Eigenfunctions of angular momentum

Equation (12-13) may be generalized to include any eigenfunction of angular momentum [Pg.731]

For the double representations, the basis functions that are eigenfunctions of angular momentum j and projection m on the z axis are noted (j, m). For T7, the basis functions transform like the products 4>(j, m) of the basis functions of T6 and those of T2 and they are noted T6 x T2. 

Edmonds classic text on the theory of angular momentum is recommended for further studies on this subject. Our presentation was very brief in many respects. For instance, we have neither introduced ladder operators to explicitly constmct the eigenfunctions of angular momentum eigenvalue equations nor have we deduced the expression for total angular momentum eigenstates in the coupled product basis. Edmonds book fills all these gaps. 

Returning now to the rigid-body rotational Hamiltonian shown above, there are two special cases for which exact eigenfunctions and energy levels can be found using the general properties of angular momentum operators. 

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

Thus () is an eigenvalue of Lz with eigenvalue The angle-dependent part of the wave equation is seen to contain wave functions which are eigenfunctions of both the total angular momentum as well as the component of angular momentum along the polar axis. 

In the presence of an electric field, this commutation is no longer true [H, J ] 7 0, though if the field is applied along the z axis, then the Hamiltonian does commute with the z component of angular momentum [H, 7 ] = 0. In this case the eigenfunctions of the Hamiltonian must also be eigenfunctions of J. This is met by functions such as those used in this work ... 

Because C y commutes with the Hamiltonian and C y can be written in terms of Lz, Lz must commute with the Hamiltonian. As a result, the molecular orbitals (f> of a linear molecule must be eigenfunctions of the z-component of angular momentum Lz ... 

The familiar theory of angular momentum, based on spherical harmonics, eigenfunctions of and operators of angular momentum, also uses the ladder operators L , which for a given value of i connect all the 2 - -1 successive states m) with m = —I, -I + 1,..., —1,0,1,..., -1,1. Section 4.2.1 illustrates the counterpart for the spheroconal harmonics or the spherical harmonics with well-defined parities under the application of the operators... 

The vector p defines a new quantization direction (40) of angular momentum as a linear combination (41) of the eigenfunctions i/ o d which is equivalent to a rotation of the axes (28-30). The contribution has no effect on the direction. 

Therefore we find that Yj is an eigenfunction of with the eigenvalue mh. The z component of the angular momentum is therefore quantized the quantum number is m = 0, + 1, 2, . Again, precise values of the z component of angular momentum are permitted since the angle 0 is totally unspecifiable. Repeating the application of on Eq. (21.75), we obtain... 

In Section 3.3 we found the eigenfunctions and eigenvalues for the linear-momentum operator p. In this section we consider the same problem for the angular momentum of a particle. Angular momentum is important in the quantum mechanics of atomic structure. We begin by reviewing the classical mechanics of angular momentum. 

To understand the origin of this power of 3 in a more fundamental way, let us consider the generalisation of our analysis to a hydrogenic wavefunction of angular momentum / in I spatial dimensions. For simplicity, we take the principal quantum number n = / + 1. Then such a hydrogenic eigenfunction has in configuration space the form... 

A tensor operator is thus created from U and V, and the analogy to the coupling of angular momentum eigenfunctions is underlined by the presence of the Clebsch-Gordan coefficient in (43) or, equivalently, the 3/ symbol in (45). If, on the other hand, we are given a double tensor, this may be represented as a sum of tensor products by... 

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