Final answer:
In quantum mechanics, commutative operators mean that the physical quantities they represent can be determined at the same time, as their measurements do not affect each other's precision and they share common eigenstates.
Step-by-step explanation:
In quantum mechanics, when we discuss two operators being commutative, it means that the sequence in which the operators are applied to a function (or state) does not affect the outcome. This is similar to the commutative property in mathematics, where for instance, the addition of scalar numbers is commutative because, for any two numbers a and b, a + b = b + a. In quantum mechanics, if two observables are represented by commutative operators, measuring one observable does not affect the uncertainty in the measurement of the other observable. They share a common set of eigenstates, and as a result, the observables can be measured simultaneously with definite precision.
The correct answer to the student's question is: a) they can be determined at the same time, which means that the measurements of these two physical quantities, such as position and momentum, can be precisely known at the same time if their corresponding operators commute.
This principle is crucial in understanding the wave-particle duality of particles like electrons, as highlighted by the uncertainty principle, which states that certain pairs of physical properties (like position and momentum) cannot both have precisely known values at the same time, except if the operators commute.