Probability Distribution in 1D Potential Box

Determine the probability distribution for a particle in a one-dimensional potential box for different quantum states $n = 1$, $n = 2$, and the classical limit as $n \to \infty$.

When exploring the probability distribution for a particle in a one-dimensional potential box, we engage with a fundamental concept in quantum mechanics known as the particle in a box model, or the infinite potential well. This model allows us to understand how particles, such as electrons, behave in confined spaces where the potential energy is zero inside the box and infinite outside it. The quantum states, denoted by the quantum number n, define the allowed energy levels and wavefunctions of the particle. For each state, the probability distribution is given by the square of the wavefunction, which describes the likelihood of finding the particle at a particular position in the box. As we analyze different quantum states, such as $n = 1$ and $n = 2$, we observe distinct probabilities, with the $n = 1$ state being the ground state and exhibiting no nodes within the well, and the $n = 2$ state having one node, highlighting how the complexity of the probability distribution increases with higher energy states. The classical limit, as the quantum number $n \to \infty$, provides an interesting convergence towards classical physics predictions. In this scenario, the probability distribution tends towards a uniform distribution across the box, resembling the classical notion where a particle is equally likely to be found anywhere in the box. This forms an important bridge between quantum and classical mechanics, illustrating how classical behaviors emerge from quantum rules as systems grow larger in scale or energy. This exercise reinforces the concept of quantization and the wave-particle duality central to quantum mechanics, while offering insights into the particle confinement in potential wells.