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Chebyshev polynomial expansion of the time-evolution operator

Here we outline an efficient numerical method of solving the time-dependent Schrödinger equation, \[i \hbar \frac{\partial | \psi(t) \rangle}{\partial t} = H | \psi(t) \rangle \,.\] The method is based on the Chebyshev polynomial expansion of the time-evolution operator. Only the basic idea is presented in these notes. Further details on the method can be found, for instance, in the following papers: An accurate and efficient scheme for propagating the time dependent Schrödinger equation Unified framework for numerical methods to solve the time-dependent Maxwell equations (in particular, section 3.3) The Hamiltonian \(H\) is assumed to have no explicit dependence on time. For instance, if the system under consideration is a particle of mass \(m\) moving in the presence of a static potential \(V(x)\), then \[\psi(x,t) = \langle x | \psi(t) \rangle\] is the particle's wave function, and \[H = -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} + V(x) \,.\] In what