In quantum theory, physical quantities are represented by mathematical operators. These operators act on a mathematical entity known as a wave function, which describes the state of a quantum system.
In classical physics, physical quantities are represented by numbers. For example, the position of a particle is represented by a specific value of position, such as x = 2 meters. However, in quantum theory, the position of a particle is described by a wave function that assigns a probability distribution to different possible positions of the particle.
Quantum operators correspond to observables in the physical world, such as position, momentum, energy, and spin. Each operator has associated eigenvalues and eigenfunctions. When an operator acts on a wave function, it yields a new wave function that represents the state of the system after the measurement of the corresponding physical quantity.
The square of the wave function, known as the probability density, gives the probability of finding the system in a particular state. This probabilistic nature of quantum theory reflects the inherent uncertainty and wave-particle duality observed at the quantum level.
The mathematical framework of quantum theory, particularly the principles of superposition and the uncertainty principle, allows for the description of phenomena that differ significantly from classical physics and provides a basis for understanding the behavior of subatomic particles.