Free-Particle Schrödinger Green's Function

Which of the following is the Green's function $G(x,y)$ for the time-dependent free-particle Schrödinger equation in one dimension?

The time-dependent free-particle Schrödinger equation in one dimension is

$-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} = i\hbar \frac{\partial \psi}{\partial t}.$

Note: Recall that a solution to the time-dependent Schrödinger equation can be written out in a basis of solutions to the time-independent Schrödinger equation

$\psi(x,t) = \sum_n c_n \phi_n (x) e^{-iE_n t /\hbar},$

where $E_n$ is the energy of the time-independent eigenfunction $\phi_n (x)$.

Notations:

• $\exp(x)$ denotes the exponential function, $\exp(x) = e^x$.

• $\text{abs}(x)$ denotes the absolute value function.