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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

Robertson-Schrödinger uncertainty relation

Consider a quantum system in a state $$| \psi \rangle$$, and let $$A$$ and $$B$$ be Hermitian operators representing a pair of observables. One can choose to perform a measurement of $$A$$ or a measurement of $$B$$ on the system. Generally, the outcomes of these measurements cannot be predicted with certainty. The outcomes are statistical in nature and characterized by the respective expectation values $$\langle A \rangle$$ and $$\langle B \rangle$$. Hereinafter, $\langle \cdot \rangle = \langle \psi | \cdot | \psi \rangle \,.$ The corresponding uncertainties are defined as $\sigma_A = \sqrt{ \langle \big( A - \langle A \rangle \big)^2 \rangle } = \sqrt{\langle A^2 \rangle - \langle A \rangle^2} \,,$ $\sigma_B = \sqrt{ \langle \big( B - \langle B \rangle \big)^2 \rangle } = \sqrt{\langle B^2 \rangle - \langle B \rangle^2} \,.$ The Robertson-Schrödinger uncertainty relation sates that \[\sigma_A^2 \sigma_B^2 \ge \langle \tfrac{i}{2} [A,B] \rangle^2 + \Big( \langle