Showing papers on "Canonical transformation published in 1967"

Free Rotation of a Rigid Body Studied in the Phase Plane

Abstract: A homogeneous canonical transformation reduces the free rotation of a solid body to a conservative Hamiltonian system with only one degree of freedom. This reduction permits the representation of all possible solutions of the Euler-Poinsot problem by isoenergetic curves in a phase rectangle. Such a diagram illustrates the stable or unstable character of a permanent rotation around an axis of inertia.

Fermions and associated bosons of one-dimensional model

Abstract: The representation of the canonical commutation relations involved in the construction of boson operators from fermion operators according to the recipe of the neutrino theory of light is studied. Starting from a cyclic Fock-representation for the massless fermions the boson operators are reduced by the spectral projectors of two charge-operators and form an infinite direct sum of cyclic Fock-representations. Kronig's identity expressing the fermion kinetic energy in terms of the boson kinetic energy and the squares of the charge operators is verified as an identity for strictly selfadjoint operators. It provides the key to the solubility ofLuttinger's model. A simple sufficient condition is given for the unitary equivalence of the representations linked by the canonical transformation which diagonalizes the total Hamiltonian.

Collective Excitations of Dipolar Systems

Abstract: A system of neutral molecules having permanent electric dipole moments should exhibit self-sustained longitudinal polarization waves analogous to the plasma vibrations of an electron gas. To study these long-wavelength collective modes (dipolar plasmons), we adapt some techniques successful in many-electron problems to systems whose Hamiltonians include kinetic energy both of center-of-mass and of rotational motion, the interaction between rigid dipoles, and short-range interactions which we need not specify in detail. A canonical transformation plus a random-phase approximation (RPA) is used to display the collective modes explicitly in the Hamiltonian, and it is shown that the ground state of the system is adequately described in the RPA. The dipolar plasmon frequency so found is also obtained by linearizing equations of motion for Fourier components of the polarization charge density, and is affected statistically by short-range forces through a constant factor. Linear dielectric response theory yields a simple exact sum rule and, in a self-consistent-field approximation, a dielectric function in which dynamical effects of short-range forces can be retained. The classical dielectric function for linear and spherical rotators is evaluated in closed form when short-range forces are neglected. Its static limit is the Onsager expression for rigid dipoles, and it vanishes at the dipolar plasmon frequency.

Second‐Order Renormalization and Phase Stability in Strontium Titanate

Abstract: The temperature dependence of the transverse optic (T.O.) mode with wave number k ∼ 0, and of its contribution to the electric susceptibility, is discussed both in zero and in finite field, in terms of the renormalization of a Hamiltonian proposed by SILVERMAN and JOSEPH. The renormalization proceeds in two steps: (a) a canonical transformation is introduced which removes linear terms and also quadratic and cubic terms nondiagonal in the normal mode coordinates for which k > 0; (b) a thermal average is made of quartic terms proportional to the square of the ferroelectric mode frequency, giving rise to an effective frequency which depends on temperature. In zero field, it is shown that the contributions of T. O. modes to the renormalized frequency become large at low-temperature, preventing this frequency from vanishing at any temperature, and thus rendering impossible a ferroelectric transition. In finite field of sufficient magnitude the optical branch frequencies are all raised, reducing their stabilizing effect. It is shown that if the stabilization is reduced sufficiently, there will exist two phases of which one is metastable, has a ferroelectric mode frequency which vanishes, and is capable of two opposed states of polarization. The second phase is stable, with a frequency which does not vanish and a susceptibility which exhibits a maximum independently of the properties of the first phase. Such a maximum has been observed by HEGENBARTH.

On the Quasi‐Harmonic Theory of Phonons

Abstract: The shift of phonon frequencies in anharmonic crystals is calculated self-consistently by a canonical transformation equivalent to Cowley's quasi-harmonic approximation. The damping of phonons is obtained from the phonon kinetic equations in the above approximation.

Canonical transformations associated with second order problems in the calculus of variations

Abstract: Our object is a systematic investigation of some of the properties of canonical transformations associated with second order problems in the calculus of variations. After the introduction of such transformations, together with the related concepts of Lagrange and Poisson brackets, the bracket relationships are found which characterize canonical transformations. This characterization is also achieved by means of so-called reciprocity relations between the original transformation and its inverse (which always exists). The effect of the canonical transformation on the underlying variational problem is discussed. It is also shown that the Jacobian of such a transformation always has the value unity.