Angular momentum spherical polar coordinates

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

Figure 2.12 Definition of the components of angular momentum in cartesian and in spherical polar coordinates.

The angular momentum operator squared L, expressed in spherical polar coordinates, is... 

Suppose that the electron is in a 2p state with angular momentum proportional to cos 6 in spherical polar coordinates. The probability density (a ) 2 of such a state would be concentrated near the z-axis, where the length of the radius vector is proportional to cos2 9. Now suppose that the whole system is physically rotated, e.g. by the application of a magnetic field (active rotation) - alternatively the axes may be thought of as rotated in the opposite direction (passive rotation). After rotation the system has a new wave function ip (x) with ip x) 2 concentrated around a displaced axis, but the value of the new wave function at a rotated point must be the same as that of the old wave function at the original point,... 

The motion of a free particle on the surface of a sphere will involve components of angular momentum in three-dimensional space. Spherical polar coordinates provide the most convenient description for this and related problems with spherical symmetry. The position of an arbitrary point r is described by three coordinates r, 0, 0, as shown in Fig. 6.2. 

R dRV dRj h R where 2 is the square of the orbital angular momentum operator given by equation (5.32). With expressed in spherical polar coordinates, equation (10.31) becomes... 

For systems of chemical interest the amplitude function ip that occurs as a solution of (4.19) is postulated to give a complete description, provided the potential energy V, is correctly specified. In reality, the only chemically significant problem that has been solved is of an electron associated with an isolated stationary proton, with potential energy V = jr, in atomic units. The differential wave equation is separable in spherical polar coordinates. Separate solutions, as functions of radial (r) and angular 9, ip) coordinates, describe the quantized energy and angular momentum of the electron as ... 

There is a third observable of interest. It is the z component of the total angular momentum, L. The relationships between angular momentum operators allow for the simultaneous knowledge of the total angular momentum (through its square, L ) and one of its Cartesian components. By convention, the z component is chosen. This is due in part to the spherical polar coordinate system, and the relatively simple definition of the z component of the angular momentum in terms of , as seen in the discussion of the 2-D rotating system. 

In the spherical polar coordinate system the angle (p sets the position away from the X-axis measured parallel to theXEplane (Figure A9.1). The normal to this plane is clearly the Z-axis. So the rp angle can be used to describe that part of the rotation which is about the Z-axis, i.e. the Z-component of the angular momentum L. The angular wavefunction must also obey the differential equation... 

In practice, it is better to work with angular momentum in an angular coordinate system rather than in a rectilinear system. This means using the spherical polar coordinate system defined as in Figure 7.4 and transforming the operators in Equations 8.48 through 8.50. The coordinate transformation to and from spherical polar coordinates is... 

With Equation 8.53b, the operator is expressed in spherical polar coordinates and in terms of the angular momentum operator, L, to give... 

Transform the components of angular momentum Mx, My, and M, found in Probleni 12, to spherical polar coordinates. 

The spherical symmetry of the interaction implies, in particular, that the angular momentum of the relative motion is conserved. That is, since the angular momentum is a vector, both direction and magnitude are conserved quantities. The collision process will, accordingly take place in the plane defined by the initial values of the radius vector and the momentum vector. This implies that only two coordinates are required in order to describe the relative motion. These coordinates are chosen as the polar coordinates in the plane (r, 0). The scattering in the center-of-mass coordinate system is shown in Fig. 4.1.7. 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

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) ... 

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