Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$
This latter definition would put the momentum and Hamiltonian operators in a nice parallel in position space, paring each conserved quantity with its Noether symmetry as a derivative.
$$\hat H = i\hbar \frac{\partial }{\partial t} \longleftrightarrow \hat p = -i\hbar \frac{\partial}{\partial x}$$
It also would put the Schrödinger equation in the following form, which parallels the classical equation.
$$\hat H \Psi = \frac{\hat p^2}{2m} \Psi + V \Psi \longleftrightarrow H = \frac{p^2}{2m} + V$$
Is there any reason why we have instead chosen to write $\hat H = \hat p^2 / 2m + V$ and the Schrödinger equation as $\hat H \Psi = i\hbar\ \partial \Psi / \partial t$? The alternative view seems a bit more enlightening to me, imho.
