How are the time dependent and time independent solutions to the Schrodinger equation related?

Question

Ok, so we know that the solution to $hat{H}psi = ihbar frac{partial psi}{partial t}$ is $psi (t) = e^{frac{-i hat{H} t}{hbar}} psi (0)$, but what exactly is $psi(0)$ here? Is it the solution to $hat{H}psi = E psi$?
Edit: What I meant here is: is the solution to the first equation related to the one of the second?

JalfredP · Accepted Answer

$psi(0)$ is what it seems, i. e. the value of $psi$ at time $0$.
Too see why, remember always that to solve a differential equation, such as Schrödinger's, you always need a "reference value". Let's go back to the basics with a simple differential equation that is qualitatively the same as Schrödinger's:
$${dx over dt} = x , .$$
To solve it, we re-write it as
$${dx over x} = dt$$
and integrate it from a given time $t_0$ to a final time $t$, meaning:
$$int_{x(t_0)}^{x(t)} {dx over x} = int_{t_0}^t dt $$
which is, very easily
$$ln(x(t))-ln(x(t_0))=t-t_0$$
and can be rewritten as
$$ln(x(t))=ln(x(t_0))+(t-t_0) , .$$
Exponentiating both sides produces
$$x(t) = e^{ln(x(t_0))+(t-t_0)}=e^{ln(x(t_0))}e^{(t-t_0)}=x(t_0)e^{t-t_0}$$
and of course, if we set $t_0=0$, then
$$x(t)=x(0)e^{t-t_0},.$$
So you see that to get the value at time $t$ we need to know the value at time $t_0$. This is because differential equation usually have infinite solutions, separated by a constant which goes away when you derive, and to specify which one we want we need to know the value of the function we are looking for in at least one point (similar concepts hold for differential equations of higher order or in higher dimensions).
So to solve Schrödinger's equation you need to know the value of $psi$ in at least one time point. Usually one chooses $0$ as a reference point, or maybe because your question is usually of the kind "I know the value of $psi$ now, which I call time $t=0$. What will be the value of $psi$ in a future time?".
That question is answered by your solution.
In the particular case of S.E., as you mentioned, yes, $psi$ is the solution of the time-independent S.E.
Once you have found your state $psi(x)$ solving the eigenvalue equation, you can propagate it in time using Schrödinger's equation, so that
$$psi(x, t)=psi(x, 0)e^{-iHt/hbar}$$

Young Kindaichi · Answer

Little Base

The Schrodinger Equation
$$ihbarfrac{d}{dt}|Psi(t)rangle =H|Psi(t)rangle$$
In Principle, This together with the initial condition is needed to solve this first-order differential equation. What's the unknown variable Here? It's $|Psi(t)rangle$, Thus we must know the state of the system at some instant in time.
The solution to this equation or rather a formal solution
$$|Psi(t)rangle =e^{-iHt/hbar}|Psi(0)rangle    mathrm{formally  !!!}$$
We define the evolution operator as
$$U(t,0)equiv e^{-iHt/hbar}$$
Now If $H$ happen to explicitly time-dependent then exponential is not the solution but even though
$$|Psi(t)rangle =U(t,0)|Psi(0)rangle$$
remain valid. The solution in this case a much-complicated formula called time-ordered evolution.

Is the solution to the time-dependent equation related to the solution of the time-independent one?

So We know what's the importance of $U(t)$ is. Once it's known the problem is solved. What one do is to construct an explicit expression for $U(t)$ in terms of $|Erangle$, the Eigenket of $H$ with eigenvalues $E$ which obey
$$H|Erangle =E|Erangle$$
A little algebra lead you to
$$U(t)=sum_{E} |Eranglelangle E|e^{-iEt/hbar}$$
You can change the basis and construct the propagator on that basis. One use position because that simple to work with. The main thing is what needs for the time-independent Schrodinger equation.
That explains the role of so-called stationary states.

How to know the initial state?

Now How do you know about initial state? Intial state can be decided from the measurement. For example take the simplest example Free particle. Suppose you have find the propagator $U(x,t;x',0)$,which is just matrix element of $U(t,0)$ in position basis. Now you want to test it. To know the initial state $psi(x',0)$  one take measurement for position of particle and localize it at $x'=x_0'$, that is, with
$$psi(x',0)=delta (x'-x'_0)$$
Then $$psi(x,t)=int U'(x,t;x',0)psi(x',0)dx'=U(x,t;x'_0,0)$$
The measurement collapse the state to some eigen state and that's what the initial state is and from there you can use propagator to see the evolution.

How are the time dependent and time independent solutions to the Schrodinger equation related?

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