Point canonical transformations

Abstract In this work an algorithm based on the point canonical transformation method to convert any general second order differential equation of Sturm-Liouville type into a Schrodinger-like equation is applied to the position-dependent mass Schrodinger equation (PDMSE). This algorithm is next applied to find potentials isospectral to Morse potential and associated to different position-dependent mass distributions in the PDMSE. Factorization of worked PDMSE are also obtained. 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

Once this divergence happens, further solution of the differential equation is not possible beyond this point, and we have to reformulate the problem. To clarify our idea, let us consider the ID problem. At the turning point, p q) = 0 and A diverges. If we invert A to A = dq/dp, the divergence is removed and the propagation of A proceeds smoothly through the caustics. This inversion is equivalent to the canonical transformation, (p,q) (—q,p). It can be easily... 

We first consider a Hamiltonian, thus deterministic, system. Denoting by oo the set of all phase space coordinates of a point in phase space (which determines the instantaneous state of the system), the motion of this point is determined by a canonical transformation evolving in time, 7), with Tq = I. The function of time TfCO thus represents the trajectory passing through co at time zero. The evolution of the distribution function is obtained by the action on p of a unitary transformation Ut, related to 7) as follows ... 

The most direct influence on the current work is the recent canonical diago-nalisation theory of White [22]. This, in turn, is an independent redevelopment of the flow-renormalization group (flow-RG) of Wegner [23] and Glazek and Wilson [24]. As pointed out by Freed [25], canonical transformations are themselves a kind of renormalization, and our current theory may be viewed also from a renormalization group perspective. 

Up to this point our discussion of canonical transformations has been exact. We now proceed to the specific approximations that characterize our formulation of CT theory and discuss their relationship with approximations commonly made in other theories involving canonical (i.e., unitary) transformations. 

A further point is of interest in the formal discussion of the canonical transformation theory. So far we have assumed that the reference function is fixed and have considered only solving for the amplitudes in the excitation operator. We may also consider optimization of the reference function itself in the presence of the excitation operator A. This consideration is useful in understanding the nature of the cumulant decomposition in the canonical transformation theory. 

Canonic transformation of a regression corresponds to transfer of coordinate beginning into a new point S and to replacement of the old coordinate axes (X,... 

The transformation from one pair of canonically conjugate coordinates q and momenta p to another set of coordinates Q = Q(p,q,t) and momenta P = P(p>qT) is called a canonical transformation or point transformation. In this transformation it is required that the new coordinates (P,Q) again satisfy the Hamiltonian equations with a new Hamiltonian H P,Q,t) [35] [43] [52]. 

Let us now set for a moment R (p,q) = 0. Then, according to the general theory discussed in Section 2.5, the Hamiltonian in normal form possesses n—dim M independent first integrals of the form geometrical considerations we conclude that any orbit with initial point po G V lies on a plane n wj(p0) through Po and parallel to M we shall call this plane the plane of fast drift. This is true in the coordinates of the normal form. If we look at the original coordinates then we must take into account the deformation due to the canonical transformations —as we already remarked while discussing the case of an elliptic equilibrium. Moreover, we must consider also the noise due to the remainder, but in this case too we have = 0(er), so that the noise causes only a slow drift that becomes comparable with the deformation only after a time T(e) l/er. 

Fig. 7.5 Classical and optimized MDS solutions calculated from the canonically transformed correlation surface from fig. 7.3. (A) The two-dimensional projection of the three-dimensional ( 2,3 = 99.2%) (or four-dimensional a2 4 = 99.98%) object found by the matrix method described by step 5 in the text. The coordinates for the points in A are the same as the first two columns of the matrix of eigenvectors in table 7.2A. The second diagram (B) is derived from the MDS optimization method discussed in step 6. The lines between points in both diagrams are obtained from the results of step 3 (table 7.1). Thus, B is more representative of the measured distance matrix. Both diagrams correspond to rotated and slightly distorted approximations to the reaction mechanism in fig.7.1. (From [1].)...

Another, slightly more technical ( ) way to understand area preservation is to recall that the coordinates of all the points on the Poincare map are specified by coordinates that are canonically conjugate. Because both the initial and final coordinates of a family of trajectories propagated for one mapping are so specified, there must exist a generating function that transforms the coordinates of the initial points into those of the final points. Such a generating function is necessarily a canonical transformation. All canonical transformations preserve the norm of the vectors they transform it can be shown that this property is equivalent to area preservation on the Poincare map. ... 

Finally, before leaving this section, we note another important aspect of the Liouville equation regarding transformation of phase space variables. We noted in Chap. 1 that Hamilton s equations of motion retain their form only for so-called canonical transformations. Consequently, the form of the Liouville equation given above is also invariant to only canonical transformations. Furthermore, it can be shown that the Jacobian for canonical transformations is unity, i.e., there is no expansion or contraction of a phase space volume element in going from one set of phase space coordinates to another. A simple example of a single particle in three dimensions can be used to effectively illustrate this point.l Considering, for example, two representations, viz., cartesian and spherical coordinates and their associated conjugate momenta, we have... 

As the reader can notice, the latter values do not differ considerably from the initially guessed ones. The Jacobi matrix evaluated at this point and transformed to canonical format is again (10.6.9), with no zero column in submatrix A . In a series of such measurements, the canonical format (with different numerical values) will remain the same. We thus conclude in addition that... 

With regard to the different points of view outlined in (a), (b) and (c), it should be pointed out that these differences arise mainly from the use of localized (a, LMO), or canonical (CMO, b, and c) molecular orbitals. In principle LMOs and CMOs are equivalent and are related by a unitary transformation. This can be illustrated by the C=C bonding in acetylene. 

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

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