Because of Heisenberg's assertion regarding the commutation relation between certain variables
it is obvious that the properties of particles such as position and momentum can no longer be represented by functions of time (as in the case presented on the previous page.) Rather they must be represented by mathematical objects in which the order of operation IS important. Matricies and Operators are two mathematical creatures which have this property. Heisenberg chose to pursue the Matrix pathway and started to associate matricies with the properties of matter. When two matricies, A and B, are multiplied tgether, the product AB is in general not the same as BA (though the situation can arise when they are the same). It is said that they do not commute, or that their commutator (as given above) is non-zero. Heisenberg constructed matricies so that they would obey the above rule. Such is matrix mechanics.
The other choice has also been developed - that is by using Operator Algebra. Indeed, Schrödinger's Equation's is now envisioned as being an operator equation. One way of choosing operators is as follows. Let the operator for position be x, defined as being the operation of multiplying by x. (This is no real change over the definition of it as a funciton.) Linear momentum, on the other hand, is to be interpreted as the operation of differentiation with respect to x and multiplication by a constant, specifically
This choice of operator satisfies the Uncertainty Principle as required.
Following these ideas, is the direction taken by Heisenberg, as assisted by Max Born and Pascual Jordan. Looked at on the surface, this approach compared to that of Schrödinger appears to be completely different. When these two approaches were brought up almost simultaneously, considerable argument arose concerning which was correct. David Hilbert was one of several mathematicians who soon helped to settle the argument.