Yes, there is a mathematical framework that describes the behavior of particles as guided by quantum waves. It is known as quantum mechanics or quantum theory, which is a fundamental theory in physics that describes the behavior of particles at the microscopic scale.
In quantum mechanics, particles are described by wave functions, which are mathematical functions that encode the probabilistic behavior of the particles. The wave function evolves over time according to a mathematical equation called the Schrödinger equation. This equation describes how the wave function changes in response to the particle's environment, including the presence of other particles or external forces.
The wave function itself contains information about the particle's position, momentum, and other physical properties. However, it is important to note that the wave function does not represent a physical wave in the usual sense. Instead, it is a mathematical representation of the particle's quantum state and contains the probabilities of different outcomes when making measurements of the particle's properties.
According to the principles of quantum mechanics, particles do not have definite positions or properties until they are measured. Instead, they exist in a superposition of states, where they can be in multiple states simultaneously. The wave function guides the probabilities of different measurement outcomes, and when a measurement is made, the wave function "collapses" to one of the possible outcomes.
This probabilistic nature of quantum mechanics is often counterintuitive compared to our everyday experiences with classical physics. However, it has been experimentally verified and successfully applied to various areas of physics, including atomic and molecular physics, solid-state physics, and quantum field theory.