An imaginary atom has just 3 energy levels: \(0 \mathrm{eV}, 1 \mathrm{eV}\), and \(3 \mathrm{eV}\). Draw an energy-level diagram for this atom. Show all possible transitions between these energy levels. For each transition, determine the photon energy and the photon wavelength. Which transitions involve the emission or absorption of visible light?

The electron volt (eV) is a unit of energy that's particularly useful in the context of atomic and particle physics. It's defined as the amount of kinetic energy gained or lost by an electron when it moves through an electric potential difference of one volt. In mathematical terms, an electron volt is equivalent to approximately \(1.602 \times 10^{-19}\) joules.

When we talk about energy levels within an atom, we are referring to the discrete amounts of energy that electrons can have when bound to an atom. These energy levels are typically quantified in electron volts. For example, in the provided exercise, the three energy levels are at \(0 \mathrm{eV}\), \(1 \mathrm{eV}\), and \(3 \mathrm{eV}\). These labels represent the energy difference relative to the ground state, which is the state of lowest energy.

Photon energy is the energy carried by a single photon, which can be calculated using the equation \(E = h \times f\), where \(E\) is the energy, \(h\) is Planck's constant \(6.626 \times 10^{-34} \mathrm{m^2 kg / s}\), and \(f\) is the frequency of the photon. Alternatively, this energy can be related to the wavelength \(\lambda\) of the photon via the equation \(E = \frac{hc}{\lambda}\), where \(c\) represents the speed of light.

In our textbook exercise, the photon energy corresponds to the difference in energy levels during a transition. If an electron falls from a higher to a lower energy level, it emits a photon with energy equivalent to that difference. Conversely, when an electron absorbs a photon, it gains energy and moves to a higher energy level. Calculating these energy differences is crucial to understanding the behavior of electrons in an atom and the properties of emitted or absorbed light.

Absorption and emission are two fundamental processes that occur when photons interact with atoms. During absorption, an atom takes in a photon, which causes one of its electrons to jump to a higher energy level, provided that the photon's energy matches the energy difference between the levels. Conversely, emission occurs when an electron in an excited state drops to a lower energy level, releasing a photon with energy equal to the difference between those levels.

These processes are beautifully illustrated by the energy-level diagram from the exercise. Transitions can represent either emission or absorption processes, depending on the direction of the electron's jump between energy levels. Emission would be indicated by a downward transition, resulting in the release of a photon, while absorption is an upward transition that consumes a photon of respective energy.

The visible light spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. It ranges approximately from \(400 \) to \(700 \text{nanometers}\) (nm) in wavelength, which corresponds to energies between \(1.65\) and \(3.1 \mathrm{eV}\). Each color of visible light has a specific range of wavelengths; for instance, violet light has the shortest wavelengths, and red light has the longest.

In the context of our exercise, we're interested in knowing which transitions between energy levels correspond to the emission or absorption of visible light. By calculating the wavelengths using the photon energies, we can determine if a transition falls within the visible spectrum. If the energy of the emitted or absorbed photons during such transitions matches the visible light energy range, then those photons contribute to the visible light that we can see.