An infinite square well is a potential energy function used in quantum mechanics to model a particle confined within perfectly rigid and impenetrable boundaries. The potential energy is zero inside the well and infinite outside, meaning the particle cannot exist outside the well.
The energy levels are quantized and determined by solving the Schrödinger equation for the system. The allowed energy levels are given by the formula \( E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \), where \( n \) is a positive integer (quantum number), \( \hbar \) is the reduced Planck's constant, \( m \) is the mass of the particle, and \( L \) is the width of the well.
The wavefunctions, or eigenfunctions, for a particle in an infinite square well are sine functions that satisfy the boundary conditions of the well. They are given by \( \psi_n(x) = \sqrt{\frac{2}{L}} \sin \left( \frac{n \pi x}{L} \right) \), where \( n \) is a positive integer, \( x \) is the position within the well, and \( L \) is the width of the well.
The quantum number \( n \) determines the energy level and the corresponding wavefunction of the particle. It must be a positive integer (1, 2, 3, ...), and each value of \( n \) corresponds to a distinct energy level and wavefunction. Higher values of \( n \) correspond to higher energy levels and more complex wavefunctions.
The infinite square well is a fundamental problem in quantum mechanics that illustrates key concepts such as quantization of energy, wavefunctions, and boundary conditions. It serves as a simple yet powerful model to understand the behavior of particles in confined spaces and provides a basis for studying more complex quantum systems.