I don't have an answer for why there is no simple Lagrangian formulation, but I can explain some of why a Hamiltonian one is easy. Part of the way to go from Classical Mechanics to Quantum is by replacing Poisson brackets with commutators, and observables with operators on Hilbert space and their expectation values. So the equation

$\frac{d}{dt} f(q, p, t) = \left\{ f,H \right\} + \frac{\partial f}{\partial t} $

becomes the quantum

$\frac{d}{dt} \langle f \rangle = -i \langle\left[f,H \right]\rangle + \langle \frac{\partial f}{\partial t} \rangle.$

So the Hamiltonian is convenient because it gives the time evolution of operators, states, and expectation values directly. Also, because the Hamiltonian is a conserved quantity, stationary states (i.e. those that do not evolve in time) will be eigenvectors of the Hamiltonian, and eigenvalue problems are easy.

This post imported from StackExchange Physics at 2014-04-05 04:38 (UCT), posted by SE-user ZachMcDargh