Significance of the hamiltonian commuting

[sridhar]

Wht is the significance of an operator commuting with the hamiltonian??
Its definitely more than them being measurable simultaneously and precisely!

 

[nrqed]

Wht is the significance of an operator commuting with the hamiltonian??
Its definitely more than them being measurable simultaneously and precisely!

It means that this physical quantity is conserved in time. In other words, if a state is an eigenstate fo this other operator at t=0, it will remain an eigenstate of that operator at later time. This is because the Hamiltonian is the operator that governs the time evolution of the system.

 

[denian]

The significance of an operator commuting with the Hamiltonian lies in the fact that it represents a conserved quantity in a physical system. This means that the operator and the Hamiltonian share the same set of eigenstates, and the corresponding eigenvalues are constant over time. This has important implications in quantum mechanics, as it allows us to make precise predictions about the behavior of a system.

Furthermore, the commutation of an operator with the Hamiltonian indicates that the operator represents a physical observable that is independent of time. This is a fundamental property of systems described by the laws of quantum mechanics, and it allows us to study the evolution of a system by simply studying the behavior of the Hamiltonian.

In addition, the commutation of an operator with the Hamiltonian also implies that the operator is a symmetry of the system. This means that the operator leaves the system unchanged under its action, and this symmetry can provide valuable insights into the underlying physical principles governing the system.

Overall, the significance of an operator commuting with the Hamiltonian goes beyond just the ability to measure them simultaneously. It represents a fundamental property of a physical system and provides us with crucial information about its behavior and underlying symmetries.