 # Why can't iℏ∂/∂t be considered the Hamiltonian operator? klepkowy7c 2022-07-19 Answered
Why can't $i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}$ be considered the Hamiltonian operator?
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If one a priori wrongly declares that the Hamiltonian operator $\stackrel{^}{H}$ is the time derivative $i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}$, then the Schrödinger equation

would become a tautology. Such trivial Schrödinger equation could not be used to determine the future (nor past) time evolution of the wavefunction $\mathrm{\Psi }\left(\mathbf{r},t\right)$
On the contrary, the Hamiltonian operator $\stackrel{^}{H}$ is typically a function of the operators $\stackrel{^}{\mathbf{r}}$ and $\stackrel{^}{\mathbf{p}}$, and the Schrödinger equation

One may then ask why is it then okay to assign the momentum operator as a gradient

(This is known as the Schrödinger representation.) The answer is because of the canonical commutation relations

On the other hand, the corresponding commutation relation for time t is

because time t is a parameter not an operator in quantum mechanics. Note that in contrast

which also shows that one should not identify $\stackrel{^}{H}$ and $i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}$