A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.
A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.
The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.
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.
The recently established existence of spherical harmonic functions, Yℓm(θ,ϕ) for half-odd-integer values of ℓ and m, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles θs,ϕs, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, S2,Sz, are represented by the usual differential operators in spherical polar coordinates (commonly used for L2,Lz), and their electron-spin eigenfunctions are sinθsexp(±ϕs/2). This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, α,β, that ``integration over spin coordinates'' is a true integration (over the angles θs,ϕs). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.
Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.
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.