What are the possible values of the principal quantum number n? What does the principal quantum number determine?

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It incorporates principles that may seem counterintuitive, differing vastly from classical physics. Concepts such as wave-particle duality and quantization of energy are cornerstones of this theory.

Within quantum mechanics, the behaviors and interactions of particles are described in terms of probabilities rather than definite outcomes. An electron's position, for instance, is not specified precisely but is given by a probability distribution. The mathematical formalism used to make predictions in quantum mechanics is complex, involving wave functions and operators that act on these wave functions to extract physically relevant information such as energy levels and probabilities of outcomes.

One of the key findings of quantum mechanics is that energy is not continuous, but rather comes in discrete packets called quanta. This quantization implies that electrons in an atom can only exist in certain discrete energy levels.

The concept of electron energy levels is central to the understanding of atomic structure in quantum mechanics. According to this theory, electrons orbit around the nucleus in specific regions called orbitals. Each orbital corresponds to a certain energy level, defined by three quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m).

The principal quantum number (n) defines the energy level of an electron and is a positive integer. As mentioned in the step-by-step solution, higher values of n mean that the electron is in a higher energy level, which also correlates to a larger average distance from the nucleus. This is crucial for understanding how atoms absorb and emit energy, as electrons can move between these levels by gaining or releasing specific amounts of energy, often observed as spectra.

Furthermore, these energy levels are not equally spaced. The energy difference between adjacent levels decreases as n increases, meaning the levels get closer together the further they are from the nucleus. In the context of the exercise, understanding these energy levels helps explain the electronic structure and chemical properties of elements.

Atomic structure describes the organization of electrons, protons, and neutrons in an atom. In the context of quantum mechanics, electrons are found in regions called orbitals within energy levels around the nucleus. The nucleus itself consists of protons and neutrons.

The principal quantum number, while indicating an atom's energy levels, also imparts valuable information regarding an atom's size and the possible arrangements of electrons, which influence the atom's chemical behavior. As n increases, the atom becomes larger, and the probability of finding an electron further from the nucleus goes up. This affects the atom's ability to interact with other atoms, thus, n plays a pivotal role in chemical bonding and the periodic properties of elements.

Moreover, the number of subshells in an energy level, as elucidated by quantum mechanics, is equal to n. Each subshell can contain a number of orbitals, which in turn host electrons. The complete set of quantum numbers (n, l, m, and spin quantum number s) defines the unique state of an electron in an atom, alluding to the complex interplay of factors determining atomic behavior. The principal quantum number’s role in defining the atomic size and energy levels is just the tip of the iceberg in the comprehensive narrative of atomic structure.