An electron is confined to a box of width \(0.25 \mathrm{nm}\). (a) Draw an energy-level diagram representing the first five states of the electron. (b) Calculate the wavelengths of the emitted photons when the electron makes transitions between the fourth and the second excited states, between the second excited state and the ground state, and between the third and the second excited states.

At its core, quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. Unlike classical mechanics, which describes the motion of objects visible to the naked eye, quantum mechanics reveals a world where particles, such as electrons, exhibit both wave-like and particle-like properties.

In the realm of quantum mechanics, concepts like the 'particle in a box' model come into play. This model is a fundamental example used to illustrate the quantized nature of energy states in systems bound by quantum rules. In the provided exercise, understanding quantum mechanics is essential as it allows us to grasp why an electron confined in a quantum box has discrete energy levels and why it can jump between these levels, emitting or absorbing energy in the form of photons.

An energy level diagram is a visual representation used to illustrate the discrete energy states that particles like electrons can occupy. For example, in the case of an electron in a quantum box, the diagram would display several lines, each corresponding to a particular energy level associated with a specific principal quantum number 'n'.

The ground state is represented by the lowest line (n=1), and as the value of 'n' increases, so does the energy level, moving further up on the diagram. When presented in step by step solutions, this diagram helps students visualize the allowed energy levels and the transitions between them. The energy level diagram in the exercise portrays the first five states, with the energy increasing as the level number goes up.

The Planck constant (denoted as 'h') is a fundamental physical constant representing the quantization of action in quantum mechanics. With a value of approximately \(6.626 \times 10^{-34} \, \text{Js}\), the Planck constant is integral in calculating energy levels and transitions in quantum systems.

In the provided exercise, the Planck constant is used within the formula to determine the energy states of an electron in a confined space. Its presence in the formula \(E_n = \dfrac{n^2 \times h^2}{8 \times m \times L^2}\) underlines how the energies are quantized - that is, they can only take on specific, discreet values rather than any value in a continuous range.

Electron transitions refer to an electron's movement between different energy levels within an atom or a quantum system, like our 'particle in a box' example. When an electron transitions from a higher to a lower energy level, it releases energy; conversely, it absorbs energy when transitioning from a lower to a higher energy level.

The energy released or absorbed is exactly equal to the difference between the energy states involved in the transition. This concept is critical for understanding phenomena such as the emission or absorption spectra in atoms and is the foundation for the calculations required in the exercise for determining the wavelengths of emitted photons during transitions.

The wavelength of an emitted photon is correlated to the energy released during an electron transition. According to the equation \(E = \dfrac{hc}{\text{λ}}\), the energy (E) of an emitted photon is inversely proportional to its wavelength (λ), with 'h' representing the Planck constant and 'c' denoting the speed of light. This principle allows us to calculate the wavelengths of photons emitted when an electron jumps between specific energy levels.

In the exercise, students are tasked with finding the wavelengths of photons released during certain electron transitions. By rearranging the equation to solve for λ, they can link the abstract concept of energy differences directly to the more tangible concept of photon wavelengths, a crucial step in connecting theory with observable phenomena.