Edmonds (1960)

Edmonds [21] is a standard reference for the theory of angular momentum. (The equation numbers below refer to the 2016 Princeton reprint of the 1960 second edition, which is the edition cited in the bibliography.)

Euler angles and spherical coordinates

In Sec. 1.3 he actually does a fair job of defining the Euler angles. The upshot is that his definition agrees with ours, though he uses the "active" definition style. That is, the rotations are to be performed successively in order:

1. A rotation $α(0 \leq α < 2π)$ about the $z$-axis, bringing the frame of axes from the initial position $S$ into the position $S'$. The axis of this rotation is commonly called the vertical.

2. A rotation $β(0 \leq β < π)$ about the $y$-axis of the frame $S'$, called the line of nodes. Note that its position is in general different from the initial position of the $y$-axis of the frame $S$. The resulting position of the frame of axes is symbolized by $S''$.

3. A rotation $γ(0 \leq γ < 2π)$ about the $z$-axis of the frame of axes $S''$, called the figure axis; the position of this axis depends on the previous rotations $α$ and $β$. The final position of the frame is symbolized by $S'''$.

I would simply write the "$y$-axis of the frame $S'$" as $y'$, and so on. In quaternionic language, I would write these rotations as $\exp[γ 𝐤''/2]\, \exp[β 𝐣'/2]\, \exp[α 𝐤/2]$. But we also have

\[\exp[β 𝐣'/2] = \exp[α 𝐤/2]\, \exp[β 𝐣/2]\, \exp[-α 𝐤/2]\]

so we can just swap the $α$ rotation with the $β$ rotation while dropping the prime from $𝐣'$. We can do a similar trick swapping the $α$ and $β$ rotations with the $γ$ rotation while dropping the double prime from $𝐤''$. That is, an easy calculation shows that

\[\exp[γ 𝐤''/2]\, \exp[β 𝐣'/2]\, \exp[α 𝐤/2] = \exp[α 𝐤/2]\, \exp[β 𝐣/2]\, \exp[γ 𝐤/2],\]

which is precisely our definition.

The spherical coordinates are implicitly defined by this statement:

It should be noted that the polar coordinates $φ, θ$ with respect to the original frame $S$ of the $z$-axis in its final position are identical with the Euler angles $α, β$ respectively.

Again, this agrees with our definition.

Angular-momentum operators

His expression for the angular-momentum operator in Euler angles — Eq. (2.2.2) — agrees with ours:

\[\begin{aligned} L_x &= -i \hbar \left\{ -\frac{\cos α}{\tan β} \frac{\partial} {\partial α} - \sin α \frac{\partial} {\partial β} + \frac{\cos α}{\sin β} \frac{\partial} {\partial γ} \right\}, \\ L_y &= -i \hbar \left\{ -\frac{\sin α}{\tan β} \frac{\partial} {\partial α} + \cos α \frac{\partial} {\partial β} +\frac{\sin α}{\sin β} \frac{\partial} {\partial γ} \right\}, \\ L_z &= -i \hbar \frac{\partial} {\partial α}. \end{aligned}\]

(The corresponding restriction to spherical coordinates also precisely agrees with our results, with the extra factor of $\hbar$.)

Spherical harmonics

Equation (2.5.5) gives the spherical harmonics as

\[Y_{ℓm}(θ, φ) = \frac{(-1)^{ℓ+m}}{2^ℓ ℓ!} \sqrt{\frac{(2ℓ+1)\,(ℓ-m)!}{4π\,(ℓ+m)!}}\; \sin^m θ\, \frac{d^{ℓ+m}}{d(\cos θ)^{ℓ+m}} \sin^{2ℓ} θ\; e^{imφ},\]

which is the Condon–Shortley form written with $ℓ+m$ derivatives rather than $ℓ-m$, and we expect it to agree with ours.

