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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?