Consider a density matrix of a free particle in non-relativistic quantum mechanics. Nice, quasi-classical particles will be well-approximated by a wavepacket or a mixture of wavepackets. The coherent superposition of two wavepackets well-separated in phase space is decidedly non-classical.
Is there a formalism I can use to call this density matrix "approximately diagonal in the overcomplete basis of wavepackets"? (For the sake of argument, we can consider a specific class of wavepackets, e.g. of a fixed width σ and instantaneously not spreading or contracting.) I am aware of the Wigner phase space representation, but I want something that I can use for other bases, and that I can use for operators that aren't density matrices e.g. observables. For instance: X, P, and XP are all approximately diagonal in the basis of wavepackets, but RXR† is not, where R is the unitary operator which maps
|x⟩→(|x⟩+sign(x)|−x⟩)/√2.
(This operator creates a Schrodinger's cat state by reflecting about x=0.)
For two different states |a⟩ and |b⟩ in the basis, we want to require an approximately diagonal operator A to satisfy ⟨a|A|b⟩≈0, but we only want to do this if ⟨a|b⟩≈0. For ⟨a|b⟩≈1, we sensibly expect ⟨a|A|b⟩ to be proportional to a typical eigenvalue.
This post imported from StackExchange Physics at 2014-06-11 21:28 (UCT), posted by SE-user Jess Riedel