# Free-Particle Schrödinger Green's Function

Classical Mechanics Level pending

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.

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