Dispersion relations for light propagation through a two-level system

Interaction of radiation with two-level system is considered using the Dicke hamiltonian. By making use of the Schwinger representation, the hamiltonian is written in a trilinear form. The hamiltonian is made bilinear by replacing one set of boson operators by C-numbers. A Bogoliubov transformation is then used to reduce the hamiltonian to diagonal form. The density matrix separates into a product of density matrices for separate modes. The energies of the photon-like modes are found to follow dispersion relations which depend on the given set of initial conditions.     

Polaron formalism applied to donor-acceptor pairs in semiconductors

The effective hamiltonian for the electronic states of a donor-acceptor pair in a polar semiconductor is formulated as a Fröhlich hamiltonian generalized to an electron and a positive hole interacting in the field of the positive donor and negative acceptor cores. The polarization of the lattice by the ion cores is simplified by a Platzman transformation; that by the electron and positive hole, by a further unitary transformation in the adiabatic approximation. The total effective hamiltonian including the lattice polarization energy is then considered for application of the variational principle to determine the parameters of the electron and positive hole effective mass functions. The Inglis-Williams analysis of the zero-phonon luminescent spactra of pairs is reconciled with the present theory by determining what single effective hamiltonian for the excited electronic state is consistent with their theoretical spactra. Within the approximations of their analysis, which are clarified, it is found to be the total effective hamiltonian. Finally, a more general and rigorously-derived effective hamiltonian is presented for the direct theoretical determination of the zero-phonon radiative transition energies of donor-acceptor pairs in compound semiconductors.     

Stability in bihamiltonian systems and multidimensional rigid body

The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is “simpler”. Thus it is natural to ask: can the dynamics of a bihamiltonian system be understood by studying its hamiltonian pair, without studying the associated first integrals?     

Nuclear hyperfine interactions in non-linear semi-rigid molecules

From the expression for the hamiltonian in molecular coordinates we obtained previously, the computation of an effective hamiltonian for a nondegenerate electronic state is performed to second order in degenerate perturbation theory. We thus obtain explicitly all dominant parameters for the spin-vibration and spin-rotation interactions; in addition, the parameters associated with the interactions between the magnetic moment induced by the molecular motion and an external magnetic field are computed. The vibrational dependence of these parameters is studied and the hamiltonian is written in a form adapted to the computation of an effective hamiltonian for an arbitrary vibrational state.     

The anomalous chiral Schwinger model and its quantization

We solve the anomalous chiral Schwinger model in the hamiltonian formulation implementing the Faddeev proposal. We diagonalize the hamiltonian by functional techniques and construct the eigenfunctionals of the quantized system. The spectrum is non Lorentz invariant since the Poincarè algebra is not closed. We point out that, relinquishing the Faddeev condition, there would be an infinite degeneration of the hamiltonian. Among the degenerate 1-particle eigenfunctionals it is possible to find a state with relativistic spectrum. We also examine the meaning of a non local modification of the hamiltonian which restores Lorentz invariance.     

A complex symplectic model for the damped oscillator

The linearly damped harmonic oscillator is described by a hamiltonian complex manifold. The master equation for the complex eigenstates of the quantized hamiltonian is found.     

Three-magnon bound states in the two-dimensional isotropic and anisotropic Heisenberg ferromagnet

The existence is shown of bound complexes of three magnons in a two-dimensional simple cubic Heisenberg ferromagnet. The Heisenberg spin hamiltonian is replaced by Dyson's boson hamiltonian. The spurious features of this hamiltonian are treated explicitly. Using Faddeev's formalism, detailed calculations of the three-magnon bound state energy curves have been performed both for the isotropic and the longitudinally anisotropic ferromagnet. Attention is paid to the possibility of observing these three-magnon bound states experimentally.     

Effective hamiltonians with relativistic corrections: The Foldy-Wouthuysen transformation versus the direct Pauli reduction

Two different methods of obtaining “effective 2 × 2 hamiltonians” which include relativistic corrections to nonrelativistic calculations are discussed. The standard Foldy-Wouthuysen transformation generates hamiltonians which order by order in 1/M decouple the upper from the lower components. The upper left-hand block then defines an effective 2 × 2 Foldy-Wouthuysen hamiltonian. In the second method the matrix element of the interaction hamiltonian of the Dirac representation is evaluated between free positive-energy states and reduced to two-component form. The resulting expression (possibly expanded in 1/M) then defines what we call the “direct Pauli reduction” effective 2 × 2 hamiltonian. We wish to investigate under which circumstances the two approaches yield the same result. Using a generic interaction with harmonic time dependence we show that differences in the corresponding effective S-matrices do arise beyond first-order perturbation theory. We attribute them to the fact that the use of the direct reduction effective hamiltonian involves the additional approximation of neglecting contributions from the negative-energy intermediate states, an approximation which is unnecessary in the Foldy-Wouthuysen case as there the 4 × 4 hamiltonian does not connect positive- and negative-energy states. We conclude that at least in the cases where the relativistic hamiltonian is known, using the direct Pauli reduction effective hamiltonian introduces spurious relativistic effects and therefore the Foldy-Wouthuysen reduction should be preferred.     

The molecular vibration-rotation hamiltonian

By use of commutation relations and sum rules Watson [1] has simplified the Darling and Dennison hamiltonian and obtained

The quantity H - U is thus identical to the analytical expression of the classical hamiltonian.

By reconsidering completely the question of the derivation of the quantum mechanical hamiltonian we intend to prove that the U term comes only from the particular method chosen to derive the quantum mechanical hamiltonian from its wave mechanical representations. From this we conclude that in the usual model, the quantum mechanical vibration-rotation hamiltonian has the same form as the classical one.     

