Classical mechanics, coordinate transformations

When the dynamics of a molecular system is studied, we have to choose a set of coordinates to describe the system. There are many possibilities and in the following we will focus on coordinates that are convenient in the analysis of molecular collisions and chemical reactions. 

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The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

The different selectivity between Sc -mont and Sc(OTf)3 has been rationahzed by the mechanism of the transformation. While the Sc(OTf)3 catalyzed reaction in water afforded 17 through the classical mechanism, when Sc -mont is used the compounds are reacting when coordinated to the Sc center in the silicate layer of Sc -mont, giving rise to the addition of two molecules of dimedone... 

However, it must be pointed out that Aiv expressed in Cartesian coordinates, is not normalised. Gaussian94 first normalises Ar, before printing and also prints the former norm or, more precisely, its inverse. This last quantity has mass units and is referred to as the normal mode reduced mass, which is not to be confused with the reduced mass defined in classical mechanics. The normal mode reduced mass appears because of the fact that, after transforming to the new normalised coordinates, one is no longer representing the energy versus Qi but versus Qi divided by the square root of this reduced mass. As a result, the resulting force constant is multiplied by this mass and the... 

The differential cross-section refers, as in the classical description, to the scattering angle in the center-of-mass coordinate system. In order to relate to experimentally observed differential cross-sections, one has to transform to the appropriate scattering angle. This transformation takes the same form as discussed previously, essentially, because the expectation value of the center-of-mass velocity V is conserved just as in classical mechanics. 

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. 

It is within the Hamiltonian formulation of classical mechanics that one introduces the concept of a canonical transformation. This is a transformation from some initial set of ps and qs, which satisfy the canonical equations of motion for H(p, q, t) as given in eqn (8.57), to a new set Q and P, which depend upon both the old coordinates and momenta with defining equations. 

For completeness it is mentioned that the transformations between different sets of coordinates describing the same motion, characterize a branch of classical mechanics named kinematics which is fundamentally mathematical methods, and is not based on physical principles. 

Equation (3.32a) implies normalization, and Eq. (3.32b) contains the essential probabilistic interpretation of the projections onto the momenta or coordinates. The last condition, a natural consequence of the definitions of the density matrix and the Wigner-Weyl transform, explicitly eliminates the singular distributions allowed in Eq. (3.31d). That is, although the completeness of the quantum PijXp, q) basis permits the construction of 8 function distributions, they make, unlike classical mechanics, no natural appearance in quantum mechanics wherein eigenfunctions of LQ are square integrable and such singular distributions are explicitly excluded in Eq. (3.32d). 

Let (p, q) be one set of canonically conjugate coordinates and momenta (the old variables) and (P,Q) be another such set (the new variables).13 (P, Q, p and q are IV-dimensional vectors for a system with N degrees of freedom, but for the sake of clarity multidimensional notation will not be used the explicitly multidimensional expressions are in most cases obvious.) In classical mechanics P and Q may be considered as functions of p and q, or inversely, P and Q may be chosen as the independent variables with p and q being functions of them. To carry out the canonical transformation between these two sets of variables, however, one must rather choose one old variable and one new variable as the independent variables, the remaining two variables then being considered as functions of them. The canonical transformation is then carried out with the aid of a generating function, or generator, which is some function of the two independent variables, and two equations which express the dependent variables in terms of the independent variables.13... 

In Chapter I we found that curvilinear coordinates, such as spherical polar coordinates, are more suitable than Cartesian coordinates for the solution of many problems of classical mechanics. In the applications of wave mechanics, also, it is very frequently necessary to use different kinds of coordinates. In Sections 13 and 15 we have discussed two different systems, the free particle and the three-dimensional harmonic oscillator, whose wave equations are separable in Cartesian coordinates. Most problems cannot be treated in this manner, however, since it is usually found to be impossible to separate the equation into three parts, each of which is a function of one Cartesian coordinate only. In such cases there may exist other coordinate systems in terms of which the wave equation is separable, so that by first transforming the differential equation into the proper... 

The time autocorrelation function can be written as a transition dipole correlation function, a form that is equally useful for an inhomogeneously broadened spectrum. This is the form that is extensively used to discuss the spectral effects of the environment (32-34). The dipole correlation function also provides for the novice an intuitively clear prescription as to how to compute a spectrum using classical dynamics. For the expert it points out limitations of this, otherwise very useful, approximation. The required transformation is to rewrite the spectrum so that the time evolution is carried by the dipole operator rather than by the bright state wave packet. The conceptual advantage is that it is easier to imagine what the classical limit will be because what is readily provided by classical mechanics trajectory computations is the time dependence of the coordinates and momenta and hence, of functions thereof. In other words, in our mind it is easier to... 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

According to this definition, a tensor of the first rank is simply a vector. As examples of second rank tensors within classical mechanics one might think of the inertia tensor 6 = (0y) describing the rotational motion of a rigid body, or the unity tensor 5 defined by Eq. (2.11). A tensor of the second rank can always be expressed as a matrix. Note, however, that not each matrix is a tensor. Any tensor is uniquely defined within one given inertial system IS, and its components may be transformed to another coordinate system IS. This transition to another coordinate system is described by orthogonal transformation matrices R, which are therefore not tensors at all but mediate the change of coordinates. The matrices R are not defined with respect to one specific IS, but relate two inertial systems IS and IS. ... 

Einstein s theory of special relativity relying on a modified principle cf relativity is presented and the Lorentz transformations are identified as the natural coordinate transformations of physics. This necessarily leads to a modification cf our perception of space and time and to the concept of a four-dimensional unified space-time. Its basic Mnematic and dynamical implications on classical mechanics are discussed. Maxwell s gauge theory of electrodynamics is presented in its natural covariant 4-vector form. 

The description of rotational motion is naturally performed in spherical coordinates. The two angular variables of rotational motion are generalized coordinates, free of additional constraints, as introduced in chapter 2. The transformation to spherical coordinates affects the definition of the angular momentum (operator) and subsequently the squared angular momentum (operator) which enters the kinetic energy (operator) expression. In order to avoid lengthy coordinate transformations of the latter containing second derivatives with respect to Cartesian coordinates, we may consider the situation in classical mechanics first and subsequently apply the correspondence principle. 

In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S are expressed in terms of the Galilean transformations. Assume that S is moving with constant speed v in the direction of the positive x axis of S. If the coordinate axes of S are parallel to those of S, the Galilean coordinate transformations are... 

These are the 3N equations of motion in Lagrangian mechanics. They generate the same solution for the motion of classical particles as Newton s equations of motion. The benefit of Lagrangian mechanics is that Eqs. 3.24 are invariant to coordinate transformations. This simply means that the functional form of the equations of motion remains... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

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