## Section6.11Evolution Equation

The simplest non-trivial ode is the first-order linear ode with constant coefficients:

$$\frac{d}{dx} f(x)= a f(x)\tag{6.11.1}$$

with solution:

$$f(x)=f(0)\, e^{ax}\tag{6.11.2}$$

We can generalize this equation to apply to solutions which are exponentials (Section 6.10), i.e.:

$$M(x)=M(0)e^{Ax}\tag{6.11.3}$$

is a solution of:

$$\frac{d}{dx}\, M(x) = A\, M(x)\tag{6.11.4}$$

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:

$$\vert \psi(x, t)\rangle = \vert \psi(x,0)\rangle\, e^{i\frac{Ht}{\hbar}}\tag{6.11.5}$$

where $$H$$ is Hermitian. Do you recognize your differential equation?