Show that the condition n e (2m e k B T) 3/2 /h 3 [cf. Eq. (20.3)] that electrons be nondegenerate is equivalent to

Show that the condition n_e ≪ (2πm_ek_BT)^3/2/h³ [cf. Eq. (20.3)] that electrons be nondegenerate is equivalent to the following statements.

(a) The mean separation between electrons, l ≡ n_e^−1/3, is large compared to the electrons’ thermal de Broglie wavelength,

(b) The uncertainty in the location of an electron drawn at random from the thermal distribution is small compared to the average inter-electron spacing.

(c) The quantum mechanical zero-point energy associated with squeezing each electron into a region of size l = ne^−1/3 is small compared to the electron’s mean thermal energy k_BT.

Equation 20.3.

^ =h//2rmek T. dB