Bad Memory

Throughout my entire schooling, I've never been very good at remembering the minutia.  This, I believe, is why I was never any good at history despite how much I enjoy it.  I also think that this gives rise to my success in mathematics and physics: I never really remember anything, but rather how to derive it.  In working over the derivations I piece together a larger world in which I can swim about and see the landscape that disparate topics create when viewed from afar.  With this in mind, I recently came across a very nice, straight-forward derivation of Schrödinger's equation that I'd like to share, plus my LaTeX could use some practice...

For Uniform Dynamics, time evolution is described by a unitary operator:
\[
|\psi(t)\rangle = U(t,t_0)|\psi(t_0)\rangle
\]And we can consider the derivative with respect to time of a state $|\psi\rangle$ at time $t$ as an operator $G$ acting on $|\psi(t)\rangle$:
\[
\begin{align}
\frac{d}{dt}|\psi(t)\rangle & = G|\psi(t)\rangle \\
\ & = \frac{d}{dt}U(t,t_0)|\psi(t_0)\rangle \\
\ & = \frac{d}{dt}U(t,t_0)\lbrack U(t,t_0)^\dagger U(t,t_0) \rbrack |\psi(t_0)\rangle \\
\end{align}
\]So, because $U(t,t_0)|\psi(t_0)\rangle=|\psi(t)\rangle$ we can say that $G = \frac{d}{dt}U(t,t_0)U(t,t_0)^\dagger$, and, keeping normalization constant,
\[\begin{align}
\frac{d}{dt}\langle\psi|\psi\rangle & = 0 \\
\ & = \lbrack\frac{d}{dt}\langle\psi(t)|\rbrack|\psi(t)\rangle + \langle\psi(t)|\lbrack\frac{d}{dt}|\psi(t)\rangle\rbrack \\
\ & = \langle\psi(t)|G^\dagger|\psi(t)\rangle + \langle\psi(t)|G|\psi(t)\rangle \\
\ & = \langle\psi(t)|G^\dagger + G|\psi(t)\rangle
\end{align}\]Therefore $G^\dagger = -G$, meaning that $G$ is Anti-Hermitian and we may write it as $i$ times a Hermitian operator, say $H$:
\[H = i\hbar G\]Here the $\hbar$ is just a constant which could be absorbed into $H$ but we need it for convention and for keeping the units squared away. Using this knowledge paired with our first consideration of the time rate of change of a given state we have ourselves what we're looking for:
\[\begin{align}
\frac{d}{dt}|\psi(t)\rangle & = G|\psi(t)\rangle \\
\ i\hbar\frac{d}{dt}|\psi(t)\rangle & = H|\psi(t)\rangle
\end{align}\]We can call $H$ the Hamiltonian, and it works out that it has units of energy, as well as the basis states being the eigenstates with energy eigenvalues. Perfect! It is working through guided derivations such as this one that really helps me keep the pieces organized, rather than just memorizing an equation and how to use it.