# What's the minimum condition for time evolution operator to be written as $U(t,t_0)=e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'}$?

Is $$\frac{d }{dt} e^{H(t)}=H(t)' e^{H(t)}$$ the minimum condition for time evolution operator to be written as $$U(t,t_0)=e^{-\frac{i}{\hbar} \int_{t_0}^t H(t') dt'}$$?

Further, what's the minimum condition for time evolution operator to be written as $$U(t,t_0)=e^{-\frac{i}{\hbar} E_n(t-t_0)}$$?(Actually, what I think I meant was $$U(t,t_0)=[$$ matrix with diagonal element of $$e^{-\frac{i}{\hbar} E_n(t-t_0)}]$$)? The same as the previous question, or something extra?

• – Qmechanic Feb 14 at 7:14

## 1 Answer

The most general "solution" to the Schrödinger equation

$$i\hbar\frac{d|\psi\rangle}{dt} = \hat{H}(t)|\psi\rangle$$

is given by the time ordered exponential

$$| \psi(t) \rangle = \mathcal{T}\left\{ e^{-\frac{i}{\hbar}\int_{t_0}^{t} dt'\ \hat{H}(t') } \right\} | \psi(t_0)\rangle,$$

where $$\mathcal{T}$$ is the time ordering operator. That is, the general time evolution operator is

$$U(t, t_0) = \mathcal{T}\left\{ e^{-\frac{i}{\hbar}\int_{t_0}^{t} dt'\ \hat{H}(t') } \right\}.$$

Therefore, the minimum condition to be able to "remove" the time ordering symbol and hence obtain

$$U(t, t_0)= e^{-\frac{i}{\hbar}\int_{t_0}^{t} dt'\ \hat{H}(t') }$$

is that the hamiltonian commutes with itself at any two times. That is, the condition is given by

$$U(t, t_0)= e^{-\frac{i}{\hbar}\int_{t_0}^{t} dt'\ \hat{H}(t') } \iff [H(t), H(t')]=0\ \ \ \forall t, t'.$$