Chapter 4: Special Matrices

### Evolution Equation

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

We can generalize this equation to apply to solutions which are matrix exponentials, i.e.: $$M(x)=M(0)e^{Ax}$$ is a solution of: $$\frac{d}{dx}\, M(x) = A\, M(x)$$ 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: $$\vert \psi(x, t)\rangle = \vert \psi(x,0)\rangle\, e^{i\frac{Ht}{\hbar}}$$ where $H$ is Hermitian. Do you recognize your differential equation?