the particles in the gas and defines a critical point at which quantum effects begin to dominate. For example, when observing the long-wavelength spectrum of black body radiation, the classical Rayleigh–Jeans law can be applied, but when the observed wavelengths approach the thermal wavelength of the photons in the black body radiator, the quantum Planck's law must be used. General definition A general definition of the thermal wavelength for an ideal gas of particles having an arbitrary power-law relationship between energy and momentum (dispersion relationship), in any number of dimensions, can be introduced. If is the number of dimensions, and the relationship between energy () and momentum () is given by (with and being constants), then the thermal wavelength is defined as where is the Gamma function. In particular, for a 3-D () gas of massive or massless particles we have and , respectively, yielding the expressions listed in the previous sections. Note that for massive non-relativistic particles (s = 2), the expression does not depend on n. This explains why the 1-D derivation above agrees with the 3-D case. Examples Some examples of the thermal de Broglie wavelength at 298 K are given below. References Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. Statistical mechanics