In the time-dependent Schrodinger equation, $ H\Psi = i\hbar\frac{\partial}{\partial t}\Psi,$ the Hamiltonian operator is given by
$$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V.$$
Why can't we consider $\displaystyle i\hbar\frac{\partial}{\partial t}$ as an operator for the Hamiltonian as well? My answer (which I am not sure about) is the following:
$\displaystyle H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ is not an equation for defining $H$. This situation is similar to $\displaystyle F=ma$. Newton's second law is not an equation for defining $F$; $F$ must be provided independently.
Is my reasoning (and the analogy) correct, or is the answer deeper than that?
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