What's the time derivative of the Annihilation operator? I've been dealing with annihilation operator recently where you can see related information


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*Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction 

*How to get the time derivative of an expectation value in quantum mechanics? 
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*"Creation and annihilation operators", Wikipedia


for definition.
Correction made by Valter Moretti and J.G.'s answer, there is no contradiction.
$$
\begin{align}
i \hbar \frac{\mathrm{d}}{\mathrm{d}t} \left(a \left| \psi(n,t) \right\rangle \right)
& ~=~ i \hbar \frac{\mathrm{d}}{\mathrm{d}t} \left(\sqrt{n} \left| \psi(n-1,t) \right\rangle \right) \\[5px]
& ~=~ H \sqrt{n} \left| \psi(n-1,t) \right\rangle \\[5px]
& ~=~ E_{n-1}\sqrt{n} \left| \psi(n-1,t) \right\rangle
\end{align}
\,.$$
and
\begin{align}
i\hbar\frac{d}{dt}(a|\psi(n,t)\rangle)
& =i\hbar\frac{d}{dt}(a)|\psi(n,t)\rangle+a(i\hbar\frac{d}{dt}|\psi(n,t)\rangle) \\[5px]
& =i\hbar \cdot iwa|\psi(n,t)\rangle+a(H|\psi(n,t)\rangle) \\[5px]
& =-\hbar w a|\psi(n,t)\rangle+a(E_n|\psi(n,t)\rangle) \\[5px]
& =(E_n-\hbar w)a|\psi(n,t)\rangle\\[5px]
& =E_{n-1} a|\psi(n,t)\rangle\\[5px]
& =E_{n-1} \sqrt{n}|\psi(n-1,t)\rangle
\end{align}
Notice $\frac{d}{dt}(a)\neq 0$ despite the fact that $a$ in matrix represtation is a constant matrix.
 A: The problem with your first answer is the identity (I prefer to  modify notations)
$$a \psi_n(t) = \sqrt{n} \psi_{n-1}(t)\:.\quad (false \:\:if \:\:t\neq 0)\tag{1}$$
This is false if $t\neq 0$ (it is trivially true for  $n=0$, but I assume $n\neq 0$ henceforth).
Since $\psi(t) :=  e^{-itH/\hbar} \psi$, the identity above can be expanded as 
$$a e^{-itH/\hbar} \psi_n = \sqrt{n}   e^{-itH/\hbar} \psi_{n-1} \quad(false \:\:if \:\:t\neq 0)\:.$$
In fact, both sides can be separately computed,
$$a e^{-itH/\hbar} \psi_n =  a e^{-itE_n/\hbar} \psi_n= e^{-itE_n/\hbar} a\psi_n= e^{-itE_n/\hbar} \sqrt{n}\psi_{n-1} =  \sqrt{n}e^{-itE_n/\hbar} \psi_{n-1} $$
whereas 
$$\sqrt{n}   e^{-itH/\hbar} \psi_{n-1} = \sqrt{n} e^{-itE_{n-1}/\hbar} \psi_{n-1}\:,$$
and
$$\sqrt{n}e^{-itE_n/\hbar} \psi_{n-1} \neq \sqrt{n} e^{-itE_{n-1}/\hbar} \psi_{n-1}$$
because 
$$E_n\neq E_{n-1}\:.$$
The fundamental reason of the failure of (1) is that  $a$ and $e^{-itH/\hbar}$ do not commute. It it were 
$$a e^{-itH/\hbar} = e^{-itH/\hbar} a\:,$$
then we would have
$$\langle \psi_n |a e^{-itH/\hbar}\psi_m \rangle  = \langle \psi_n | e^{-itH/\hbar} a \psi_m\rangle$$
i.e.
$$\langle a^*\psi_n | e^{-itH/\hbar}\psi_m \rangle  = \langle e^{itH/\hbar}\psi_n |  a \psi_m\rangle\:.$$
taking the time derivative for  $t=0$,
$$\langle a^*\psi_n | H\psi_m \rangle  = \langle H\psi_n |  a \psi_m\rangle\:,$$
so that
$$\langle \psi_n | aH\psi_m \rangle  = \langle \psi_n |  Ha \psi_m\rangle\:,$$
so that 
$$\langle \psi_n | (aH-Ha)\psi_m \rangle  = 0$$
namely, since the Hermite functions $\psi_n$ span a dense space, 
$$aH\psi_m= Ha\psi_m\:.$$
By linearity, we would have
$$aH= Ha\:.$$
at least restricting the domains to  the span of Hermite functions $\psi_n$. The identity above is false in view of the commutation relations of $a$ and $a^*$ and $H = \hbar \omega (a^*a + I/2)$ over the said space.
Coming back to your question, the correct result  is the second one.
A: You have to be careful not to mix up implications specific to different pictures. For example, $\partial_t\langle m|a|n\rangle$ doesn't imply $\dot{a}=0$ in the Heisenberg picture because $|n,\,t\rangle=\exp -i\omega_n t|n,\,0\rangle,\,\omega_n:=\frac{E_n}{\hbar}$. While $\dot{a}=0$ in the Schrödinger picture, it can be shown $\dot{a}=-i\omega a$ in the Heisenberg picture, where the Hamiltonian is $-\frac{\hbar^2}{2m}\partial_x^2+\frac{m\omega^2}{2}x^2$. I'll give two proofs; there are plenty of others.
The first is one I mentioned in a comment: since $\partial_t|n\rangle=-i\omega_n|n\rangle$ and $a=\sum_{n\ge 1}\sqrt{n}|n-1\rangle\langle n|$, $\dot{a}=-i(\omega_n-\omega_{n-1})a=-i\omega a$.
Another approach uses $\dot{x}=\frac{p}{m},\,\dot{p}=-m\omega^2 x,\,a=\frac{m\omega x+ip}{\sqrt{2m\hbar\omega}}$.
