The quantum concentration 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.
Quantum effects become appreciable when the particle concentration is greater than or equal to the quantum concentration, which is defined as:
:
where:
* is the mass of the particles in the system
*
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
* is the temperature as measured in
kelvin
The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
s
*
is the
reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
The quantum concentration for room temperature protons is about 1/cubic-Angstrom.
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
A white dwarf is a Compact star, stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very density, dense: in an Earth sized volume, it packs a mass that is comparable to the Sun. No nuclear fusion takes place i ...
.
For an ideal gas the
Sackur–Tetrode equation can be written in terms of the quantum concentration as
: