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When studying the behavior of particles at the quantum level, we encounter distinct 'steps' or 'rungs' of energy, much like a ladder. These are known as the energy levels. In a quantum mechanical square well, a particle can only occupy these distinct energy states. The energy of each level is not continuous but quantized, meaning that the particle cannot possess energy in between these levels.

The formula for calculating the energy of a particle in a one-dimensional infinite potential well is given by \(E_n = \frac{n^2h^2}{8mL^2}\), where \(E_n\) represents the energy of the nth level, \(n\) is the principal quantum number, and only takes on positive integer values, \(h\) is Planck's constant, \(m\) is the mass of the particle (an electron in our case), and \(L\) is the width of the potential well. The square of \(n\) indicates that energy increases with the square of the principal quantum number — the higher the 'rung' on the 'ladder,' the greater the energy of the particle.

The ground state of a particle in a quantum mechanical square well is its lowest possible energy state, referred to as the baseline energy level. For the ground state, the principal quantum number \(n\) is equal to 1. Using the formula given for energy levels, we can deduce that \(E_1 = \frac{h^2}{8mL^2}\), since \(n = 1\). This represents the minimum energy the electron must have to be within the well.

Understanding ground state energy is crucial in quantum mechanics, as it forms the foundation for calculating excitation. Any energy added to the system will potentially 'lift' the electron to a higher energy level. The ability of an electron to exist at this basic level yet still be confined within the potential well underscores the fundamental principle of quantization in quantum mechanics.

Photon absorption is a process wherein an electron absorbs a photon, gaining its energy. This gain in energy can prompt the electron to jump to a higher energy level or, given enough energy, break free from the confines of the potential well entirely — a concept known as 'photoelectric effect' in a broader sense.

When an electron is bound within a potential well, it requires a precise amount of energy to escape, equal to the depth of the well, \(U_0\). The energy of a photon is determined by its wavelength and is given by the formula \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. To calculate the maximum wavelength of a photon that can still liberate an electron from the square well, we find the wavelength that corresponds with the energy equal to the well depth. Any longer wavelength would not have enough energy to free the electron, and thus, the electron will remain bound within the well.