Green's function for the wave function

  • #1
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Main Question or Discussion Point

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) ?
 
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  • #2
Orodruin
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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.
 
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