Quantum concentration

The quantum concentration n_Q is the particle concentration (i.e. the number of particles per unit volume) of a system where the interparticle distance is equal to the thermal de Broglie wavelength or equivalently when the wavefunctions of the particles are touching but not overlapping.

Quantum effects become appreciable when the particle concentration is greater than or equal to the quantum concentration, which is defined as:

$n_Q=\left(\frac{M k T}{2 \pi \hbar^2}\right)^{3/2}$

where:

M is the mass of the particles in the system

k is the Boltzmann constant

T is the temperature as measured in kelvins

$\hbar\,$ is the reduced Planck constant

As the quantum concentration depends on temperature, high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf.