Free-Particle Schrödinger Green's Function

Which of the following is the Green's function G(x,y)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

22m2ψx2=iψt.-\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

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

where EnE_n is the energy of the time-independent eigenfunction ϕn(x)\phi_n (x).

Notations:

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

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

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