# Time evolution operator in quantum mechanics

One of the postulates of quantum mechanics is that, given a quantum state $$\psi_{0}$$ at time $$t=0$$, the state of the system at a posterior time $$t > 0$$ is given by $$\psi_{t} = e^{-iHt}\psi_{0}$$, where $$H$$ is the Hamiltonian operator of the system.

The operator-valued function $$\mathbb{R}^{+}\ni t \mapsto U(t) = e^{-iHt}$$ is an example of what mathematicians call a strongly continuous one-parameter unitary group. As a matter of fact, I have seen in many mathematical physics books the above postulate being restated as follows: given an initial quantum state $$\psi_{0}$$, the state at a posterior time $$t > 0$$ is given by $$\psi_{t} = U(t)\psi_{0}$$, where $$U(t)$$ is a strongly continuous one-parameter unitary group. It turns out that Stone's Theorem ensures that for each strongly continuous one-parameter unitary group $$U(t)$$, there exists a self-adjoint operator $$H$$ such that $$U(t) = e^{-iHt}$$, so we are now back to the previous formulation by taking $$H$$ to be the Hamiltonian of the system.

This second approach, however, made me think about the nature of this postulate. Stone's Theorem ensures the existence of some self-adjoint operator $$H$$, which is not necessarily the Hamiltonian of the system. So, my question is pretty basic and straightfoward: is there any reason or interpretation why the time-evoution is defined in terms of the Hamiltonian $$e^{-iHt}$$, and not any other self-adjoint operator/observable? Or is it just another postulate of quantum mechanics that we should accept without any apparent reason?

Note that, by setting $$U(t) = e^{-iHt}$$, the associated Schrödinger equation is then reduced to finding eigenvalues of $$H$$; of course, if another observable, say $$A$$, was used instead of $$H$$, one would find eigenvalues of $$A$$ instead. Hence, I would expect that this postulate has some sort of "we want to find a basis of eigenvectors of $$H$$" explanation, but I am not quite sure if this is the reason behind the postulate, since it is not always possible to find such a basis of eigenstates anyways.

• As a motivation: In classical mechanics the Hamiltonian is the generator of time-evolution (too). Commented Mar 25, 2023 at 19:03
• What do you mean by the Hamiltonian of the system, if not the operator which generates time-evolution? Commented Mar 25, 2023 at 19:10
• The usual sign convention is $U(t)=e^{-iHt}$ not $e^{+iHt}$
– hft
Commented Mar 25, 2023 at 19:16
• @hft fixed the sign convention. Thanks! Commented Mar 25, 2023 at 19:26
• That's wrong. As @TobiasFünke wrote above, that's also what happens classically. The Hamiltonian is a function on phase space that describes the dynamical evolution of the system through the EoM (Hamilton's equations classically, Schrödinger equation in a nR quantum system). In other words, the Hamiltonian generates time evolution. Commented Mar 25, 2023 at 19:41

You should re-read the statement of Stone's theorem : it doesn't ensure the existence of some self-adjoint operator $$H$$ associated to a given strongly continuous one-parameter unitary group $$U(t)$$, but precisely a unique one, and vice versa, hence the unambiguous correspondence between the Hamiltonian and time evolution in the present case.
• The Schroedinger equation is $i \frac{\partial}{\partial t} \left | \psi \right > = H \left | \psi \right >$. Nothing about this is limited, unphysical or different from time evolution. Commented Mar 26, 2023 at 11:45