Why can't $iℏ ∂/∂t$ be used when calculating commutators of $H$?

Question

$[x, \hat{H}]$ or $[\hat{p}, \hat{H}]$ can be computed by substituting $\frac{\hat{p}^2}{2m} + V(x)$ for $\hat{H}$ and doing some simple calculations.

e.g

$$[x, \hat{H}] = [x, \frac{\hat{p}^2}{2m} + V(x)] = \frac{1}{2m}[x, \hat{p}^2] = \frac{1}{2m}(\hat{p}[x, \hat{p}] + [x, \hat{p}]\hat{p}) = \frac{i\hbar\hat{p}}{m}$$

$$[\hat{p}, \hat{H}] = [\hat{p}, \frac{\hat{p}^2}{2m} + V(x)] = [\hat{p}, V(x)] = -i\hbar\frac{\partial V}{\partial x}.$$

However, when I use Schrodinger's equation and substitute $\hat{H} = i\hbar\frac{\partial}{\partial t}$, I get completely different results like

$$[x, \hat{H}]f = i\hbar(x\frac{\partial f}{\partial t} - \frac{\partial}{\partial t}(xf)) = -i\hbar\frac{\partial x}{\partial t}f, [x, \hat{H}] = -i\hbar\frac{\partial x}{\partial t}$$

and $$[\hat{p}, \hat{H}] = \hbar^2\left(\frac{\partial^2}{\partial x \partial t} - \frac{\partial^2}{\partial t \partial x}\right) = 0$$

As far as I know, the former answers are correct and the latter(w/ the Schrodinger equation) are wrong. Why does using the Schrodinger equation give wrong answers here?

Answer 1

$i\hbar\partial_t$ is not the Hamiltonian operator - otherwise the Schrödinger equation would be a trivial statement, like $\hat{x}f(x)=xf(x)$. Instead the equation states that the rate of time change of the wave function equals to the result of action of the Hamiltonian operator. In other words, like all meaningful equation it states a relation between two independently defined quantities.

Answer 2

The Hamiltonian operator is not the time derivative $i\hbar\partial/\partial t$ (see this excellent answer by qmechanic, that also explains why it is fine for momentum to be proportional to the gradient operator and the Hamiltonian cannot $i\hbar\partial_t$). In fact, time derivative isn't even an operator on Hilbert space (see this answer by ACuriousMind)

The form of the Hamiltonian operator depends on your particular system. For example $$\hat{H}=\frac{\hat{p}^2}{2m}$$ is the free particle Hamiltonian. Schrödinger's equation, on the other hand, encloses the fact the the Hamiltonian governs the time evolution of your system $$\hat{H}|\Psi,t\rangle=i\hbar\frac{\partial|\Psi,t\rangle}{\partial t}.$$ The equation above tells you the the action of the Hamiltonian is equivalent to the action of time derivative. Solving this equation you determine how your system evolves over time. If the Hamiltonian were defined as you proposed, Schrödinger equation would be a tautology.