential energy (x) and the other for kinetic energy (v). In case of electromagnetic wave, these two degree of freedom are derived from electric field and magnetic field.” ( http://www.pa.uky.edu/~kwng/phy361/class/class10.pdf )

-=average total Energy per unit degree of freedom =212kT=kT

degree of freedom

 Since k is a function of n and n is an integer, it is easy to count n. Therefore, we calculate the degrees of freedom using n.

4πn2dn=4πkL2π2L2πdk=4πL2π3k2dk   kL=2πn, n=kL2π =4πL2π32πcυ22πcdυ    ω=ck=2πυ, k=2πcυ =4πL3υ2c3dυ

 This completes the calculation except that we must multiply these results by a factor of 2. because, for each of the allowed frequencies we have enumerated, there are actually two independent waves corresponding to the two possible states of polarization of electromagnetic radiation.

uυ,Tdυ=-2degree of feedomVdυ=kT2degree of feedomVdυ=kT8πυ2c3dυ     V=L3