Evolution Equation

The simplest non-trivial ode is the first-order linear ode with constant coefficients: \begin{equation} \frac{d}{dx} f(x)= a f(x) \end{equation} with solution: \begin{equation} f(x)=f(0)\, e^{ax} \end{equation}

We can generalize this equation to apply to solutions which are matrix exponentials, i.e.: \begin{equation} M(x)=M(0)e^{Ax} \end{equation} is a solution of: \begin{equation} \frac{d}{dx}\, M(x) = A\, M(x) \end{equation} where $A$ is a suitable constant matrix. (Show that if $A$ is anti-Hermitian, then $M(x)$ is unitary.)

Example Problem: Find the matrix differential equation that has the solution: \begin{equation} \vert \psi(x, t)\rangle = \vert \psi(x,0)\rangle\, e^{i\frac{Ht}{\hbar}} \end{equation} where $H$ is Hermitian. Do you recognize your differential equation?

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