The quantum concentration can be derived from the canonical
partition function of an ideal gas particle with the free particle
Hamiltonian. For a single particle in a box of volume V, the partition function in the classical limit is given by integrating over all momentum states: :Z_{\text{ideal, monatomic}} = \frac{1}{h^3} \int d^3\vec{r} \int d^3\vec{p} \, e^{-\beta \vec{p}^2/2m} where \beta = 1/(k_B T), m is the particle mass, and h is Planck's constant. The factor 1/h^3 accounts for the quantum density of states in phase space. The spatial integral gives the volume V, while the momentum integral is evaluated in spherical coordinates: :Z_{\text{ideal, monatomic}} = V \left(\frac{1}{h^3}\right) \int_0^{2\pi} d\phi \int_0^{\pi} \sin\theta \, d\theta \int_0^{\infty} p^2 \, dp \, e^{-\beta p^2/2m} The angular integrals give 4\pi, and the radial momentum integral is a standard
Gaussian integral: :\int_0^{\infty} p^2 \, e^{-\beta p^2/2m} dp = \frac{1}{2}\sqrt{\frac{\pi}{\beta^3}} (2m)^{3/2} Combining these results: :Z_{\text{ideal, monatomic}} = V \left(\frac{1}{h^3}\right) (4\pi) \cdot \frac{1}{2}\left(\frac{2\pi m}{\beta}\right)^{3/2} = V \left(\frac{2\pi m k_B T}{h^2}\right)^{3/2} The quantum concentration is defined as the coefficient with units of inverse volume that relates the partition function to the volume: :Z_{\text{ideal, monatomic}} = V \cdot n_Q :n_Q \equiv \left(\frac{2\pi m k_B T}{h^2}\right)^{3/2} = \left(\frac{m k_B T}{2\pi\hbar^2}\right)^{3/2} Note that, when n>n_Q we shall have energy quantization and the ideal gas approximation shall not hold no further. ==References==