Cartesian coordinates operator transformation from

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... 

The particles are numbered from 1 to N with Mi the mass of particle i, R, = [A, Yt Zi) a column vector of Cartesian coordinates for particle i in the external, laboratory fixed, frame, Vr the Laplacian in the coordinates of R, and Ri — Rj the distance between particles i and j. The total Hamiltonian, eqn.(l), is, of course, separable into an operator describing the translational motion of the center of mass and an operator describing the internal energy. This separation is realized by a transformation to center-of-mass and internal (relative) coordinates. Let R be the vector of particle coordinates in the laboratory fixed reference frame. 

A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a mapping from the minimal surface to a Riemann surface. The mapping is a composite one first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except flat points), characterised by cartesian coordinates (x,y,z) is described by the complex number (o, which... 

Here we would like to add some comment of a general character. Dirac (1958) argued that the transformation from classical to quantum mechanics should be made, first by constructing the classical Hamiltonian in the Cartesian coordinate system and then by replacing the positions and momenta by their quantum-mechanical operator equivalents, which are determined by the particular representation chosen. The important point is that this transformation should be performed in the Cartesian coordinate system, for it is only in this system that the Heisenberg uncertainty principle for the positions and momenta is usually enunciated. In this connection, notice that some momentum wave functions such as those obtained by Podolsky and Pauling (1929) are correct wave functions that are useful in calculations of the expectation value of any observable, but at the same time they have a drawback in that the momentum variables used there are not conjugate to any relevant position variables (see also, Lombardi, 1980). 

We will need the unitary transformations exp (lA) and exp (iS). They are very convenient, since when starting from some set of the orthonormal functions (spinorbitals or Slater determinants) and applying this transformation, we always retain the orthonormality of new spinorbitals (due to A) and of the linear combination of determinants (due to S). This is an analogy to the rotation of the Cartesian coordinate system. It follows from the above equations that exp (/A) modifies spinorbitals (i.e., operates in the one-electron space), and exp (i S) rotates the determinants in the space of many-electron functions. 

The solution of Eq. (6.7) can be conveniently accomplished in a new set of spatial coordinates. Since the potential energy operator V r) depends on a radial distance coordinate and hence not on any orientational coordinate, solution of Eq. (6.7) should be attempted in spherical coordinates (r, i , cp). The transformation equations from Cartesian coordinates to spherical coordinates and its inverse transformation have already been introduced in section 4.4, Eqs. (4.95)-(4.100). 

This operator is related to the Laplacian operator, (del squared), in spherical polar coordinates. From the definition of in Cartesian coordinates, and from the coordinate transformation given previously. 

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