The Schrodinger partial differential equation of quantum mechanics is the heart of atomic physics. This elegant PDE governs the behavior of all particles under the fundamental forces, but, unlike other PDEs, it cannot be derived from simpler principles. Like time, space, matter and energy, it “just is”. To quote from one of my PDE books, “Schrodinger’s equation is most easily regarded as simply an axiom that leads to the correct physical conclusions, rather than as an equation that can be derived from simpler principles…In principle, elaborations of it explain the structure of all atoms and molecules and so all of chemistry.”

The Schrodinger equation contains a parameter, h, called Planck’s constant, which is one of the many constants of Nature that is very “fine-tuned”: change it a little bit and you get a universe that cannot support any imaginable forms of life. Now I know enough mathematics and physics to be sure that most changes to this equation itself would result in a universe that could not have supported life; the properties of the elements in the periodic table certainly depend sensitively on the properties of this magnificent PDE. There may be some ways to modify it without disasterous results (I doubt it); but there is no doubt that the Schrodinger equation itself is very fine-tuned for life.

So I think to explain our existence without design, we not only have to imagine some cosmic random-number generator which churns out values for Planck’s constant and the other constants, but also a cosmic random-equation generator. Are we to assume that in all these other universes imagined by man to explain our existence, the behavior of particles is still governed by the Schrodinger equation, but the forces, masses and charges, and Planck’s constant have random values? Or perhaps the behavior of particles is governed by random types of PDEs in different universes, but there are still many universes in which Schrodinger’s equation holds, with random values for Planck’s constant? No doubt there were some universes which couldn’t produce life because the governing equation looked just like the Schrodinger equation, but with first derivatives in space where there should be second derivatives, or a second derivative in time where there should be a first derivative, or the complex number i was missing, or the mass was in the numerator, or the probability of finding a system in a given state was proportional to |u| rather than |u|^2??