system
Given the following post, respond as if you're commenting on Reddit:
human
Why is there a time-energy uncertainty relation when time is not an operator in Quantum Mechanics?

The Heisenberg Uncertainty Principle is based on the property of the commutators of two Hermitian operators. But time is not an operator in Quantum Mechanics, so what gives? How fundamental is this relation? Where does it come from and what does it mean?
gpt
It's not really the same kind of uncertainty relationship, since you are correct that time is not an operator.

Instead think more as a time evolution on states. If a state is an eigenstate of the Hamiltonian, it is stable in time. If a state is not an eigenstate of the Hamiltonian, it can be written as a linear combination of Hamiltonian eigenstates, with a time dependence determined by the difference in energy levels from the spectrum of the Hamiltonian. It is from this time dependence that the time energy uncertainty comes from.