Hamiltonian Fields

Going now to check whether the Schrodinger equation may be recovered from these Lagrangian or Hamiltonian field formalisms, one may employ the so-called Schrodinger Lagrangian... 

It is clear that locally Hamiltonian fields from a linear subspace H oc(M) in V M) and that H oc M) D H M) inasmuch an any Hamiltonian field is obviously locally Hamiltonian. The converse is false. This means that there exist fields that admit in the neighbourhood of each point x on M a representation in the form v tt = sgrad Fu but do not admit a unique representation in the form t = sgrad F, where F is a single-valued function defined at all points of M. In other words, in such cases, the local Hamiltonians defined on separate neighbourhoods U cannot be "sewed into one smooth single-valued function defined on the entire M. 

It can be shown that the symplectic structure a is invariant under the group of diffeomorphisms (3 generated by the Hamiltonian field v. We shall prove the general assertion. 

In the case of a two-dimensional Riemannian manifold Af with a Riemannian metric gij and with the form of the Riemannian area w = y/det(gij)dx A dy as a symplectic structure (see above), the condition that the group (3 of diffeomorphisms gt preserve the form cj is equivalent to the condition that the domain areas be preserved on the surface when these domains are shifted by the diffeomorphisms gt. Thus, the shifts along integral trajectories of a Hamiltonian field on a two-dimensional symplectic manifold preserve the domain areas. 

In the case of a two-dimensional symplectic manifold, the condition of the locally Hamiltonian character of the field admits another vivid geometrical interpretation. Let gij be a Riemannian metric on and let u) = y/det gij)dx A dy be the form of the Riemannian area. By virtue of the Darboux theorem, one can always choose local coordinates p and q such that the form oj be written in the canonical form dp A dq. Here p and q are certain functions of x and y (and vice versa). Let t be a locally Hamiltonian field t = (i (a , y),Q(x,y)), where P and Q are coordinates of the field in the local system of coordinates p and q. Let us interpret the field v as a velocity field of the flow of liquid of constant density (equal to unity) on the surface M. Let us investigate the variation of the mass of the liquid bounded by an infinitesimal rectangle on the surface when it is shifted along integral trajectories of the field v. It is clear that the mass of this liquid is equal to the area of the rectangle. Therefore, the mass of the liquid contained in a bounded (sufficiently small) domain on is equal to the area of the domain. 

We have given an example of a locally Hamiltonian vector field on R 0 which is not a Hamiltonian field. The symplectic manifold was not compact here. But such kind of examples may be constructed on compact closed (i.e., without a boundary) manifolds as well. Take, for instance, an ordinary flat torus supplied with a Euclidean metric gij = Si, Then the 2-form of the area on this torus will be written in Cartesian coordinates x, y as follows dx A dy, i.e., this 2-form is canonical. On the torus, consider a vector field v = (1,0) given by a uniform liquid motion along its parallels (Fig. 7). 

The obstacle which prevents an arbitrary local Hamiltonian field from being a Hamiltonian one is that the manifold is not simply-connected. In our first example,t he manifold R 0 is not simply-connected because it punctures the point out of the plane (a closed detour of this point is not contracted in the point on R 0). In our... 

Thus, the commutator [ui,V2 is a globally defined Hamiltonian field, because the smooth function o (t i,U2) well defined on the entire manifold (and does not depend on the choice of the neighbourhood U). 

Thus, we have proved not only item 4, but item 5 as well. Indeed, as has become clear, the commutator of two locally Hamiltonian vector fields Ui and V2 is a globally Hamiltonian field. Therefore, Hamiltonian fields form the ideal in the space of locally Hamiltonian fields. This completes the proof of the theorem. 

Let us consider a Hamiltonian field t = sgrad / on M. The Hamiltonian / may be interpreted as an element of the Lie algebra (M) of smooth functions on M. Then we may analyze a linear subspace 0(f) of all functions g G C° (M) which commute with the element / with respect to the Poisson bracket, i.e., functions... 

FVom proposition 1.2.5, it follows that the Poisson bracket /, is a locally constant function, and if the manifold Af is connected, then the function is constant on the entire manifold Af. Thus, we have proved that two Hamiltonian field t and w commute (like vector fields) if and only if the Poisson bracket of their Hamiltonians, /) ) is locally constant (or constant, in case the manifold is connected). 

Since the projection n Q Q may be thought of as smooth, it follows that the symplectic structure w, the Hamiltonian field sgradff, and the integral /,... 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

The possible types of symmetry for the Hamiltonian of an isolated molecnle in field-free space (all of them are discussed in more detail later on in the article) can be listed as follows ... 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

The translational linear momentum is conserved for an isolated molecule in field free space and, as we see below, this is closely related to the fact that the molecular Hamiltonian connmites with all... 

We now add die field back into the Hamiltonian, and examine the simplest case of a two-level system coupled to coherent, monochromatic radiation. This material is included in many textbooks (e.g. [6, 7, 8, 9, 10 and 11]). The system is described by a Hamiltonian having only two eigenstates, i and with energies = and Define coq = - co. The most general wavefunction for this system may be written as... 

This expression may be interpreted in a very similar spirit to tliat given above for one-photon processes. Now there is a second interaction with the electric field and the subsequent evolution is taken to be on a third surface, with Hamiltonian H. In general, there is also a second-order interaction with the electric field through which returns a portion of the excited-state amplitude to surface a, with subsequent evolution on surface a. The Feymnan diagram for this second-order interaction is shown in figure Al.6.9. 

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

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