Black-Body Radiation: Classical vs. Quantum Statistical Mechanics Approach
Branislav K. Nikolić
Department of Physics & Astronomy, University of Delaware, Newark, DE 19716
EMAIL: bnikolic@udel.edu

1. Formulation of the Problem

“Blackbody radiation” or “cavity radiation” refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity held at temperature T which is radiating. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency.

Connection to experiment: An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. All skymap show below is obtained from Wilkinson Microwave Anisotropy Probe NASA satellite launched in 2001.

We want to compute flux of radiation emitted out of the small hole drilled on the side of the cavity:

Φ=⟨(Subscript[c, ⊥])⟩(E/V)= (c/4)(E/V)

c_perp=c/(4*Pi)*Integrate[2*Pi*Sin[θ]*Cos[θ],{θ,0,Pi/2}]

The experimental curve shown above plots Φ(k,T), where the flux of emitted radiation in the interval [k,k+dk] is Φ(k,T)dk. We obtain this quantity from:

E=∫E(k,T)dk

Φ(k,T)=(c/4)((E(k,T))/V)

Thus, to compare theory with experiment we have to find statistical average of energy density of radiation inside the cavity and resolve it in wavevector k.

2. Classical Statistical Mechanics Approach (leads to the ultraviolet catastrophe)

The algorithm of classical statistical mechanics takes Hamiltonian of a physical system, H(p,q), as an input for calculation of the partition function (shown here in the canonical ensemble):

Z=(1/(N!(h^(3N))))∫(e^(-βH(p,q)))dpdq

and free energy:

F=-(Subscript[k, B])Tln Z

Hamiltonian of electromagnetic field is the sum of its “kinetic” and “potential” energy:

H(E, B) =(1/2)(Subscript[ϵ, 0])∫(E^2)dV + (1/(2(Subscript[μ, 0])))∫(B^2)dV

In practical calculations in electrodynamics, it is always more advantageous to switch to scalar and vector potential (which is up to four numbers, instead of six vector components of electric and magnetic field). We use gauge fixing in which:

E=- ((∂A)/(∂t)), B=∇×A

and assume periodic boundary conditions for the vector potential which makes it possible to expand it into Fourier series:

A(x,y,z)=A(x+L,y,z)=A(x,y+L,z)=A(x,y,z+L)

k =((2π)/L)((Subscript[n, x]),(Subscript[n, y]),(Subscript[n, z]))

A(r,t)=(1/2)(∑_(k,α))( (Subscript[a, k,α])(t)(e^(ik·r))+ (Subscript[a, k,α]^( * ))(t)(e^(-ik·r))) 

By changing variables to the so-called normal modes:

(Subscript[P, k,α])=((-(Subscript[iω, k,α]))/2)(Sqrt[(V/((c^2)(Subscript[μ, 0])))])((Subscript[a, k,α]) - (Subscript[a, k,α]^( * )))

(Subscript[Q, k,α])=(1/2)(Sqrt[(V/((c^2)(Subscript[μ, 0])))])((Subscript[a, k,α]) + (Subscript[a, k,α]^( * )))

(Subscript[P, k,α])= ((d(Subscript[Q, k,α]))/(dt))

(Subscript[ω, k,α])=ck;  ∇·E=0  ⇒ E ⊥ k, α=1,2

We can now re-write the Hamiltonian of electromagnetic field in cavity as the sum over energy of independent harmonic oscillators:


H=(Underscript[(1/2) ∑,       k,α])(|(Subscript[P, k,α])|2 + (Subscript[ω, k,α]^2)|(Subscript[Q, k,α])|2)

Since (Subscript[n, x]),(Subscript[n, y]),(Subscript[n, z]) can be arbitrary large, we arrive at the ultraviolet catastrophe (i.e., divergence) of radiation intensity at high frequencies, after (Subscript[k, B])T energy is assigned per each mode. The low frequencies are cut off by the size of the box.

