A particle is confined to a 1.0 -nm-wide infinite square well. If the energy difference between the ground state and the first excited state is \(1.13 \mathrm{eV},\) is the particle an electron or a proton?

Quantum mechanics is the branch of physics that deals with the behavior, properties, and energy of matter and light on the atomic and subatomic levels. Unlike classical mechanics, which describes the motion of objects much larger than atoms with classical laws of physics, quantum mechanics employs a mathematical framework to explain phenomena that cannot be explained by classical physics, such as the behavior of electrons within an atom.

At the core of quantum mechanics is the principle that energy, instead of being continuous, is quantized. This means that certain physical quantities, such as energy, can take on only discrete values. In the context of the exercise, a particle confined within an infinite square well can only possess certain energy levels, much like how an electron in an atom can only inhabit specific orbitals.

Particle confinement refers to the condition where a particle cannot move freely in all directions due to physical constraints or boundaries. In the exercise, the particle is placed in an 'infinite square well', a theoretical construct in which a particle is trapped in a one-dimensional well with infinitely high walls. This means it cannot escape and is confined to a space of defined boundaries. The infinite walls provide an idealized scenario where the particle's motion is strictly restricted to a finite region of space, allowing it to have only certain allowed energy states. This model is crucial because it provides a simple system to study quantum mechanical effects and helps develop an understanding of more complex systems used in quantum computing, semiconductors, and other fields of modern physics.

Energy level calculations in quantum mechanics involve determining the permitted levels of energy that a confined particle can have. For the infinite square well problem, these levels are derived from the Schrödinger equation, which provides a way to calculate the wave function of the particle and, consequently, its energy states.

In the exercise, the energy difference between the ground state (the lowest energy state) and the first excited state is used to calculate the mass of the particle. This difference is a characteristic of the quantum system and depends on Planck's constant, the mass of the particle, and the size of the confinement region. By applying the given values and constants, it's possible to calculate the energy levels and then deduce the type of particle within the well, whether it's an electron, a proton, or any other subatomic particle. Such calculations are foundational in fields like quantum computing and nanotechnology, where understanding the energy properties of quantum systems is essential.