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.