Reconstruction of spin states and its conceptual implications ∗

State reconstruction for quantum spins is reviewed. Emphasis is on nontomographic approaches which are based on measurements performed with a Stern-Gerlach apparatus. Two consequences of successfully implemented state reconstruction are pointed out. First, it allows one to determine experimentally the expectation value of an arbitrary operator Ô without a device measuring it. Second, state reconstruction suggests a reformulation of Schrödinger’s equation in terms of expectation values only, without explicit reference to a wave function or a density operator. In a footnote of his article on quantum mechanics for the “Handbuch der Physik,” Pauli hides a question which, apparently, he does not know to answer [1]: Die mathematische Frage, ob bei gegebenen Funktionen W (x) und W (p) die Wellenfunktion ψ stets eindeutig bestimmt ist, wenn es eine solche zugehörige Wellenfunktion überhaupt gibt, (d. h. wenn W (x) und W (p) physikalisch vereinbar sind), ist noch nicht allgemein untersucht worden. Knowledge of the probability distributions W (x) ≡ |ψ(x)|2 und W (p) ≡ |ψ(p)|2 for position and momentum is equivalent to the knowledge of all moments 〈ψ|x̂n|ψ〉 and 〈ψ|p̂n|ψ〉, n = 1, 2, . . . Therefore, one can immediately rephrase Pauli’s question in terms of quantities which are measurable in experiments, at least in principle. Is it possible to reconstruct the state |ψ〉 (of a particle in a potential V (x) on the real line, say) on the basis of a collection of expectation values, i.e., {〈ψ|x̂|ψ〉, 〈ψ|p̂|ψ〉, n = 1, 2, . . .} ⇒ |ψ〉 ? (1) ∗Contribution to the proceedings of the conference New Insights in Quantum Mechanics, 9/98 D-Goslar (to appear)