In Heisenberg Picture, for a free particle, $[x_i(t),x_i(0)]=\frac{-i\hbar t}{m}$.
This relation implies that even if the particle is well localized at t=0, its position becomes more and more uncertain with time. (Sakurai)
Applying the same approach to Schrodinger picture,
Now, the position operators do not depend upon time ($x_i(t)=x_i(0)$, for any time t) and $[x_i(t),x_i(0)]=0$. This means if the particle is well localized at t=0, its position remains localized with time.
But this is the violation of Heisenberg's Principle as we're providing some amount of certainty to the location of the particle. Even if we measure the location at t=0, leaving the particle free for some time changes its location because the state kets change with time in Schrodinger's Picture and hence the location changes.
Where am I wrong?