Section 5.12 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)\tag{5.12.1}
\end{equation}
with solution:
\begin{equation}
f(x)=f(0)\, e^{ax}\tag{5.12.2}
\end{equation}
We can generalize this equation to apply to solutions which are exponentials of matrices (Section 5.8), i.e.:
\begin{equation}
M(x)=M(0)e^{Ax}\tag{5.12.3}
\end{equation}
is a solution of:
\begin{equation}
\frac{d}{dx}\, M(x) = A\, M(x)\tag{5.12.4}
\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}}\tag{5.12.5}
\end{equation}
where \(H\) is Hermitian. Do you recognize your differential equation?