vela said:Going back to AB-BA=C, what is C' equal to? How does it compare to what you derived so far?
A Hermitian operator is a mathematical operator that has a special property known as self-adjointness. This means that the operator is equal to its own adjoint, or complex conjugate transpose. In other words, for a Hermitian operator A, A* = A. Hermitian operators are commonly used in quantum mechanics to represent physical observables such as energy, momentum, and angular momentum.
An operator is Hermitian if it satisfies the condition A* = A, where A* is the adjoint of A. To determine if this condition is met, one can use the Hermiticity test, which involves taking the complex conjugate transpose of the operator and comparing it to the original operator. If they are equal, then the operator is Hermitian.
Hermitian operators are important in quantum mechanics because they represent physical observables, such as energy and momentum, which are measured in experiments. The eigenvalues of Hermitian operators correspond to the possible outcomes of these measurements, and the corresponding eigenvectors represent the states of the system. Additionally, Hermitian operators play a crucial role in the mathematical formulation of quantum mechanics.
No, not all physical observables are represented by Hermitian operators. Some observables, such as position and time, are represented by non-Hermitian operators. However, in order for a physical observable to be meaningful and have real-valued measurements, its corresponding operator must be Hermitian.
Yes, two Hermitian operators can commute, meaning that their corresponding operators can be multiplied in either order without changing the result. This is because Hermitian operators share a common set of eigenvectors, which allows them to commute. However, not all Hermitian operators commute with each other.