Show explicitly that the difference between adjacent energy levels in an infinite square well becomes arbitrarily small compared with the energy of the upper level, in the limit of large quantum number \(n.\)

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. Unlike classical physics, which describes the world we see every day, quantum mechanics reveals a world where particles can exist in multiple states at once, have definite properties only when measured, and are interconnected in seemingly impossible ways.

While classical physics uses concrete values to predict the behavior of objects (such as a ball thrown into the air), in quantum mechanics, particles like electrons exhibit wave-particle duality – they behave as both particles and waves. This fundamental concept is at the heart of exercises dealing with systems like the infinite square well, where the behavior of particles is dictated by wave functions and probability rather than deterministic paths.

Schrodinger's equation is a key mathematical formula in quantum mechanics that describes how the quantum state of a physical system changes over time. For stationary problems, the time-independent form of Schrodinger's equation is used. It relates the wave function (which encapsulates the probabilities of finding a particle in certain states) to the energy of the system.

In the context of the infinite square well problem, where a particle is confined within perfectly rigid walls, the time-independent Schrodinger's equation is employed to find the allowed energy levels of the particle. These levels are quantized, meaning that the particle can only occupy certain discrete energy states.

A quantum number is an integer that quantifies the value of a conserved quantity in the quantum system. Essentially, quantum numbers serve as addresses for electrons or other particles, specifying their location and behavior within an atom or a quantum system, such as an infinite square well.

For the infinite square well scenario specifically, the principal quantum number, typically denoted by 'n', identifies the energy level of a particle. It determines the quantization of energy, as seen in the energy levels formula \(E_n = \frac{n^2 \pi^2 \hbar^2}{2 ma^2}\), where \(n\) is the quantum number, and it takes on positive integer values (1, 2, 3, ...).

Energy quantization is a phenomenon in quantum mechanics where systems such as atoms or a particle in an infinite square well can have only certain discrete energy values. This contrasts with classical systems, where objects can possess any value of energy.

The process to demonstrate energy quantization involves calculations using Schrodinger's equation, which results in distinct energy levels that depend on quantum numbers. Our exercise dealt with deriving the difference between energy levels—known as the energy gap—and showing its relative insignificance for large quantum numbers. As the quantum number \(n\) becomes very large, the energy gap \(E_{n+1} - E_n\) between adjacent levels becomes negligible compared to the energy of the levels themselves, which is an example of energy quantization at work.