Quantum mechanics and visualization

Tim Mauldin has a double book review in the Boston Review, "The Defeat of Reason". One book is about quantum mechanics and the other is about Thomas Kuhn. I found the quantum mechanics section more interesting. Here's a few paragraphs:

In 1925 Werner Heisenberg had invented matrix mechanics. Heisenberg’s mathematical formalism got the predictions that Bohr had been seeking. But the central mathematical objects used in his theory were matrices, rectangular arrays of numbers. The predictions came out with wonderful accuracy, but that still left the old puzzle in place: how does the electron get from one orbit to another? You can stare at a matrix from morning to night, but you will not get a clue.

Bohr took an unexpected approach to this question: instead of asking if the theory was too young to be fully understood, he declared that the theory was complete; you cannot visualize what the electron is doing because the microworld of the electron is not, in principle, visualizable (anschaulich). It is unvisualizable (unanschaulich). In other words, the fault lay not in the theory, it lay in us. Bohr took to calling any visualizable object classical. Quantum theory had passed beyond the bounds of classical physics: there is no further classical story to tell. This became a central tenet of the Copenhagen interpretation of quantum theory.

Imagine Bohr’s motivation to adopt this extreme conclusion. For over a decade, he had been seeking exact, visualizable electron trajectories and failed. He concluded that his failure was rooted in the impossibility of the task.

But in 1926 Erwin Schrödinger produced a mathematically different theory, wave mechanics. Schrödinger’s mathematics was essentially just the classical mathematics of waves. The atomic system was not designated by a matrix, it was described by a wavefunction. And waves may not be particles, but they are certainly visualizable objects from everyday life...

So the situation in 1926 was rather confused. Matrix mechanics and wave mechanics were, in some sense, thought to be the same theory, differently expressed. But if you use the mathematics to derive a certain matrix yet have no notion of how the physical situation associated with the matrix would appear, how do you get a prediction about what you will observe? And wave mechanics is not much better off. Waves are certainly visualizable, but the world we live in, the world of laboratory experiments, does not present itself as made of waves. It presents itself, if anything, as made of particles. How do we get from waves to recognizable everyday stuff?

This, in a nutshell, is the central conundrum of quantum mechanics: how does the mathematical formalism used to represent a quantum system make contact with the world as given in experience? This is commonly called the measurement problem, although the name is misleading. It might better be called the where-in-the-theory-is-the-world-we-live-in problem.