Only results and calculation formulas are shown on this page. You can find the derivations for them and other useful sources on the bottom of this page.
1-D Wave Equation | 2-D Wave Equation | 3-D Wave Equation | Calculation formula | |
Eigenfunction solutions | - | |||
Dispersion relation | - | |||
Density of states, D(k) | ||||
Density of states, D(ω) | ||||
Density of states, D(λ) | ||||
Density of states, D(E) | ||||
Intensity distribution, I(λ) | ||||
Wien's law | ||||
Stefan - Boltzmann law | ||||
Internal energy distribution, u(λ) | ||||
Internal energy | ||||
Helmholtz free energy | ||||
Entropy | ||||
Specific heat | ||||
Pressure | ||||
c = velocity of propagation. Photons: c = 299792458 m/s. u = wave function. k = wave number. = wave vector. Ω = the wave's frequency. p = number of polarizations. Photons: p = 2, Phonons: p = 3. = the area of the surface in reciprocal space (k-space) with given k. Vrez = the volume of the brillouin zone in the reciprocal space. . V = volume of our box. . λ = wavelength. E = energy. h = Planck constant. h = 6.62606896E-34 Js. = reduced Planck constant. . kB = Boltzmann constant. kB = 1.3806504E-23 J/K. = the vector pointing out of the infinitesimally small area dA. ΩR = the solid angle of the space. In 3D, ΩR = 4π. ΩR/2 = the solid angle of the half-space. = the unitary vector pointing in the direction specified by Ω. λmax = the peak wavelength of the intensity. T = temperature. . ζ(s) = the Riemann Zeta function. |
How to derive these results
Wikipedia article about the Stefan Boltzmann law
Wikipedia article about Planck's Black Body radiation law
The Bose-Einstein distribution in short
German Wikipedia article containing formulas on Planck's Black Body radiation law
German Wikipedia article about Planck's Black Body radiation law