Before getting into problems relating to the free particle schroedinger equation, let’s review the full Schroedinger equation. The energy operator E_hat appears in the first equation below. Thus far, the euqation relates only kinetic energy. Potential energy however when considered would allow the Schroedinger equation to be applied in a wide range of possible applications, being able to describe the interactions of atoms and molecules and their interactions in free space, wells, and other environments due to the linearity of quantum mechanics. One major point to take from discovering the free particle Schroedinger equation is how important it is in Quantum Mechanics to create energy operators. An operator can be as simple as a constant or as complicated as a partial differential. By allowing an ‘operator’ to take on this wider range of features as opposed to a basic variable makes for the basis of many quantum mechanical calculations. It then follows that the portential, V(x,t) can also be treated as an operator that modifies the system.
Consider an operator X_hat that when multiplied by a function, results in the function being multiplied by x. Remember that although this may look like a variable, it is useful to consider this as an operator in Quantum mechanics.
Does the order in which operators are multiplied matter?
Considering that operators are not always constants or variables, but also sometimes differentials, the order of operations for operators does matter.
A communtator is understood as the difference of linear operators. The commutator of x_hat and p_hat is i*h_bar.