In quantum theory, physical quantities are represented by mathematical operators. These operators act on quantum states, which describe the state of a physical system. The eigenvalues and eigenvectors of these operators provide the possible values and corresponding states of the physical quantity being measured.
Here are a few key points about how physical quantities are represented in quantum theory:
Observables: Physical quantities, such as position, momentum, energy, and angular momentum, are referred to as observables in quantum mechanics. Each observable is associated with a mathematical operator that represents it.
Hermitian Operators: In quantum mechanics, observables are represented by Hermitian operators. Hermitian operators have the property that their eigenvalues are real, and their eigenvectors are orthogonal. The eigenvalues correspond to the possible values that can be measured for the observable, while the eigenvectors represent the states associated with those values.
Measurement: When a measurement is performed on a quantum system, the corresponding observable's operator is applied to the system's quantum state. The result of the measurement is one of the eigenvalues of the operator, and the system collapses into the corresponding eigenstate.
Superposition: Quantum theory allows for superposition, meaning that a quantum system can exist in a combination of multiple eigenstates simultaneously. This superposition leads to the wave-like behavior and interference effects observed in quantum phenomena.
Uncertainty Principle: Quantum theory introduces the uncertainty principle, which states that certain pairs of observables, such as position and momentum, cannot be simultaneously known with arbitrary precision. This inherent uncertainty is a fundamental aspect of quantum mechanics.
Mathematically, the representation of physical quantities in quantum theory relies on the formalism of linear algebra and the manipulation of wavefunctions or state vectors. These mathematical tools, along with the principles and postulates of quantum mechanics, provide a framework for describing and predicting the behavior of quantum systems.
It's worth noting that the mathematical representation of physical quantities in quantum theory can be abstract and different from classical physics. However, this formalism has been remarkably successful in explaining and predicting the behavior of particles and systems at the microscopic scale.