Quantum mechanical effects become significant for physics in the range of the
de Broglie wavelength. For condensed matter, this is when the de Broglie wavelength of a particle is greater than the spacing between the particles in the lattice that comprises the matter. The de Broglie wavelength associated with a massive particle is :\lambda = \frac{h}{p} where h is the Planck constant. The momentum can be found from the
kinetic theory of gases, where :p = mv_p = m\sqrt{2\frac{k_B T}{m}} = \sqrt{2 m k_B T} Here, the temperature can be found as :k_BT = \frac{p^2}{2m} Of course, we can replace the momentum here with the momentum derived from the de Broglie wavelength like so: :k_BT = \frac{h^2}{2m\lambda^2} Hence, we can say that quantum fluids will manifest at approximate temperature regions where \lambda > d, where d is the lattice spacing (or inter-particle spacing). Mathematically, this is stated like so: :k_B T = \frac{h^2}{2m\lambda^2} It is easy to see how the above definition relates to the particle density, n. We can write :k_B T as n = \frac{1}{d^3} for a three dimensional lattice The above temperature limit T has different meaning depending on the quantum statistics followed by each system, but generally refers to the point at which the system manifests quantum fluid properties. For a system of
fermions, T is an estimation of the
Fermi energy of the system, where processes important to phenomena such as superconductivity take place. For
bosons, T gives an estimation of the Bose-Einstein condensation temperature. == See also ==