Entangled Particles and Spin Measurement

In an experiment like the one Einstein proposed, two particles are entangled. If one particle is measured to have spin up, the other measured in the same direction must have spin down. What happens if their spins were vertical and opposite, but both are measured in the horizontal direction? Explain how this relates to the conservation of angular momentum and the concept of entanglement.

This problem delves into the fascinating realm of quantum entanglement, a key phenomenon in quantum mechanics that challenges our classical intuition about the physical world. Entanglement suggests that particles can be correlated in such a way that the state of one particle can instantaneously influence the state of another, regardless of the distance separating them. This experiment, reminiscent of the Einstein-Podolsky-Rosen (EPR) paradox, is a classical illustration of these nonlocal correlations.

When particles are entangled, measuring one particle in a particular direction affects the measurement outcome of its partner in the corresponding direction, showcasing the property of spin correlation in entangled particles. The measurement in a new, perpendicular direction, as posed by this problem, introduces an additional layer of complexity involving quantum superposition. Rather than having a predetermined outcome, quantum mechanics tells us that when spins oriented vertically and oppositely are measured in the horizontal direction, each particle exists in a superposition of spin states. The act of measurement causes each particle to collapse into one of the possible states, again dependent on its partner due to their entangled nature.

This process underscores the non-intuitive elements of quantum measurement, where the observer plays a crucial role in determining the state of a quantum system. Additionally, this problem touches on the conservation of angular momentum, which remains a vital principle even in the quantum domain. Despite the inherent uncertainties in individual measurements, the total angular momentum of an isolated system is conserved. The results of measurements on entangled spins reflect this conservation, as they always add up to preserve the initial total spin, even when measured along different axes. This illustrates the beautifully consistent yet seemingly paradoxical nature of quantum mechanics.