{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Mechanics \u2192 Energy Levels and Lifetimes", "refinement": "This problem involves the relationship between the energy difference of quantum states and their lifetimes, specifically in the context of resolving energy levels.", "specific_scope": "The focus is on determining the minimum energy difference required to distinguish between two quantum states with given lifetimes.", "goal": "Identify the energy difference that allows for clear resolution of the two quantum states based on their lifetimes."}, "summary": "In quantum mechanics, the ability to distinguish between two energy levels is influenced by their lifetimes. The energy uncertainty principle states that the uncertainty in energy (\u0394E) is inversely related to the uncertainty in time (\u0394t), given by \u0394E * \u0394t \u2248 \u0127/2. For the two states with lifetimes of 10^-9 sec and 10^-8 sec, we can calculate the corresponding energy uncertainties. To clearly resolve the energy levels, the energy difference must be significantly larger than the uncertainty. Among the given options, 10^-4 eV is the only value that exceeds the uncertainties derived from the lifetimes, making it the correct choice for distinguishing the two states."}