The solution of spherical function is given in the paper,and it is applied reasonably in the problem of eigenvalue of angular momentum square operator.
The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.
The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
Angular momentum theory provides a mathematical foundation for the rotation of the eigenvectors of the angular momentum operator, the spherical harmonics functions. A useful property of spherical harmonics is that their rotation in 3D space corresponds to rotation in the virtual space of their azimuthal degrees of freedom. In addition to that, spherical harmonics functions that have different orbital indices do not mix. Given that spherical harmonics functions have known rotation properties, and that any arbitrary function can be expanded in a series of spherical harmonics, this property allows the rotation of an electrostatic potential when it is expanded in the series of spherical harmonics. In this chapter, the exponential form of the rotation operator, the rotation matrix for a 3D rotation of the coordinate system, and the principles of rotation of vectors and functions are described. The chapter provides a detailed self-contained introduction to the theory of angular momentum, derives the commutative properties of the angular momentum operator, and explains shift operators, while minimizing references to quantum mechanics and entirely avoiding group theory.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.
Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.
The Fermion Spherical harmonics [Yℓm(θ,ϕ) for half-odd-integer ℓ and m - presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [Yℓm(θ,ϕ) for integer ℓ and m]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators M+ (M−) that ascend (descend) the sequence of harmonics over the values of m for a given value of ℓ, do not produce the expected result {\em in just one case}: when the value of m changes from ±1/2 to ∓1/2; i.e. when m changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take m outside the range: −ℓ≤m≤+ℓ. The unexpected result in the one case does not invalidate this scalar coordinate representation of spin angular momentum, because the eigenfunction property is essential for a valid quantum mechanical state, whereas ladder operators relating states with different eigenvalues are not essential, and are in fact known only for a few physical systems; that this coordinate representation of spin angular momentum differs from the abstract theory of angular momentum in this respect, is simply an interesting curiosity. This new representation of spin angular momentum is expected to find application in the theoretical description of physical systems and experiments in which the spin-angular momentum (and associated magnetic moment) of a particle is oriented in space, since the orientation is specifiable by the spherical polar angles, θ and ϕ.