Diagonalization of Hamiltonians, uncertainty matrices and Robertson inequality. (English) Zbl 1061.81521

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 294-312 (2001).
Summary: The problem of diagonalization of Hamiltonians of \(N\)-dimensional boson systems by means of time-dependent canonical transformations (CT) is considered, the case of quadratic Hamiltonians being treated in greater detail. The unitary generator of time-dependent CT which can transform any Hamiltonian to that of a system of uncoupled stationary oscillators is constructed. The close relationship between methods of canonical transformations, time-dependent integrals of motion and dynamical symmetry is noted.
The diagonalization and symplectic properties of the uncertainty matrix for \(2N\) canonical observables are studied. It is shown that the normalized uncertainty matrix is symplectic for the squeezed multimode Glauber coherent states and for the squeezed Fock states with equal photon numbers in each mode. The Robertson uncertainty relation for the dispersion matrix of canonical observables is shown to be minimized in squeezed coherent states only.
For the entire collection see [Zbl 0957.00038].


81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81R30 Coherent states
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