Schrodinger equation

In plain terms

The Schrodinger equation is the law of motion for quantum states, so given the state of a system now it says what the state will be a moment later. The textbook justification for it is that it works, which is fair enough as physics but circular as explanation, and it leaves you asking why this equation and not another, and what it is about quantum states that makes this the only way they could move.

CSD's answer is that the equation is a theorem rather than a choice, and two ideas are needed to see how. A quantum state is the complete description a physicist uses to predict what measurements will show, and between any two states there's a number, the transition probability, often called the overlap, which is the chance that a system prepared in the first state passes a test for the second. Now suppose only two things about the underlying motion, that it preserves every overlap between every pair of states, and that it doesn't jump, meaning the state changes continuously in time.

The first assumption is enough to force the motion, at every instant, to be a symmetry of a very restricted kind, a unitary transformation, a rotation-like map of the state space that keeps all overlaps intact. The second assumption forces the family of these transformations to have a generator, a single fixed object that determines the whole motion, much as a fixed velocity determines a whole journey. Assemble the pieces and the familiar equation appears, with the Hamiltonian, the energy operator, in the role of generator. Nothing about waves went in and nothing about energy went in, so the entire input is that the motion preserves the overlaps and doesn't jump.

In CSD

The Schrodinger equation is the second of the programme's two pillars, the Born rule being the first, and the corpus proves both on a single sector, one arena of states, so they're one story about one object. A reconstruction that derived probability on one arena and dynamics on another wouldn't have shown that the two belong to the same theory, and here they demonstrably do.

The route chains two classical necessities. Wigner rigidity says any map preserving all transition probabilities acts as a unitary symmetry, and Stone's theorem, in its finite-dimensional form, says any continuous family of unitary symmetries indexed by time has a generator. The caveat is stated where it belongs, as the sector itself is posited and not derived. What's proved is that any flow on such a sector must take the Schrodinger form, and whether the world contains such a sector is exactly the kind of question a formal corpus can't 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, which is 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. He wrote the equation 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, in the same year that he left Germany in disgust at the regime, though he wasn't 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, wasn't an illustration. It was an attack, a reductio against the standard reading of his own equation, and history's small joke is that it's now the standard classroom illustration of the very thing it was built to refute.

Related entries

• Schrodinger from recordsCSD doesn't postulate the Schrodinger equation, as preserving the statistics of records forces the dynamics to be unitary, unitarity together with continuity forces a Hamiltonian, and the exponential form of the equation is a theorem at the end of that chain.
• Stone's theoremAsk only that time evolution never jumps, and differentiability, a generator and the Hamiltonian all follow for free.
• Hermitian operatorA single symmetry condition on an operator guarantees that every number it can produce in a measurement is real, and it delivers the whole machinery of outcomes and eigenstates at no extra cost.
• Lindblad equationThe master equation of open quantum systems is formalised end to end, as the evolution generated by every Lindblad generator is proved to be completely positive, with the bystander-system argument carried out rather than gestured at.
• Poisson bracketThe Poisson bracket is the one rule that tells classical mechanics how fast any quantity changes, and the quantum commutator obeys exactly the same algebra, so that shared grammar is what lets a deterministic flow present itself in quantum form.