Physics Asked on December 26, 2020
In Heisenberg representation we have:
$$frac{d}{dt}A_H(t)=frac{1}{ihbar}left[A_H(t),H_H(t)right]+frac{partial A_H(t)}{partial t} (1)$$
where I use the subscript $H$ to make as explicit as possible that we are working in Heisenberg representation. If we apply $(1)$ to the position operator $hat{q}_H$ we get:
$$frac{d}{dt}hat{q}_H=frac{1}{ihbar}left[hat{q}_H,H_H(t)right]$$
or at least this is what my lecture notes said. This derivation uses the fact that the position operator has no explicit time dependence:
$$frac{partial hat{q}_H}{partial t}=0$$
I do not understand why this has to be true. How do we know that the partial derivative of $hat{q}_H$ has to be zero in Heisenberg representation?
Since in the Schrödinger representation the states can explicitly evolve with time, changing their "relation" with the position operator, I would expect the same in the Heisenberg representation but mirrored onto the operators.
It literally means that this operator doesn't have an explicit time dependence, as opposed, for example to something like $$ V(x, t) = hat{x} cos(omega t). $$
Correct answer by Vadim on December 26, 2020
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