Concerning the sparcity of the real-time quantum propagator

1. Jun 27, 2014

rudaguerman

Considering the real-time quantum propagator originated by the green's function of the time-dependent Schrödinger equation:
$i\hbar \frac{d\psi}{dt} =H\psi$
namely,
$\psi(t) =e^{-i\hbar H t}\psi(0)$
it is straightforward to see that using as basis set the eigenvector of
$H\phi_k =\varepsilon_k\phi_k$
The matrix representation of the propagator is diagonal (maximaly sparse)
$e^{-i\hbar H t}=\sum_{k}e^{-i\hbar\varepsilon_k}|\phi_k><\phi_k|.$
My question is: To which extent one may claim that the matrix representation of the propagator is sparse, if a basis set different from the eigenbasis (e.g. a overcomplete family of Hermite functions) is employed to represent the propagator considering that the potential in the Hamiltonian is an analytic function?

2. Jun 27, 2014

vanhees71

I don't understand what you mean with your question. You can write the propgator in any representation you like. E.g., in the Schrödinger picture you can write the propgator in position representation as
$$U(t,x,x')=\langle x|\exp(\mathrm{i} \hat{H} t/\hbar)|x' \rangle.$$
Then the solution of the time-dependent Schrödinger equation, given the initial condition for the wave function reads
$$\psi(t,x)=\int_{\mathbb{R}} \mathrm{d} x' U(t,x,x') \psi(0,x').$$
It's clear that $U(t,x,x')$ fulfills the Schrödinger equation with the initial condition
$$U(0,x,x')=\langle x|x' \rangle=\delta(x-x').$$

3. Jun 27, 2014