Particle in a One Dimensional Box

Consider a particle that can move only in one dimension between two impenetrable walls at $x = 0$ and $x = a$. The potential is zero between the walls and infinite outside. Solve the time-independent Schrödinger equation to find the allowed wave functions and corresponding energies of the particle.

This classic problem in quantum mechanics involves understanding how a particle behaves in a confined space, leading to quantization of energy levels. The particle-in-a-box model is a fundamental example used to illustrate key principles such as quantization and boundary conditions in quantum mechanics. Here, you will be solving the time-independent Schrödinger equation, a cornerstone of quantum theory, which describes how the quantum state of a physical system changes over time.

In this scenario, the particle is subject to potential energy that is zero in the region it can move and infinite where it cannot move, symbolizing rigid boundaries or walls. The problem involves finding solutions, known as wave functions, that satisfy the Schrödinger equation within these boundaries. These wave functions must satisfy specific boundary conditions, leading to discrete energy levels, which is characteristic of quantum systems.

The solutions to this problem reveal that only certain wave functions, and thus energies, are allowed. This results from the mathematical requirement that the wave function must be zero at the boundaries of the box, a concept tied to the physical impossibility of the particle existing outside the defined region. Analyzing these wave functions and energies provides insight into fundamental phenomena, such as quantization and normalization, which have profound implications across quantum mechanics and related fields.