(a) Using the particle-in-the-box model for the hydrogen atom and treating the atom as electron in a one-dimensional box of length 150 . pm, predict the wavelength of radiation emitted when the electron falls from the level with \(n=3\) to that with \(n=2\). (b) Repeat the calculation for the transition from \(n=4\) to \(n=2\).

Quantum mechanics is a fundamental theory in physics that provides the description of the physical properties of nature at the scale of atoms and subatomic particles. It departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. Quantum mechanics introduces a major change in how we understand measurements and predictions concerning the behavior of energy and material particles.

In the context of the particle-in-a-box model, quantum mechanics tells us that an electron is restricted to specific, quantized energy levels, which are determined by the electron's wave properties. This model is a simplification used for solving quantum problems where the potential energy is constant inside a box and infinite outside it, preventing the particle from escaping.

The exercise given is a classic example of applying quantum principles to predict the behaviors of an electron under constrained conditions. Instead of moving freely in space, the electron is imagined to move in a straight line back and forth between the walls of an imaginary box, changing its energy in discrete amounts when it transitions between levels. This is a quintessential quantum mechanical scenario, showcasing the dual nature of particles that possess both wave-like and particle-like properties.

The calculation of energy levels is essential in understanding the electronic transitions in atoms and molecules within quantum mechanics. Energy level calculation applies well-defined mathematical models to predict the possible energy states an electron can occupy.

The particle-in-a-box model gives us a simplified way to calculate these energy levels. Using the formula,
\( E_n = \frac{n^2h^2}{8mL^2} \), where each symbol represents a physical constant or a condition of the system, we can determine the specific, quantized energies that an electron bound within a one-dimensional box (or a potential well) can have.

Each \( n \) value represents a quantum number corresponding to a unique energy level. The exercise you tackled demonstrates how to calculate the energy difference between two quantized states, and this energy difference correlates with the energy of the photon emitted when an electron transitions between these levels. As such, these energy differences are crucial for understanding the resulting spectral lines we observe, a concept that is central to fields like spectroscopy.

Wave-particle duality is one of the most fascinating and fundamental concepts in quantum mechanics. It proposes that every particle or quantum entity exhibits both wave and particle properties. This concept is at the heart of the famous double-slit experiment, which revealed that light can display characteristics of both classically defined waves and particles; in other words, photons can interfere with themselves as waves, yet be counted as particles.

The particle-in-a-box model that was used in the given exercise demonstrates this duality. By treating the electron as both a particle confined to a box and as a wave that has a specific wavelength determined by the box's size, the exercise beautifully showcases the dual nature of quantum entities.

The reality that an electron constrained to a box has quantized energy levels can be explained by its wave-like behavior: only certain wavelengths (and therefore energies) can fit into the box 'just right', leading to standing waves that correlate with stable energy states. The wave-particle duality further enriches our understanding of the electronically emitted radiation when these electrons make transitions between different quantized energy levels.