In Hilbert Space, All Things Are Quantumly Possible

Quantum mechanics has a strange rule: don't track where an object is now, only what it could become later. To compute those possibilities, physicists use a mathematical arrow called a vector, pointing not to a location in space but to a direction in a 'space of possibilities.'

Two great minds, one quantum elephant

In 1925 Heisenberg devised 'matrix mechanics,' using tables of numbers to calculate the odds of an electron jumping between orbits. The next year Schrödinger came up with 'wave mechanics,' using waves to track where a particle might be found. The pictures looked totally different, yet they gave identical predictions.

They were the same theory. Von Neumann united them: a quantum state is an arrow in a Hilbert space, and the space's size depends on how many possible futures the object has.

Observe it, and it collapses

Before you look, the arrow rotates smoothly through the space, representing shifting possibilities. The moment you measure, it snaps randomly onto an axis, and the superposition collapses into a definite outcome, like a show suddenly reaching its finale.

Is Hilbert space real?

A free particle could be anywhere in the universe, so its Hilbert space has infinite dimensions. Von Neumann required the space be 'complete' and support an 'inner product,' i.e., the shadow an arrow casts on an axis.

The space is a human invention, but very useful. Carroll calls Hilbert space the fundamental theatre of reality; Sorce calls it a handy tool. Even von Neumann later wrote, 'I no longer believe in Hilbert space.' Yet to this day, it remains the handiest stage for quantum theory.