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The Schrödinger equation is a fundamental part of quantum mechanics. It describes how the quantum state of a particle changes over time. In the "particle in a box" problem, we use the time-independent form of the Schrödinger equation. This describes the stationary states of particles confined within a potential box. The equation looks like this:

• \[-\frac{\hbar^2}{2m}abla^2 \psi = E \psi\]

Here, \(\psi\) represents the wave function of the particle, \(E\) is the energy of the particle, and \(abla^2\) is the Laplacian operator, which represents the spatial part of the wave function within the well. The boundary conditions for a cubical box entail that outside the box, the potential is infinite, hence the wave function must be zero outside. Inside the box, the potential is zero which allows for separation of variables, resulting in solvable one-dimensional problems. The core idea is that the equation allows quantum mechanics to determine possible states and corresponding energies of a confined particle.

Quantum mechanics is the branch of physics that explores the behavior of matter on very small scales, such as atoms and subatomic particles. In quantum mechanics, the position and momentum of particles cannot be precisely determined. Instead, we use probabilities to describe where a particle is likely to be found. For a particle in a box, quantum mechanics comes into play to describe energy levels that are quantized. This means a particle can only exist in certain discrete energy states. Some key aspects are:

• Wave functions: They describe the probability distribution of a particle's location.
• Quantization: Only specific, discrete energy levels are allowed.
• Uncertainty Principle: Exact positions and velocities cannot be known simultaneously.

In the infinite cubical well or box scenario, quantum mechanics dictates specific patterns and rules for how the particle behaves and the allowed energy levels that stand out distinctively as being quantized.

In quantum mechanics, degeneracy refers to the number of quantum states that have the same energy. When dealing with a particle in a three-dimensional box, degeneracy becomes a crucial concept. For example, if multiple combinations of quantum numbers \((n_x, n_y, n_z)\) result in the same energy calculation \(E_{n_x,n_y,n_z}\), those energy levels are degenerate. Degeneracy is significant because:

• It describes symmetry in the system. If more states share the same energy, it indicates a high level of symmetry.
• Leads to complex behavior as different states with the same energy can mix.

In our exercise, for energies like \(E_1\), where \(n_x = n_y = n_z = 1\), there is no degeneracy. Meanwhile, higher energy levels might show degeneracy due to combinations like \((n_x = 2, n_y = 1, n_z = 1)\), \((n_x = 1, n_y = 2, n_z = 1)\), and \((n_x = 1, n_y = 1, n_z = 2)\) leading to the same energy state, indicating a degenerate energy level.