Green's function for the wave function

[Higgsono]

We want to solve the equation.
$$H\Psi = i\hbar\frac{\partial \Psi}{\partial t} $$ (1)

If we solve the following equation for G

$$(H-i\hbar\frac{\partial }{\partial t})G(t,t_{0}) \Psi(t_{0}) = -i\hbar\delta(t-t_{0})$$ (2)

The final solution for our wave function is,

$$\Psi(t) = G(t,t_{0})\Psi(t_{0})$$ (3)I don't understand the steps. How do we get from (2) to (3) ?

 

[Orodruin]

First: (2) is not the differential equation for the Green's function, you need to remove the $\Psi(t_0)$.

Second: The differential equation on its own does not specify the Green's function, you need to add the condition that $G(t,t_0) = 0$ for $t < t_0$. This will imply that $G(t,t_0) \to 1$ as $t \to t_0^+$. With that, you will find that (3) satisfies the Schrödinger equation with $\Psi(t) \to \Psi(t_0)$ as $t \to t_0^+$.

If you have access to my book, this is discussed in section 7.2.1 for a one-dimensional ODE, but it generalises directly to your case.