Potency and safety analysis of hemp-derived delta-9 products: The hemp vs. cannabis demarcation problem
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In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics , which are well-defined at the origin and the irregular solid harmonics , which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:
Derivation, relation to spherical harmonics
Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that is a (smooth) function , we can write the Laplace equation in the following form
It is known that spherical harmonics Y^{m}
_{ℓ} are eigenfunctions of l^{2}:
Substitution of Φ(r) = F(r) Y^{m}
_{ℓ} into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,
The particular solutions of the total Laplace equation are regular solid harmonics:
Racah's normalization
Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions
Addition theorems
The translation of the regular solid harmonic gives a finite expansion,
The similar expansion for irregular solid harmonics gives an infinite series,
The addition theorems were proved in different manners by several authors.^{[1]}^{[2]}
Complex form
The regular solid harmonics are homogeneous, polynomial solutions to the Laplace equation . Separating the indeterminate and writing , the Laplace equation is easily seen to be equivalent to the recursion formula
If we combine the degree basis and the degree basis with the recursion formula, we obtain a basis of the space of harmonic, homogeneous polynomials (in three variables this time) of degree consisting of eigenvectors for (note that the recursion formula is compatible with the -action because the Laplace operator is rotationally invariant). These are the complex solid harmonics:
Plugging in spherical coordinates , , and using one finds the usual relationship to spherical harmonics with a polynomial , which is (up to normalization) the associated Legendre polynomial, and so (again, up to the specific choice of normalization).
Real form
By a simple linear combination of solid harmonics of ±m these functions are transformed into real functions, i.e. functions . The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order in x, y, z. The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical atomic orbitals and real multipole moments. The explicit Cartesian expression of the real regular harmonics will now be derived.
Linear combination
We write in agreement with the earlier definition
The following expression defines the real regular solid harmonics:
z-dependent part
Upon writing u = cos θ the m-th derivative of the Legendre polynomial can be written as the following expansion in u
(x,y)-dependent part
Consider next, recalling that x = r sin θ cos φ and y = r sin θ sin φ,
In total
List of lowest functions
We list explicitly the lowest functions up to and including ℓ = 5. Here
The lowest functions and are:
m | A_{m} | B_{m} |
---|---|---|
0 | ||
1 | ||
2 | ||
3 | ||
4 | ||
5 |
References
- Steinborn, E. O.; Ruedenberg, K. (1973). "Rotation and Translation of Regular and Irregular Solid Spherical Harmonics". In Lowdin, Per-Olov (ed.). Advances in quantum chemistry. Vol. 7. Academic Press. pp. 1–82. ISBN 9780080582320.
- Thompson, William J. (2004). Angular momentum: an illustrated guide to rotational symmetries for physical systems. Weinheim: Wiley-VCH. pp. 143–148. ISBN 9783527617838.