In quantum mechanics, observables and operators are closely related concepts, but they have distinct meanings.
An observable in quantum mechanics is a physical quantity that can be measured or observed. Examples of observables include position, momentum, energy, angular momentum, and various other properties of a quantum system. Each observable is associated with a specific mathematical operator.
An operator, on the other hand, is a mathematical operation that acts on a quantum state to produce another state or a numerical result. Operators represent the mathematical machinery of quantum mechanics and are used to describe the behavior of observables. Each observable has a corresponding operator, and the eigenvalues and eigenvectors of the operator provide the possible values and states that can be measured for that observable.
When an operator is applied to a quantum state, it returns another quantum state or a numerical result. If an observable corresponds to a Hermitian operator, the eigenvalues of that operator correspond to the possible outcomes of a measurement of the observable. In this sense, the observable can be seen as the value of a physical quantity obtained when the operator is used on a state and the system is measured.
For example, consider the observable of position. The position operator is represented by the operator "x," and when it acts on a quantum state, it returns the position of the particle. The position eigenvalues are the possible values of the particle's position that can be observed in a measurement.
In summary, observables are physical quantities that can be measured, while operators are mathematical operations that describe the behavior of observables in quantum mechanics. The application of an operator to a quantum state yields another state or a numerical result, which corresponds to the observable value when the system is measured.