Quantum mechanics from a Heisenberg-type equality

The usual Heisenberg uncertainty relation, ∆X∆P ≥ h̄/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exact uncertainty relation, δX∆Pnc ≡ h̄/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation. 1 An exact uncertainty relation The well known Heisenberg inequality ∆X∆P ≥ h̄/2 has a fundamental significance for the interpretation of quantum theory as argued by Heisenberg himself, it provides “that measure of freedom from the limitations of classical concepts which is necessary for a consistent description of atomic processes” [1]. It might be asked whether this “measure of freedom” from classical concepts can be formulated more precisely. The answer is, surprisingly, yes and, as a consequence, the Heisenberg inequality can be replaced by an exact equality, valid for all pure states. To obtain this equality, note that for a classical system, the position and momentum observables can be measured simultaneously, to an arbitrary accuracy. For a quantum system we therefore define the classical component of the momentum to be that observable which is closest to the momentum observable, under the constraint of being comeasurable with the position of the system. More formally, for the case that we have maximal knowledge about the system, i.e., we know its wavefunction ψ(x), the classical component P ψ cl of the momentum is defined by the properties [X,P ψ cl ] = 0, 〈 (P − P ψ cl ) 〉ψ = minimum. (1)