Module 1 : Quantum Mechanics
Chapter 2 : Introduction to Quantum ideas
 
  2.4 Black-body radiation  
  Historically, Planck's explanation (1900) of black-body radiation was the first quantum description of a physical phenomenon. Black-body is a perfect absorber of incident radiation. Because of thermal equilibrium, it is also the best emitter of radiation. It may be simulated by a hole grilled in a cavity. Experimentally, the radiation emitted by a black-body has a well-defined intensity distribution as a function of wavelength $\lambda$, which has a maximum at some wavelength $\lambda_m$. As the temperature of the black-body is raised, the radiation intensity increases at all wavelengths and $\lambda_m$ shifts to a smaller value such that
  \begin{displaymath} \lambda_m T=const. \end{displaymath} (2.20)
  known as Wien's displacement law (1993). To start with we have e.m. radiation described by the wave equation
  \begin{displaymath} \nabla^2 \vec E-\frac{1}{c^2}\frac{\partial^2 \vec E}{\partial t^2}=0 \end{displaymath} (2.21)
  which follows from the Maxwell's equations. The solutions to this equations are of the type
 
  \begin{displaymath} \vec E=\vec E_0\;cos(2\pi\nu t)sin(2\pi k_x x)sin(2\pi k_y y) sin(2\pi k_z z) \end{displaymath} (2.22)
  where $k=1/\lambda$ is the wave number. For the radiation confined to a cubic box of length $l$, the boundary conditions imply  
  \begin{displaymath} 2\pi k_x l=n_x \pi \quad \Rightarrow \quad k_x=\frac{n_x}{2l}, \quad k=\frac{n}{2l} \end{displaymath} (2.23)