What is the difference between a parameter, a variable, and an operator in QM? On the question why time isn't an operator, people will usually say that time is a parameter  in QM (Time as a Hermitian operator in QM?) and not a variable.


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*Can someone please distinguish between a parameter, a variable and an operator as it is used in QM?

*Oh by the way, if someone can resolve why time cannot be an operator but the derivative of time multiplied by some complex number namely $i\hbar d/dt$ is totally cool.
 A: Coordinates and momenta of the system are called variables by convention.
Time is sometimes called variable, after all its nature is to keep changing.
But to distinguish time from coordinates and momenta, people call it parameter in some cases.
Parameter is a word used instead of variable when the quantity is a different kind of argument of a function.
For example, if 
$$
\psi_t(q_1,q_2)
$$
is a function of both coordinates $q_1,q_2$ and time $t$, but we want to stress that we focus on the dependence on $q_1,q_2$ and think of value of time $t$ as fixed constant during discussion, we call it a parameter.
Operator in quantum theory is used in a specific sense "operator that acts on functions of coordinates", or "operator that acts on functions of momenta." In this sense $i\hbar \partial/\partial t$ is not an interesting QM operator because it always results in function equal to 0 everywhere. It is a differential operator of functions of time, but not a QM operator in the usual sense.
The reason for this is partially that it makes no sense to use this differential operator on functions of $q$ in a typical QT role
$$
\int \psi^* \hat{A}\psi dq
$$
to get expected average of quantity modeled by $A$. Sometimes people think that this quantity for $i\hbar \partial/\partial t$ is energy, and then the integral gives expected average energy. But that is correct only if $\psi_t(q)$ obeys time-dependent Schroedinger's equation, while $\hat{H}$ works for any function of $q$.
The difference in treating and naming $t$ stems also from the fact that while coordinates and momenta describe state of the atomic system and are difficult to measure accurately, time $t$ describes state of a clock, which is not microscopic but can be measured simply by taking a look at the clock.
