Minimum Energy of a System of 4 Particles in a 2D Square Box

Introduction:
The problem involves finding the minimum energy of a system consisting of 4 particles inside a 2D square box. Each particle has the same mass 'm', and the side length of the square box is 'L'. We need to determine the minimum energy that can be obtained for this system.

Explanation:

Step 1: Setting up the Hamiltonian:
To solve this problem, we need to set up the Hamiltonian operator for the system. The Hamiltonian represents the total energy of the system. In this case, the Hamiltonian operator for the 2D square box can be written as:

H = (-ħ²/2m) ∇²

where ħ is the reduced Planck's constant, m is the mass of each particle, and ∇² is the Laplacian operator.

Step 2: Solving the Schrödinger Equation:
Next, we need to solve the Schrödinger equation for the system. The Schrödinger equation is given by:

Hψ = Eψ

where H is the Hamiltonian operator, ψ is the wave function, E is the energy of the system, and ħ is the reduced Planck's constant.

Step 3: Wave Function and Energy States:
Solving the Schrödinger equation for a 2D square box yields a set of allowed wave functions and corresponding energy states. The wave functions are given by:

ψ(x, y) = A sin(nπx/L) sin(mπy/L)

where A is a normalization constant, (x, y) are the coordinates of the particles inside the box, and n, m are positive integers representing the quantum numbers.

The corresponding energy states are given by:

E(n, m) = (ħ²π²/2mL²)(n² + m²)

Step 4: Occupying the Energy States:
According to the Pauli exclusion principle, each energy state can only be occupied by one particle. Since there are 4 particles in the system, we need to find the 4 lowest energy states that can accommodate these particles.

Step 5: Determining the Minimum Energy:
To find the minimum energy of the system, we need to consider the 4 lowest energy states. By substituting different values of n and m into the energy equation, we can calculate the energy for each state. The minimum energy will be the sum of the 4 lowest energies.

Conclusion:
In summary, the minimum energy of a system consisting of 4 particles in a 2D square box can be determined by solving the Schrödinger equation and finding the 4 lowest energy states. The minimum energy is obtained by summing the energies of these states.