{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Mechanics \u2192 Wave Functions", "refinement": "This problem involves understanding the properties of wave functions and normalization in quantum mechanics, particularly for identical particles in a confined space.", "specific_scope": "The focus is on determining the normalization constant 'a' for the wave function of particles confined between specific boundaries (x=1 and x=3).", "goal": "Calculate the value of 'a' such that the wave function is properly normalized over the specified range."}, "summary": "This problem is rooted in quantum mechanics, specifically in the normalization of wave functions for identical particles confined to a specific region. The wave function provided must be normalized over the interval from x=1 to x=3, ensuring that the total probability of finding a particle within this range equals 1. The task is to find the appropriate value of the constant 'a' that satisfies this normalization condition. The correct answer is 0.85, which ensures the wave function is properly normalized."}