A key part of the traditional derivation above is that the wave function must be single-valued. This is now recognised by many as not being completely correct: a wave function ψ ψ is not observable and only the probability density ψ ψ ∗ ∗ ψ ψ is required to be single-valued. The possible double-valued half-integer wave functions have a single-valued probability density. This was recognised by Pauli in 1939 (cited by Japaridze et al)