Momentum transformation

The Coordinate-Momentum Transformation.—We shall first derive the matrix expressions for the operators Pk on the j-represen-tation, and for the operators Qk on the p-representation. From these we shall then be able to derive the transformation matrices connecting the q- and the -representations. We start with the evident relationships ... 

The probability that J has a wave vector K relative to I in HD + is given by the momentum transform of the wave function for the vibrational and rotational interactions in HD +. The probability that I is captured by X with a wave vector k is given by the momentum transform of the wave function for the rotational and vibrational interactions in XI+. 

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. 

Using this coordinate transformation, and the conjugate momentum transformation, the internal Hamiltonian for the problems we are considering (cases 1 and 2 above) can be written as,... 

Until this stage all discussions have been based on classical mechanics. However, in the present formulation the translation to quantum mechanics is quite straightforward, since quantum momenta can be defined from the momentum transformation with only slight modifications. Furthermore, the resulting expressions are general in the sense that they can be derived without considering any particular representation of the momenta as differential operators. They apply equally well in a wave mechanical context. 

In quantum mechanics the momentum transformation [Eq. (2.23)] and its inverse should read... 

The Hamiltonian form of the kinetic energy is most easily obtained by considering the momentum transformation directly,... 

The usual angular velocities, ojx, coy and cjz, are inconvenient since the redundancy among them demands special precautions59. However, we can circumvent this problem completely by considering the momentum transformation. 

This rather extended discussion of the theory of a very special type of rigid molecule has been presented here, since it well illustrates the advantages obtained by a reformulation of the basic principles in terms of the momentum transformation. It applies to classical as well as quantum mechanical considerations. Furthermore, these examples, applying the method to wellknown molecular models, should make it easier to follow the derivations of the following section. 

We generalize the procedure from Sect. 3.5 and use the momentum transformation as expressed by Eq. (3.49). However, when the range of g is extended to include p as well, another generalized quantity appears, namely... 

Angular momentum transformations of Lam6 spheroconal harmonic polynomials... 

In Eq. (99), the momentum transformation vector function s(p) has the same role in momentum orbit V as f f) in position orbit of. 

Still another idea is introduced in the contributions by Cooper and Allan.They removed the core electron density dominance problem by using momentum transformations. A similar expression exists for the electron density in momentum space ... 

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