= Solution
The equation is $i\hbar\partial_t\psi=[-\hbar^2\partial_x^2/(2m)+V(x)]\psi$. The generator of <time evolution> is the sum of kinetic and <potential energy> operators. For a sharp stationary <energy>, insert $\psi(x,t)=\phi(x)e^{-iEt/\hbar}$ to obtain $[-\hbar^2\phi''/(2m)+V\phi]=E\phi$. The <probability> in $dx$ is $|\psi|^2dx$, normalized to one; hence a normalized one-dimensional <wavefunction> has dimensions length$^{-1/2}$. Its overall phase is unobservable, but <relative phase> affects interference.
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