Schrödinger equation and propagator

Question

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?)

Answer 1

There are some basic misconceptions in your post: first the assumption that states can be time-dependent and time-independent within the same Schrodinger picture. The state merely represents the properties of the system at a given time, and whether it changes or not depends on the Hamiltonian ($H$) that it is subject to.

As for your questions, yes, you do need an initial state. It is just like solving your classical dynamics, you need to know the initial position and momentum of your particle to predict the future trajectory.

I'm not sure I get your question 2, where you want a representation of $U$ -- but earlier in the post you have (correctly, if $H$ does not depend explicitly on $t$) written it as $exp(iHt/\hbar)$. The Feynman propagator is a different construction altogether. The propagator connects wavefunctions at particular events in spacetime as such:

$$\psi(y,t')=\int_{x} \psi(x,t) K(y,t';x,t) dx$$

There is a way to link it to the Schrodinger picture, but for that you should look up a textbook.