The Schrödinger equation is a foundational equation in quantum mechanics that describes the behavior of quantum systems. It is a partial differential equation, and its solutions give the wave functions that describe the probability amplitudes of particles.
While the Schrödinger equation is continuous in time and space, the discrete nature of particles and their quantized properties emerge through the interpretation of the wave function. When a measurement is made on a quantum system, the wave function collapses to a specific value corresponding to the measured property (e.g., position, momentum, energy). This collapse results in a discrete outcome, such as the detection of a particle at a specific location or with a particular energy.
The process of wave function collapse is often described by the Copenhagen interpretation, which states that the wave function represents the probabilities of different outcomes, and the act of measurement "selects" one of these possibilities. This interpretation is probabilistic, meaning that the outcome of a measurement cannot be predicted with certainty but can be described in terms of probabilities.
It is worth noting that other interpretations of quantum mechanics exist, such as the many-worlds interpretation or the pilot-wave theory. These interpretations propose different ways to reconcile the continuous nature of the Schrödinger equation with the discrete measurement outcomes, but they are subject to ongoing debate and philosophical considerations.
In summary, while the Schrödinger equation is continuous, the discrete nature of particles and their quantized properties arise through the interpretation of the wave function and the process of measurement in quantum mechanics.