I took a quantum mechanics course many years ago and I'm currently revising it and learning some more advanced subjects of it. Now, I'm trying to understand the use of propagator to find the wave function at some time $t$. Let me elaborate.

Consider the Schrödinger equation: \begin{eqnarray} i\hbar \frac{\partial \psi}{\partial t}(x,t) = \hat{H}\psi(x,t) \tag{1}\label{1} \end{eqnarray} Here, the state can either be time dependent or time independent. Suppose it is time independent. Then, we use separation of variables $\psi(x,t) = \varphi(t)\psi_{0}(x)$ to conclude that (1) $\varphi(t) = \varphi_{0}e^{\frac{i}{\hbar}\lambda t}$ and (2) $\psi_{0}(x)$ is an eigenstate of $\hat{H}$. Once we specify $\hat{H}$ and an initial condition $\psi(x,0)$, we can solve $\hat{H}\psi = \lambda \psi$ and then obtain the general solution $\psi(x,t)$.

If the state is time-dependent, then we can no longer apply the above, but one of the basic axioms of quantum mechanics states that if one knows the state $\psi(x,t_{0})$ at a given time $t_{0}$ then the state at each time $t> t_{0}$ is given by $U(t)\psi(x,t_{0}) = e^{\frac{i}{\hbar}\hat{H}}\psi(x,t_{0})$.

Question 1: I think (but I'm not sure) that the Schrödinger equation (\ref{1}) can be solved by only specifying $\psi(x,t)$ at some $t_{0}$. Is it right? The whole point then is to find such particular solution and use $U(t)$ to 'propagate' and find the solution $\psi(x,t)$ at all times $t>t_{0}$.

Question 2: $U(t)$ can be written as an integral operator, and its Kernel can be obtained (in principle) by Feynmann's path integral. I don't have much experience at this topic, but is it the only way to obtain its kernel or even a representation of $U(t)$ (and consequently of $\psi(x,t)$)? If it is not, what are the alternatives (and what are the advantages of each method?)

I took a quantum mechanics course many years ago and I'm currently revising it and learning some more advanced subjects of it. Now, I'm trying to understand the use of propagator to find the wave function at some time $t$. Let me elaborate.

Consider the Schrödinger equation: \begin{eqnarray} i\hbar \frac{\partial \psi}{\partial t}(x,t) = \hat{H}\psi(x,t) \tag{1}\label{1} \end{eqnarray} Here, the state can either be time dependent or time independent. Suppose it is time independent. Then, we use separation of variables $\psi(x,t) = \varphi(t)\psi_{0}(x)$ to conclude that (1) $\varphi(t) = \varphi_{0}e^{\frac{i}{\hbar}\lambda t}$ and (2) $\psi_{0}(x)$ is an eigenstate of $\hat{H}$. Once we specify $\hat{H}$ and an initial condition $\psi(x,0)$, we can solve $\hat{H}\psi = \lambda \psi$ and then obtain the general solution $\psi(x,t)$.

If the state is time-dependent, then we can no longer apply the above, but one of the basic axioms of quantum mechanics states that if one knows the state $\psi(x,t_{0})$ at a given time $t_{0}$ then the state at each time $t> t_{0}$ is given by $U(t)\psi(x,t_{0}) = e^{\frac{i}{\hbar}\hat{H}}\psi(x,t_{0})$.

Question 1: I think (but I'm not sure) that the Schrödinger equation (\ref{1}) can be solved by only specifying $\psi(x,t)$ at some $t_{0}$. Is it right? The whole point then is to find such particular solution and use $U(t)$ to 'propagate' and find the solution $\psi(x,t)$ at all times $t>t_{0}$.

Question 2: $U(t)$ can be written as an integral operator, and its Kernel can be obtained (in principle) by Feynman's path integral. I don't have much experience at this topic, but is it the only way to obtain its kernel or even a representation of $U(t)$ (and consequently of $\psi(x,t)$)? If it is not, what are the alternatives (and what are the advantages of each method?)