The hyperspherical harmonics

5.2 The hyperspherical harmonics

While there is only one HH with L = 0, namely ₀ = π^−3∕2, as L increases, the number of HH grows quite fast. There are N(L) = (L + 1)₅∕5! independent homogeneous polynomials of degree L with 6 variables, and N(L) −N(L− 2) independent linear combinations of these are harmonic. So, if one operates without care, one would end up with a very large number of HH, even if the cut-off L_max in the expansion (5.6) is small.

A complete set of HH is provided by the following functions which are eigenstates of the individual orbital momenta l_ρ and l_λ:

P = N (sin φ)lρ(cos φ)lλPlρ+1∕2,lλ+1∕2Y mρ(ω )Y mλ (ω ) , [L] n lρ ρ lλ λ

(5.10)

where one has introduced the Jacobi polynomials P_n^α,β [28] and the normalization coefficient

{ 2(2n + lρ + l + 2)n!(n + lρ + l + 1)!}1∕2 N = ------------λ-----------------λ------ ⋅ π [2(n + lρ) + 1]!![2(n + lλ ) + 1]!!

(5.11)

The integer n, which corresponds to the degree of the Jacobi polynomial, is submitted to the constraint

L = lρ + lλ + 2n .

(5.12)

Harmonics of given total (ordinary) angular momentum = ρ + λ (not to be confused with the grand orbital momentum L) are constructed by Clebsch–Gordan coupling of the HH (5.10). One should further restrict the expansion (5.6) to these HH having the desired permutation properties.

There are several ways of constructing HH of given angular momentum, parity and permutation symmetry. One method consists of using harmonic-oscillator wave functions and removing the Gaussian factor exp(−2 −2). More details will be provided in Section 8.2 where we come back to the link between the harmonic oscillator and the HH expansion.

A direct construction of symmetrized HH has been given by Simonov [40], who made great use of the complex vectors λ + iρ and λ−iρ to solve the Laplace equation (5.3). For the case of scalar harmonics (l^P = 0⁺), he arrived at the compact expressions

∘ ------ ν L-+-2- ν ν,0 2 V L = N ν π3 cos ανA Pn (1 − 2A ) ∘ ------ , W ν= L-+--2sin ανA νP ν,0(1 − 2A2) L π3 n

(5.13)

where the scalar variables A and α are introduced

λ 2 − ρ 2 + 2iλ ⋅ ρ Aei α = ------2-----------. λ + ρ 2

(5.14)

The integer ν runs from L∕2 to 0, in steps of 2, so that the degree of the Jacobi polynomial,

( ) n = 1- L-− ν , 2 2

(5.15)

remains acceptable, i.e., n ≥ 0. The normalization factor is N_ν = 1 if ν≠0 and N_ν = 2^−1∕2 if ν = 0. All permutation properties of the above harmonics are contained in the cos αν or sin αν factors: if ν = 3ν′ (ν′ integer) then _L^ν is symmetric and _L^ν, if non-vanishing, antisymmetric; otherwise, _L^ν and _L^ν form a pair of mixed symmetry.

In principle, one can write down the HH of any given angular momentum l and projection l_z = M, parity P and permutation properties. Barma and Mandelzweig [54], using earlier work [57] on the permutation group, propose

∑ l P [L] ∝ exp iα ν DlMm (α1, α2,α3)Fm (A ) , m=− l

(5.16)

where α₁,α₂ and α₃ are the Euler angles defining the position of the particle plane (,) with respect to fixed axes and are thus expressable in terms of ωˆρ and ˆωλ ; D_Mm^l are the Wigner rotation functions; F_m(A) are given by a differential equation that is deduced from the original equation (5.3) or (5.4). However, no concise expression as simple as (5.13) can be exhibited for general l.

[next] [prev] [prev-tail] [front] [up]