Time Evolution of Measurables(Hamiltonian)

[aim1732]

Supposing a physical quantity f whose operator commutes with the Hamiltonian operator H, and supposing it has no explicit time dependence, then the result regarding the time derivative of the operators gives us that the quantity is conserved and its mean value does not change with time.The commutator being zero will also mean that the two quantities, f and energy can be measured simultaneously.Now if the system were subject to a time varying potential then the energy will not be conserved(because the Hamiltonian will have an explicit time dependence) but the quantity f is still conserved(according to the theorem).And the commutator should still be zero so that the energy changes but f does not, and they are still simultaneously measurable.How will the energy evolve in time?Will it become indefinite(considering we start with a definite-energy state) or will it continue to remain a definite energy(and also definite-f) state?Or does this depend on the potential the system is subject to?Is what i said above correct?

 

[Simon Bridge]

The measurement of f does not affect the statistics of the measurements of energy, so what is different - what are you asking? The time evolution of energy measurements would be the same as if you didn't know about f.

 

[aim1732]

My bad.I mixed up questions and was talking something else.I think I have got it now,though.