Final answer:
The ground-state energy of ten non-interacting spin-1/2 fermions in a one-dimensional box is the sum of the energies of the first five filled levels. The Fermi energy is the energy of the highest filled level at absolute zero. The calculations are based on the quantized energy levels of particles in a one-dimensional potential well.
Step-by-step explanation:
The question concerns the ground-state energy and Fermi energy of ten non-interacting spin-1/2 fermions in a one-dimensional box, a topic within the realm of quantum mechanics. In a one-dimensional potential well or box, the energy levels of a particle are quantized. For non-interacting spin-1/2 fermions, such as electrons, each energy level can be occupied by two particles due to their two possible spin states, following the Pauli exclusion principle.
The ground-state energy is the sum of the energies of the lowest states that all the fermions occupy. For the ten fermions to be in the lowest possible energy states, the first five levels would be filled, since each level can hold two fermions. Calculating these energies requires using the formula for the energy levels in a one-dimensional box, E_n = (n^2 π^2 ĭ^2)/(2mL^2), where ĭ is the reduced Planck constant, m is the particle mass, n is the principal quantum number, and L is the length of the box. To find the total ground-state energy, you sum the energies of each occupied level.
The Fermi energy is the energy of the highest occupied single-particle state at absolute zero. This corresponds to the energy of the highest filled level when all the lower energy states are fully occupied.