In Hilbert space, all things are quantumly possible

Quantum mechanics doesn’t primarily describe particles moving through familiar three-dimensional space but rather tracks the set of possible futures those systems can have, represented by arrows (vectors) in an abstract Hilbert space. John von Neumann showed that the apparently different formalisms developed by Heisenberg (matrix mechanics) and Schrödinger (wave mechanics) are two mathematical views of the same vector-based state living in that space. A quantum state is a superposition pointing in some direction among axes that stand for possible measurement outcomes; the inner product (the “shadow” of the arrow on an axis) determines the probability of a given outcome. Before measurement the state evolves smoothly and predictably through Hilbert space; upon observation it snaps randomly onto one outcome axis with probabilities set by that alignment.

Von Neumann formalized Hilbert space by demanding completeness and an inner product, which allows both finite cases (a qubit’s two-dimensional space) and infinite ones (a particle’s continuum of possible positions). The choice of axes corresponds to the observable being measured, so different pictures - rotating axes versus rotating vectors - are merely different mathematical perspectives on the same underlying structure. Treating quantum theory as dynamics of vectors in Hilbert space both unifies historical approaches and makes clear that quantum theory is fundamentally about possibilities encoded in an abstract, highly structured mathematical arena.