Proving Ehrenfest's Theorem: Diff. vs. Partial Diff.

[calculus_jy]

In proving the Ehrenfest Theorem
This is the typical first line:

\frac{d }{dt}&lt;O&gt; = \frac{\partial}{\partial t} &lt;\psi|O|\psi&gt; = &lt;\dot{\psi}|O|\psi&gt; + &lt;\psi|O|\dot{\psi}&gt;+&lt;\psi|\dot{O}|\psi&gt; <br />

My question is how can the exact differential
\frac{d }{dt}&lt;O&gt;
be changed the partial differential
\frac{\partial}{\partial t} &lt;\psi|O|\psi&gt;
in the first equality. would it not be
\frac{d }{dt}&lt;O&gt;=\frac{\partial}{\partial x} &lt;\psi|O|\psi&gt; \frac{dx}{dt}+\frac{\partial}{\partial t} &lt;\psi|O|\psi&gt;

Have we assumed that \frac{dx}{dt}=0
If so why?

 

[facenian]

I think that if we are working in abstrac state space the only variable is time since |\psi(t)&gt; is only function of time as well as the operator