The extension of a bound particle's wave function to infinity is a mathematical property rather than a physical object that physically extends to infinity. The wave function is a mathematical representation used in quantum mechanics to describe the probabilistic behavior of particles.
In the context of bound particles, such as electrons in an atom, the wave function represents the probability distribution of finding the particle in different locations. The square of the wave function, known as the probability density, provides the likelihood of finding the particle at a specific position.
For bound particles, the wave function typically describes an exponentially decaying envelope that extends to infinity. This does not mean that the particle itself physically extends to infinity. Rather, it reflects the possibility of finding the particle in different regions of space, including regions with very low probabilities.
The mathematical extension of the wave function to infinity is a consequence of the mathematical formalism used in quantum mechanics and the specific solutions to the Schrödinger equation or other quantum equations for the system. These solutions allow for the consideration of a wide range of possible locations for the particle, including regions with very low probabilities.
It's worth noting that the wave function is a mathematical tool that provides useful predictions about the behavior of particles in quantum systems. While it represents the probabilities associated with the particle's location, it does not represent a physical object or imply that the particle itself physically occupies all the space described by the wave function. The wave function is primarily a mathematical representation that helps us understand and predict the behavior of quantum systems.