The distinction between deterministic classical systems and non-deterministic quantum systems arises from the fundamental principles and mathematical formalism underlying each system.
Classical systems, governed by classical physics, are described by classical mechanics. In classical mechanics, the behavior of a system is determined by well-defined equations of motion, such as Newton's laws of motion. Given the initial conditions of a classical system, its future behavior can be precisely predicted, allowing for deterministic outcomes. In other words, in a classical system, the state of the system at any given time uniquely determines its state at all future times.
On the other hand, quantum systems are described by quantum mechanics, a mathematical framework developed to explain the behavior of particles at the microscopic scale. In quantum mechanics, the state of a system is represented by a mathematical entity called a wavefunction. The wavefunction provides a probabilistic description of the system, giving the probabilities of different outcomes when a measurement is made.
One of the fundamental principles in quantum mechanics is the principle of superposition, which states that quantum systems can exist in multiple states simultaneously. This means that a quantum system can be in a superposition of different states, each with an associated probability amplitude. When a measurement is made, the wavefunction "collapses" into one of the possible states, and the outcome is probabilistic. The specific state that the system collapses into is determined by the probabilities encoded in the wavefunction.
Additionally, quantum mechanics introduces the concept of indeterminacy or uncertainty. The Heisenberg uncertainty principle, for example, states that there are inherent limits to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. This intrinsic uncertainty in quantum systems contributes to their non-deterministic nature.
It's important to note that while quantum systems are inherently probabilistic and non-deterministic, the probabilistic outcomes they produce are still subject to precise mathematical rules and can be statistically predicted. These probabilistic predictions, when averaged over many measurements, can exhibit precise patterns and reproduce deterministic behavior on a macroscopic scale, known as the correspondence principle.
In summary, the distinction between deterministic classical systems and non-deterministic quantum systems arises from the foundational principles and mathematical formalism of classical and quantum mechanics. Classical systems are governed by deterministic equations of motion, while quantum systems, due to principles such as superposition and uncertainty, exhibit probabilistic behavior at the microscopic level.