There is a variety of
literature on the relationship between the two equations
where P and Q are self-adjoint operators on a Hilbert space. Von Neumann has
characterized the solutions of (1.2) as those pairs (P,Q) which are unitarily
equivalent to a direct sum of a number of copies of the Schrödinger pair (p,q) where
p is −i(d∕dx) and q is multiplication by x on L2(R). Hence any pair which satisfies
(1.2) and is irreducible in the obvious sense is unitarily equivalent to the
Schrödinger pair. It is well known that any pair satisfying (1.2) satisfies (1.1) in the
following strong sense:
there is a dense subspace Ω which is a core for both P and Q, is invariant
under P and Q and PQf − QPf = −if for all f in Ω.