Hamiltonian formulation of the anomalous chiral Schwinger model

We construct the hamiltonian formulation of the anomalous chiral Schwinger model, which has recently been shown to yield a consistent unitary theory. The impact of the anomaly on the constraints of the system is exhibited and the system is quantized using an appropriate hamiltonian consistent with the constraints.     

Yang-Mills theory on a cylinder

Yang-Mills theory on a two dimensional cylinder is studied in the hamiltonian formalism, without using gauge conditions. Since the only gauge invariant variable is the Wilson loop (holonomy) this system is equivalent to a finite dimensional system. The eigenstates and eigenvalues of the hamiltonian are found exactly.     

Hamiltonian walks and polymer configurations

Kasteleyn's treatment of the hamiltonian walk problem on lattice graphs is briefly reviewed. The asymptotic behaviour of the number of hamiltonian walks on the kth covering of a closed oriented lattice graph is expressed in terms of the asymptotic behaviour of the number of oriented trees on the lattice graph. Asymptotic results on the enumeration of hamiltonian walks are presented for the covering and underlying lattices of the Manhattan oriented square lattice, and the covering lattices of certain orientations of the diamond and cubic lattices. The effect of boundary conditions is examined.Relations are established between hamiltonian walks and close-packed polymer configurations. An analysis of the assumptions in lattice models of the melting and glass transitions in polymers shows that the Flory-Huggins approximation underestimates the total number of polymer configurations, but that corrections can be made without altering the principal features of the models thus permitting better agreement with the experimental data.     

A string-like equation for a U(N) gauge theory

A change of variables is made in the hamiltonian of a U(N) gauge theory so that the independent variables are the path dependent phase factors. The resulting hamiltonian is similar in form to that of the Nambu-Gato string     

Die dekorationstechnik zur untersuchung des mischzustandes von typ-II-supraleitern mit höheren Ginzburg-Landau-parametern

We show that the Ising model hamiltonian is a special case of a matrix Toda lattice hamiltonian with operator variables from a Clifford algebra of dimension equal to the size of the lattice.     

Low-energy Compton scattering by a proton: Comparison of effective hamiltonians with relativistic corrections

We discuss low-energy Compton scattering by a proton using two different methods of obtaining “effective 2 × 2 hamiltonians” of the electromagnetic interaction which include relativistic corrections. One is the standard Foldy-Wouthuysen transformation which we compare with the hamiltonian obtained from a direct reduction of the matrix element of the interaction hamiltonian between positive-energy solutions of the free Dirac equation. It is found that the Foldy-Wouthuysen hamiltonian yields the same result as a covariant calculation. However, an identification of the Z-diagrams of the usual Dirac representation with the contact graphs of the Foldy-Wouthuysen representation is incorrect beyond the order of the low-energy theorem. The direct-reduction method will, in general, lead to incorrect results. Although part of the problem may be cured by restoring the gauge invariance property of the hamiltonian, gauge invariance alone is not sufficient to reproduce the correct result.     

Non-abelian Cheshire cat bag models in 1+1 dimensions

Non-abelian Cheshire cat models are investigated in their lagrangian and hamiltonian formulations. The lagrangian bag boundary conditions are used to derive the form of non-abelian soliton operators, through which fermions are represented in bosonic language. These soliton operators are then used to construct the boundary interaction in the hamiltonian picture, wherein the bosonic sector is formulated by means of a current algebra involving anomalous commutators. The hamiltonian and the momentum operator are shown to commute, thus implying that the Cheshire cat criterion — the independence of the energy spectrum on the bag wall position — is fulfilled by the system.     

Effective hamiltonians

Non-linear equations, generalizing those determining Bloch's transformation to an effective hamiltonian, are set up in a very simple way. It is shown how they immediately determine the results of Van Vleck, Bloch and Soliverez. Treating the Foldy-Wouthuysen transformation as a special case of the Van Vleck transformation, sixth-order expressions are obtained by means of the non-linear equations. The transformations of Soliverez, Roussy and Kvasnicka are shown to be strictly identical—because they all have a simple property which is proven to be unique for that of Soliverez. Kvasnicka relates his transformation to that of Primas'. We shall see that it is also very closely related to that of Van Vleck's. A certain non-hermitian effective hamiltonian also given by Roussy is proven to be identical to that of Bloch's. Primas' level shift transformation gives an effective hamiltonian for each level in zeroth order. This effective hamiltonian is seen to be identical to that of Soliverez's up to and including third order—and to all orders if one makes a simple change in ‘normalization’. Computer treatment of finite matrices is discussed. The sixth-order Van Vleck hamiltonian recently given by Jørgensen and Pedersen in terms of commutators is derived in a more simple way.     

Dissipative hamiltonian systems: A unifying principle

The concept of hamiltonian sysem is generalized to include a wide class of dissipative processes. Evolution of any observable is generated jointly by a hamiltonian, with an entropy-conserving Poisson bracket, and an entropy, with an energy-conserving dissipative bracket. This approach yields many of the standard kinetic equations, such as those representing particle collisions, three-wave interactions, and wave-particle resonances.     

Relativistic fluid dynamics as a Hamiltonian system

The equations of ideal relativistic fluid dynamics in the laboratory frame form a noncanonical hamiltonian system with the same Poisson bracket as for nonrelativistic fluids, but with dynamical variables and hamiltonian obtained via a regular deformation of their nonrelativistic counterparts.