![]() For comparison, we estimate the average thermal energy per particle in a system, which may be approximated by the expression k B T, where k B is Boltzmann's constant and T the temperature of the system in degrees Kelvin (K). (8) is already quite formidable, and it increases at a rate proportional to the two-thirds power of the matter density. The average kinetic energy per electron given by Eq. (8) as the insertions of the c factors are explicitly displayed. This scheme is employed in the evaluation of Eq. The units for h can also be simplified if they are combined with a factor of c, since hc = 1.24 × 10 −4 eV cm. Thus, instead of expressing the electron mass in grams it is converted into energy units by multiplying by a factor of c 2, and m e c 2 = 0.511 × 10 6 eV. These two units are picked to be electron volts (eV) for energy and centimeters (cm) for length. ![]() (8), we shall follow a scheme that reduces all units needed in the problem to two by inserting factors of h or c at appropriate places. Let the radius of the sphere be given by p F called the Fermi momentum, then the integral for the volume of the sphere with the momentum variable expressed in the spherical polar coordinates is given by: The summation of all occupied quantum states is equivalent to finding the volume of the sphere, since all states are equally spaced from each other inside the sphere. ![]() Graphically, the momenta of the occupied states plotted in a three-dimensional momentum space appear like a solid sphere centered about the point of zero momentum. ![]() In other words, there exists a fixed momentum magnitude p F and, in the ground-state configuration, all states with momentum magnitudes below p F are occupied while the others with momentum magnitudes above p F are unoccupied. The momenta of all electrons are completely fixed if it is further required that the system exist in its ground state, which is the lowest possible energy state for the system because there is just one way that this can be accomplished, which is for the electrons to occupy all the low-momentum quantum states before they occupy any of the high-momentum quantum states. According to Pauli's exclusion principle, all electrons of the same spin orientation in a system must possess momenta that differ from each other by the amount specified above. ![]()
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