Stationary states vs. the unitary time evolution operator

[LarryS]

In QM, states evolve in time by action of the Time Evolution Unitary Operator, U(t,t₀). Without the action of this operator, states do not move forward in time. Yet even stationary states, like an eigenstate of energy, still contain a time variable – they oscillate in time at a fixed frequency.

To me, the above implies that there are TWO types of “time”, one for the fixed oscillations of stationary states and the other as arguments for that unitary operator.

Comments?

As always, thanks in advance.

 

[PeterDonis]

In QM, states evolve in time by action of the Time Evolution Unitary Operator, U(t,t0). Without the action of this operator, states do not move forward in time.

"Without the action of this operator" is meaningless; there's no way to somehow stop the operator from working so we can see what happens without it. The operator is a description of how states evolve in time; it's not a thing that could either be applied to the states or not. (This, btw, is a fundamental difference between the time evolution operator and operators describing things that are done to quantum systems in experiments; we can choose to apply or not apply the latter, but we can't choose to apply or not apply the former.)

even stationary states, like an eigenstate of energy, still contain a time variable – they oscillate in time at a fixed frequency.

More precisely, their phases change with time at a fixed frequency. But multiplying a quantum state by a phase doesn't change any observables, so all of the observables of a stationary state do not change with time.

To me, the above implies that there are TWO types of “time”, one for the fixed oscillations of stationary states and the other as arguments for that unitary operator.

No, they're the same thing. The argument of the unitary operator is the same variable that governs the phases of stationary states. Why? Because the change of phase of a stationary state is the action of the unitary time evolution operator. It's not some separate phenomenon.

 

[LarryS]

No, they're the same thing. The argument of the unitary operator is the same variable that governs the phases of stationary states. Why? Because the change of phase of a stationary state is the action of the unitary time evolution operator. It's not some separate phenomenon.

If the changes of phase of a stationary state is the action of the unitary time evolution operator, then why have a unitary time evolution operator at all? The presence of the "t" variable in e^iωt would be sufficient.

 

[PeterDonis]

The presence of the "t" variable in eiωt would be sufficient.

$e^{i \omega t}$ is the unitary time evolution operator. You've just limited it to the case of a single stationary state, i.e., an eigenstate of the Hamiltonian with eigenvalue $\omega$ (in "natural" units where $\hbar = 1$). The general unitary time evolution operator is $e^{i \hat{H} t}$, which can be applied to any state; applying it to an eigenstate of $\hat{H}$ gives what you wrote.