Eigenvalues spherical coordinates

In principle, knowledge of Eqs. [18]—[22] is sufficient to set up differential equations for the orbital angular momentum operators and to solve for eigenvalues and eigenfunctions. The solutions are most easily obtained employing spherical coordinates r, 0,< ) (see Figure 7). The solutions, called spherical harmonics, can be found in any introductory textbook of quantum chemistry and shall be given here only for the sake of clarity. 

V r, V, z r, V- 6 Tl. V, z Tl, V, z T1,0, V 0 0 0,1 0, oo OQ 1,0 1, OO 1,2 1,2,3 12 ID constant volume condition cylindrical coordinates spherical coordinates elliptical cylinder coordinates bicylinder coordinates oblate and spheroidal coordinates zero thickness limit based on centroid temperature zeroeth order, first order value on the surface and at infinity infinite thickness limit first eigenvalue value at zero Biot number limit first eigenvalue value at infinite Biot number limit solids 1 and 2 surfaces 1 and 2 cuboid side dimensions net radiative transfer one-dimensional conduction... 

Of course, there is nothing special about the z axis. All directions of space are equivalent. If we had chosen to specify L and (rather than L, we would have gotten the same eigenvalues for as we found for L. However, it is easier to solve the eigenvalue equation because has a simple form in spherical coordinates, which involve the angle of rotation (p about the z axis. 

Solving the eigenvalue equation = E P is most often achieved in spherical coordinates r= (r, d, rotational invariance of the potential, bound states can be chosen with a well defined angular momentum /. As explained in any standard textbook [21], introducing... 

An alternative approach consists of using spherical coordinates, as in section 2.1. This provides a new labelling of the eigenvalues... 

There is nothing special about the z-axis, since all directions in space are equivalent. The only reason why it is easier to solve the Lz eigenvalue equation (compared to Lx and Ly) is because it has a simpe form in spherical polar coordinates, which involves only the angle of rotation about the z-axis. 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

Hence there exists a complete set of common eigenfunctions for L2 and any one of its components. The eigenvalue equations for L2 and Lz are found to be separable in spherical polar coordinates (but not in Cartesian coordinates). Using the chain rule to transform the derivatives, we can find... 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

The eigenfunctions of the free electron confined in the same prolate spheroids are expressed as products of regular radial and angular spheroidal wave functions [16] Chapter 21, in the respective coordinates u and v, and the eigenfunctions of Equations (34) and (35). The radial functions are expressed as infinite series of spherical Bessel functions of order m + s and argument kfu. Its eigenvalues are determined by the boundary condition on the radial factor,... 

For example, the action of K is just multiplication by the eigenvalue —Kj. The action of the Dirac matrices / and a in the partial wave subspace is described by (110). Likewise, we can compute the action of a spherically symmetric potential in one of the angular momentum subspaces. It remains to observe that due to the factor r in (102) the operator djdr - 1/r in (which is part of expression for the Dirac operator in polar coordinates) simply becomes d/dr in L (0,oo) ... 

Show that the spherical harmonic function Poo eigenfunction of the inversion operator with eigenvalue 1, while the spherical harmonic function Y is an eigenfunction with eigenvalue — 1. In spherical polar coordinates the inversion operator replaces 0by tt — 0 and replaces by n + (j). Show that rotation of 180° around an axis perpendicular to the bond axis gives the same result. 

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