Hamiltonian systems canonical transformation

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 coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

The above Hamiltonian is a result of a canonical transformation of the coordinate system applied to the Hamiltonian given in Eq.(3), when the external potential is homogeneous. This approach has been first utilized by Linderberg and Shull [42] in the study of the QC correlation energy for high-enough Z, small a, with a=Z l. Notice that HaQC does not keep the density fixed as a changes in contrast to H [n]. ... 

An alternative procedure to generate a canonical transformation is to use the Hamiltonian flow itself. Consider an arbitrary Hamiltonian system of the same dimension as the original system. The associated functional dependence of the final state ait = tf on the initial state at t = t, can be represented by... 

Further reduction of the constrained reaction path model is possible. Here we adopt a system-bath model in which the reaction path coordinate defines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the motion along the reaction coordinate. In particular. Miller showed that a canonical transformation of the reaction path Hamiltonian T + V) yields [38]... 

Briefly, the aim of Lie transformations in Hamiltonian theory is to generate a symplectic (that is, canonical) change of variables depending on a small parameter as the general solution of a Hamiltonian system of differential equations. The method was first proposed by Deprit [75] (we follow the presentation in Ref. 76) and can be stated as follows. 

The state of a classical system is specified in terms of the values of a set of coordinates q and conjugate momenta p at some time t, the coordinates and momenta satisfying Hamilton s equations of motion. It is possible to perform a coordinate transformation to a new set of ps and qs which again satisfy Hamilton s equation of motion with respect to a Hamiltonian expressed in the new coordinates. Such a coordinate transformation is called a canonical transformation and, while changing the functional form of the Hamiltonian and of the expressions for other properties, it leaves all of the numerical values of the properties unchanged. Thus, a canonical transformation offers an alternative but equivalent description of a classical system. One may ask whether the same freedom of choosing equivalent descriptions of a system exists in quantum mechanics. The answer is in the affirmative and it is a unitary transformation which is the quantum analogue of the classical canonical transformation. 

Here // < is the Hamiltonian for the radiation field in vacuo, flmo the field-free Hamiltonian for molecule , and //m( is a term representing molecular interaction with the radiation. It is worth emphasising that the basic simplicity of Eq. (1) specifically results from adoption of the multipolar form of light-matter interaction. This is based on a well-known canonical transformation from the minimal-coupling interaction [17-21]. The procedure results in precise cancellation from the system Hamiltonian of all Coulombic terms, save those intrinsic to the Hamiltonian operators for the component molecules hence no terms involving intermolecular interactions appear in Eq. (1). 

The SRTS sequence consists of a preparatory pulse and an arbitrary long train of the phase-coherent RF pulses of the same flip angle applied with a constant short-repetition time. As was noted above, the "short time" in this case should be interpreted as the pulse spacing T within the sequence that meets the condition T T2 Hd. The state that is established in the spin system after the time, T2, is traditionally defined as the "steady-state free precession" (SSFP), ° and includes two other states (or sub-states) quasi-stationary, that exists at times T2effective relaxation time) and stationary, that is established after the time " 3Tie after the start of the sequence.The SSFP is a very particular state which requires a specific mechanism for its description. This mechanism was devised in articles on the basis of the effective field concept and canonical transformations. Later approaches on the basis of the average-Hamiltonian theory were developed. ... 

In well-known papers [22,23] Nose showed how a deterministic canonical ensemble N, V, T) MD simulation scheme could be constructed. Again the system phase space variables were augmented with an additional degree of freedom s. The physical system variables (qiiP ) were related to virtual variables (q,-,p,-,f) by a non-canonical transformation q, - = q,-, p = p,/i and = J df/i. The Hamiltonian of the extended system is... 

There are many physieal systems whieh are modelled by Hamiltonians, which can be transformed through a canonical transformation to a quadratic form ... 

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

Canonical transformations are used to reformulate the Hamiltonian of a system in new coordinates that render it more amenable to analysis [101]. The transformation of H is defined by a unitary operator, U, that acts to produce a new Hamiltonian, H = U HU. Because U is unitary, it is guaranteed to preserve the eigenvectors and eigenvalues of the original Hamiltonian. When H is written in terms of some set of variables (operators) A, then H will have the same form when written in terms of the transformed variables A = U AylJ, each of which is in general a function of all the untransformed variables [i.e., A. =/( Ay )]. To make progress, then, we use the latter relationship to write the transformed Hamiltonian H in terms of the untransformed variables, A (see Ref. 101). 

Furthermore, the canonical transformation W of the coordinate system minimizes the angular dependencies of the new Hamiltonian H, thereby making the new action variables J as nearly constant as possible. If H can be obtained altogether independent of the angle 0 (at least, at the order of the perturbative calculation performed), then... 

As shown in Appendix B, it can easily be proved for any transforms described by the functional form of Eq. (2.15), that if z(0) are canonical, z(s) are also canonical (and vice versa), as the time evolution of any Hamiltonian system is regarded as a canonical transformation from canonical variables at an initial time to those at another time, maintaining the structure of Hamilton s equations. 

The ensemble (27) is readily obtained by transforming the usual grand canonical ensemble to a moving coordinate system. In making this transformation we have neglected relativistic effects the hamiltonian H then transforms according to... 

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