{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Spin Systems \u2192 Time Evolution of Quantum States", "refinement": "This problem involves the time evolution of a spin-1/2 particle in a magnetic field, specifically how the state of the particle changes over time due to the influence of the magnetic field.", "specific_scope": "The focus is on calculating the probability of measuring the particle in the |down\u27e9 state with respect to the z-axis after a certain time t, given its initial state and the direction of the magnetic field.", "goal": "Determine the probability of finding the particle in the |down\u27e9 state after time t, using the principles of quantum mechanics and the effects of the magnetic field on the spin state."}, "summary": "This problem deals with the time evolution of a spin-1/2 particle in a magnetic field. Initially in the |up\u27e9 state along the z-axis, the particle's state changes due to the magnetic field applied in the +X direction. The task is to calculate the probability of finding the particle in the |down\u27e9 state along the z-axis after a time t. The correct answer is derived from the quantum mechanical principles governing spin dynamics, leading to the result sin\u00b2(\u03b3Bt/2), which reflects the oscillatory nature of the spin state under the influence of the magnetic field."}