Introduction:
In this problem, we have 10 identical particles, each with mass m, that need to be accommodated in a cubical box of side L. We are asked to determine the lowest energy of the system if the particles obey Fermi-Dirac (FD) statistics.

Explanation:

Step 1: Understanding Fermi-Dirac Statistics
Fermi-Dirac statistics describe the behavior of particles with half-integer spins, such as electrons, at low temperatures. According to these statistics, no two identical fermions can occupy the same quantum state simultaneously.

Step 2: Identifying the Energy Levels
In a cubical box, the energy levels are quantized due to the confinement of the particles. To determine the energy levels, we can consider the particles as standing waves inside the box, with each wave having a specific wavelength and energy. The energy levels are given by the equation:

E = (h^2 * n^2) / (8 * m * L^2),

where E is the energy, h is the Planck's constant, n is the principal quantum number, m is the mass of the particle, and L is the length of the side of the cube.

Step 3: Filling the Energy Levels
According to FD statistics, each energy level can be occupied by at most one particle. We need to find the lowest energy configuration by filling the energy levels one by one.

To minimize the energy, we start by filling the lowest energy level (n = 1) with one particle. The next particle is then placed in the next available energy level (n = 2), and so on, until all 10 particles are accommodated.

Step 4: Calculating the Lowest Energy
To calculate the lowest energy of the system, we sum up the energies of all the occupied energy levels. Since each particle occupies a unique energy level, there are no degeneracies to consider.

The lowest energy is given by:

E_lowest = (h^2 / (8 * m * L^2)) * [1^2 + 2^2 + 3^2 + ... + 10^2].

Step 5: Simplifying the Expression
To simplify the expression, we can use the formula for the sum of the squares of the first n natural numbers:

1^2 + 2^2 + 3^2 + ... + n^2 = (n * (n + 1) * (2n + 1)) / 6.

Applying this formula, we can simplify the expression for the lowest energy:

E_lowest = (h^2 / (8 * m * L^2)) * [(10 * 11 * 21) / 6].

Step 6: Final Result
By evaluating the expression, we can find the numerical value of the lowest energy of the system.

E_lowest = (h^2 / (8 * m * L^2)) * 385.

This is the lowest energy of the system when 10 identical particles, each with mass m, are accommodated in a cubical box of side L and obey Fermi-Dirac statistics.