I would say because of the way you efficiently solve problems as well as pedagogy. Both are used in both cases though.

The Hamiltonian operator approach emphasises the spectrum aspects of quantum mechanics, which the student is introduced to at this point $-$ but here is a Lagrangian

$$\mathcal{L}\left(\psi, \mathbf{\nabla}\psi, \dot{\psi}\right) = \mathrm i\hbar\, \frac{1}{2} (\psi^{*}\dot{\psi}-\dot{\psi^{*}}\psi) - \frac{\hbar^2}{2m} \mathbf{\nabla}\psi^{*} \mathbf{\nabla}\psi - V( \mathbf{r},t)\,\psi^{*}\psi$$

for the Schrödinger equation
$$\frac{\partial \mathcal{L}}{\partial \psi^{*}} - \frac{\partial}{\partial t} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial t}} - \sum_{j=1}^3
\frac{\partial}{\partial x_j} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial x_j}} = 0.$$

The Lagrangian (density) is especially relevant for the path integral formulation, and in some way closer to bring out symmetries of a field theory. Noether theorem and so on. $-$ but I remember Peskin & Schröders book on quantum field theory starts out with the Hamiltonian approach and introduces path integral methods only 300 pages in.

This post imported from StackExchange Physics at 2014-04-05 04:39 (UCT), posted by SE-user NiftyKitty95