In quantum mechanics, Hermitian operators play a significant role because they correspond to observables, which are measurable quantities in physical systems. The mathematical representation of observables in quantum mechanics is through Hermitian operators, and their eigenvalues represent the possible outcomes of measurements.
The spectral theorem states that every Hermitian operator in a complex Hilbert space can be decomposed into a set of orthogonal projectors associated with its eigenvalues. These projectors are called the spectral projectors, and they provide the complete set of possible measurement outcomes for the corresponding observable.
However, it's important to note that not every Hermitian operator corresponds to a physically meaningful observable in a given physical system. The reason is that not all Hermitian operators are directly associated with measurable quantities that can be observed experimentally.
In order for a Hermitian operator to represent a physically meaningful observable, it needs to satisfy additional conditions. One important condition is that the operator should be bounded, meaning that it has a finite range of possible eigenvalues. Physical observables must have a well-defined range of possible values in order to be physically meaningful.
Moreover, the operator should also commute with the Hamiltonian of the system, ensuring that it represents a quantity that is conserved throughout the dynamics of the system. For example, in a system with time-translation symmetry, the energy operator (corresponding to the Hamiltonian) commutes with observables related to conserved quantities such as momentum and angular momentum.
Therefore, while every Hermitian operator in Hilbert space can be associated with an observable mathematically, not all of them have physical significance in a given system. Physically meaningful observables must satisfy additional conditions to represent measurable quantities that are conserved and have a well-defined range of values.