{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Mechanics \u2192 Quantum States", "refinement": "This problem pertains to the behavior of a quantum particle in a potential well, specifically focusing on the probability density of the particle in its excited state.", "specific_scope": "The context is a one-dimensional infinite potential well, where the particle's wave function and corresponding probability density are analyzed at a specific point (the center of the well).", "goal": "Determine the probability density of the particle at the center of the well when it is in its first excited state."}, "summary": "In this quantum mechanics problem, we analyze a particle in a one-dimensional infinite potential well. The focus is on finding the probability density at the center of the well when the particle is in its first excited state. The wave function for the first excited state has a node at the center of the well, leading to a probability density of zero at that point. Thus, the answer to the question is 0."}