## Proceedings of the International Conference on Geometry, Integrability and Quantization

### Diagonalization of Hamiltonians, Uncertainty Matrices and Robertson Inequality

#### Abstract

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.

#### Article information

Dates
First available in Project Euclid: 5 June 2015

https://projecteuclid.org/ euclid.pgiq/1433528481

Digital Object Identifier
doi:10.7546/giq-2-2001-294-312

Mathematical Reviews number (MathSciNet)
MR1815646

Zentralblatt MATH identifier
1061.81521

#### Citation

Trifonov, Dimitar A. Diagonalization of Hamiltonians, Uncertainty Matrices and Robertson Inequality. Proceedings of the Second International Conference on Geometry, Integrability and Quantization, 294--312, Coral Press Scientific Publishing, Sofia, Bulgaria, 2001. doi:10.7546/giq-2-2001-294-312. https://projecteuclid.org/euclid.pgiq/1433528481.