What is the minimum frequency of a photon required to ionize: (a) a \(\mathrm{He}^{+}\) ion in its ground state? (b) \(\mathrm{A} \mathrm{Li}^{2+}\) ion in its first excited state?

Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous phase. In simpler terms, it is the energy needed to overcome the attraction between the electron and the nucleus, thereby freeing the electron and creating an ion. This concept is critical in understanding photoionization, where a photon's energy has to match or exceed the ionization energy to strip an electron from an atom.

As illustrated in the exercise, different atoms and ions have unique ionization energies. Knowing the ionization energy allows us to calculate the threshold frequency of a photon capable of ionizing an atom — this brings us to the photoelectric effect observed in quantum mechanics. When the frequency of the incident photon is above this threshold, the excess energy is given to the ejected electron as kinetic energy.

Atomic physics is the field that studies atoms as isolated systems, which mostly consist of electrons orbiting a nucleus. Understanding atomic structure is fundamental to explaining how atoms interact with light, particularly how they absorb or emit photons. When we consider ionization in atomic physics, we focus on interactions involving the transfer of energy from photons to electrons.

The exercise uses the model of an atom where electrons reside in quantized energy levels or orbits, each described by a principal quantum number, n. The equations provided in the solution calculate the energy of an electron at a specific level for the ions He+ and Li^2+. For ionization to occur, an incoming photon must supply energy equal to the difference between the electron's present energy level and the energy at an infinite distance away from the nucleus, essentially the ionization energy.

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. The Planck-Einstein relation from quantum mechanics, given by the equation E = hν, where h is Planck's constant and ν is frequency, connects energy and frequency of a photon. This relation is used in the exercise to find the minimum frequency of a photon that can ionize an atom.

Understanding quantum mechanics is crucial when working with the concept of ionization, as it deals with quantization of energy, probabilistic nature of particle properties, and wave-particle duality. In the solution for the exercise, quantum mechanics tells us that if the photon's energy meets or exceeds the ionization energy of the He+ or Li^2+ ion, the electron can be ejected, resulting in ionization.