3. Quantum Statistical Mechanics Approach (by Planck using heuristic formulas)

Planck guessed that eigenvalues of quantum-mechanical Hamiltonian of electromagnetic field (not known at the time) are quantized:

(Subscript[H, QM])=(Underscript[ ∑, k,α]) ℏck ((Subscript[n, k,α])+(1/2)), (Subscript[n, k,α])=0,1,2,...

so that microstates of electromagnetic radiation in the cavity are counted by specifying collection of numbers {(Subscript[n, k,α])} telling us how many quanta of energy are excited in a given normal mode. Partition function used by Planck is heuristic (rather than rigorously derived) generalization of the classical one to take into account quantized energy levels:

Z=(Underscript[ ∏ , k,α])  (Underscript[ ∑, (Subscript[n, k,α])])e-βℏck((Subscript[n, k,α])+(1/2))

where, as usual, β=1/((Subscript[k, B])T).

Z1=Sum[Exp[-β*ℏ*c*k*(n+1/2)],{n,0,Infinity}]

D[-Log[Z1],β]//FullSimplify

%/(((c k ℏ)/2)+((c k ℏ )/((E^(c k β ℏ))-1)))//FullSimplify

Statistical average of energy is obtained from the logarithmic derivative of Z:

E=- (∂/(∂β))lnZ=(Underscript[ ∑, k,α]) ((ℏck)/2) + (Underscript[ ∑, k,α]) ((ℏck)/((e^(βℏck))-1))

The first sum is zero-point energy which is infinite, but as only energy differences are measured experimentally, it can be ignored. Note that in the second term we can also identify Bose-Einstein distribution function for non-interacting bosons with zero chemical potential (as is the case of photons in the cavity):

nBE(ℏck)=(1/((e^(βℏck))-1))

Instead of trying to sum over all possible values of k-vector, we usually switch to integration over dk via the following general prescription:

(Underscript[ ∑,  k])f(k) = (1/(Δk))(Underscript[ ∑,  k]) f(k) Δk ⇒ limΔk→0(Underscript[ ∑,  k]) f(k) Δk = (1/(Δk))∫f(k)dk = (V/(((2π))^3))∫f(k)dk

Esmart=2*V/(2*Pi)^3*Integrate[ℏ*c*k/(Exp[β*ℏ*c*k]-1)*k^2*Sin[θ],{k,0,Infinity},{θ,0,Pi},{ϕ,0,2*Pi}]

Ebrute=2*V/(2*Pi)^3*Integrate[ℏ*c*Sqrt[kx^2+ky^2+kz^2]/(Exp[β*ℏ*c*Sqrt[kx^2+ky^2+kz^2]]-1),{kx,-Infinity,Infinity},{ky,-Infinity,Infinity},{kz,-Infinity,Infinity}]//FullSimplify

Using average energy density E/V, we recover Stefan-Boltzmann law for the intensity of electromagnetic radiation as a function of cavity temperature:

Φ=⟨(Subscript[c, ⊥])⟩(E/V)= (c/4)(E/V)=(((π^2) )/(60(c^2) (ℏ^3)))((((Subscript[k, B])T))^4)=σT4

so that:

Φ(k,T)=(c/4)E(k,T)=(((ℏc)^2)/(4(π^2))) ((k^3)/((e^(βℏck))-1))

The characteristic wavevector (SuperStar[k])= (((Subscript[k, B])T)/(ℏc)) separates classical (i.e., Rayleigh-Jeans law):

k≪ (SuperStar[k])⇒I(k,T)=(((Subscript[ck, B])((Tk)^2))/(4π)) 

from quantum regime:

k≫ (SuperStar[k])⇒I(k,T)=((((ℏc)^2)(k^3))/(4(π^2)))(e^(-βℏck))

by providing the upper cutoff that eliminates the ultraviolet catastrophe produced by classical statistical mechanics.

We can use formulas above to find spectrally-resolved intensity of cosmic microwave background radiation:

FundamentalConstants={ℏ->1.055*10^(-34),c->3*10^8,(Subscript[k, B])->1.38*10^(-23)}

Flux[k_,T_]=ℏ*c^2/(4*Pi^2)*k^3/(Exp[ℏ*c*k/((Subscript[k, B])*T)]-1)/.FundamentalConstants

Plot[Flux[k,2.72],{k,0,20000}] (*Radiation spectrum at the temperature T=2.72 K of CMB*)

Another example of relevance to modern research is the so-called solar irradiance: