Schrodinger equation

In plain terms

Here is the problem. The Schrodinger equation is the law of motion for quantum states, and the textbook justification for it is: it works. Fair enough as physics. But as an explanation it is circular -- why this equation and not another? What about quantum states makes this the only way they could move?

CSD's answer is that the equation is a theorem, not a choice. Suppose only two things about the underlying motion: it preserves the measurable relations between states, and it does not jump. Rigidity then forces it to act as a unitary symmetry at every moment. Continuity forces those symmetries to have a generator. Assemble the pieces and the familiar equation appears, with the Hamiltonian in the role of generator.

Nothing about waves went in. Nothing about energy went in. Preserve the overlaps, do not jump -- that is the entire input.

In CSD

This is the second of the programme's two pillars, the Born rule being the first, and the corpus proves both on a single sector so that they are one story about one object. A reconstruction that derived probability on one arena and dynamics on another would not have shown they belong to the same theory. Here they demonstrably do.

The route is Wigner rigidity followed by finite-dimensional Stone -- the two unconditional necessities, chained. And the honest caveat is stated where it belongs: the sector itself is posited, not derived. What is proved is that any flow on such a sector must take the Schrodinger form. Whether the world contains such a sector is exactly the kind of question a formal corpus cannot settle.

Mathematically

For a strongly continuous projective flow preserving transition probabilities, the induced evolution is a one-parameter unitary group, hence generated by a Hamiltonian -- the usual equation of motion, up to phase.

The phase is where a residue lives. Lifting a projective flow to a genuine unitary group requires killing a cocycle, and Bargmann's theorem, which does that from continuity alone, is beyond the proof assistant's current reach. So the lift carries a coboundary datum as an explicit hypothesis, recorded rather than hidden.

The name

Erwin Schrodinger (1887-1961) was born in Vienna, held positions at Jena, Stuttgart, Breslau and Zurich, and succeeded Planck in Berlin in 1927. The equation was written over Christmas 1925 and the weeks after, at the Villa Herwig in Arosa in the Swiss Alps. He shared the 1933 Nobel Prize with Dirac -- the same year he left Germany in disgust at the regime, though he was not himself Jewish.

After an ill-judged return to Austria and a hurried escape from Graz in 1938, he settled at the Dublin Institute for Advanced Studies, created for him by de Valera, and stayed seventeen years.

The famous cat, from 1935, was not an illustration. It was an attack -- a reductio against the standard reading of his own equation. History's little joke is that it is now the standard classroom illustration of the very thing it was built to refute.