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

Sadie: 3 days ago

Ok, so we know that the solution to $\hat{H}\psi = i\hbar \frac{\partial \psi}{\partial t}$ 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?

Luca: 3 days ago

$\psi(0)$ 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 $$\psi(x, t)=\psi(x, 0)e^{-iHt/\hbar}$$