Schrödinger and Heisenberg are known to be some of the founders of quantum revolution in Physics; quantum physics. What it is not so apparent is that they came from competing views on the nature of quantum physics. The second was a strong supporter of the “Copenhagen Interpretation” of quantum physics and Schrödinger hated it.
Copenhagen interpretation following Hume’s epistemology believes that there is no reason to argue if the physical “reality” exists beyond our experience, and goes one step further stating that physical reality needs consciousness in order to exist as what we experience.
Schrödinger did not like that view at all. For that reason he tried to base the mathematics of quantum mechanics on well known idea of the physical word the wave. Once he learned about particle-wave dualism (matter is both a wave and a particle but we can experience only one nature; either the wave either the particle) he shared Debye’s view that “You cannot have waves without a wave equation” and worked hard on producing his famous equation.
Heisenberg on the other hand, as a strong supporter of the Copenhagen Interpretation, he tried to discover the mathematics behind quantum physics by working on the other fundamental idea of the new physics; the arbitrary “quantum jump” (in quantum physics the particles cannot be wherever they like in space, they have to have a certain energy and occupy a specific “trajectory” that corresponds to that energy. We cannot determine the exact time that a particle will “jump” from a higher energy level to a lower one, this is the uncertainty in “quantum jumps”). The mathematics that he used were not based in a physical idea it was just algebra (vector algebra in Hilbert Space).