Rotation matrices

Here is where the disagreement lies. Edmonds defines the rotation operator in Eq. (4.1.9) as

\[𝒟(α β γ) = \exp\left( \frac{iγ}{\hbar} J_z\right) \exp\left( \frac{iβ}{\hbar} J_y\right) \exp\left( \frac{iα}{\hbar} J_z\right),\]

and its matrix elements in Eq. (4.1.12) as

\[𝒟^{(j)}_{m'm}(α β γ) = e^{im'γ}\, d^{(j)}_{m'm}(β)\, e^{imα},\]

with the explicit formula of Eq. (4.1.15),

\[d^{(j)}_{m'm}(β) = \left[\frac{(j+m')!\,(j-m')!}{(j+m)!\,(j-m)!}\right]^{1/2} \sum_σ \binom{j+m}{j-m'-σ} \binom{j-m}{σ} (-1)^{j-m'-σ} \left(\cos\frac{β}{2}\right)^{2σ+m'+m} \left(\sin\frac{β}{2}\right)^{2j-2σ-m'-m}.\]

Compared with our $U(𝐑_{α,β,γ}) = e^{-iαL_z} e^{-iβL_y} e^{-iγL_z}$, Edmonds' operator is exactly $U(𝐑_{α,β,γ}^{-1}) = U(𝐑_{-γ,-β,-α})$: the signs in the exponents are reversed and the order of the angles is reversed. This is the natural consequence of his "active" description, in which the Euler angles move the frame of axes to its new position — that is, the rotation of the coordinates rather than of the field. Since $𝔇(𝐑^{-1}) = 𝔇(𝐑)^\dagger$, we expect

\[𝒟^{(j)}_{m'm}(α, β, γ)\big|_{\text{Edmonds}} = 𝔇^{(j)}_{m',m}(-γ, -β, -α) = \overline{𝔇^{(j)}_{m,m'}(α, β, γ)}, \qquad d^{(j)}_{m'm}(β)\big|_{\text{Edmonds}} = d^{(j)}_{m,m'}(β),\]

which the tests below confirm. Note that this is not simply the complex conjugate of our $𝔇$: the indices are also transposed. (Several later sources — e.g., SymPy — cite Edmonds for their conventions but transcribe him inexactly.)

Implementing formulas

We begin by writing code that implements the formulas from Edmonds. We encapsulate the formulas in a module so that we can test them against the SphericalFunctions package.

module Edmonds

We'll use some predefined utilities to make the code look more like the equations, and ForwardDiff to evaluate the derivatives in the angular-momentum operators.

import ..ConventionsUtilities: 𝒾, ❗, dʲsin²ᵏθdcosθʲimport ForwardDiff

Edmonds includes $\hbar$, so we will include it in the expressions, but we will set it to 1 to match the conventions of the SphericalFunctions package.

const ħ = 1

Equation (2.5.5). We capture the floating-point type T to ensure that we don't lose precision when converting π and the factorials to floating-point numbers.

function Y(ℓ, m, θ::T, φ::T) where {T<:Real}    (-1)^(ℓ+m) / T(2^ℓ * (ℓ)❗) * √T((2ℓ+1) * (ℓ-m)❗ / (4big(π) * (ℓ+m)❗)) *    sin(θ)^m * dʲsin²ᵏθdcosθʲ(j=ℓ+m, k=ℓ, θ=θ) * exp(𝒾 * m * φ)end

Equation (4.1.15). The sum runs over all $σ$ for which the binomial coefficients are nonzero, $\max(0, -m'-m) \leq σ \leq \min(j-m', j-m)$.

function d(j, m′, m, β::T) where {T<:Real}    √T((j+m′)❗ * (j-m′)❗ / ((j+m)❗ * (j-m)❗)) *    sum(        T(binomial(big(j+m), j-m′-σ) * binomial(big(j-m), σ)) * (-1)^(j-m′-σ)        * cos(β/2)^(2σ+m′+m) * sin(β/2)^(2j-2σ-m′-m)        for σ ∈ max(0, -m′-m):min(j-m′, j-m);        init=zero(T)    )end

Equation (4.1.12):

function 𝒟(j, m′, m, α, β, γ)    exp(𝒾 * m′ * γ) * d(j, m′, m, β) * exp(𝒾 * m * α)end

Equation (2.2.2), the angular-momentum operators in Euler angles. Note that we defined $\hbar = 1$ above. Each operator takes a function f(α, β, γ) and returns a new function, with the derivatives evaluated by forward-mode automatic differentiation.

∂α(f) = (α, β, γ) -> ForwardDiff.derivative(α′ -> f(α′, β, γ), α)∂β(f) = (α, β, γ) -> ForwardDiff.derivative(β′ -> f(α, β′, γ), β)∂γ(f) = (α, β, γ) -> ForwardDiff.derivative(γ′ -> f(α, β, γ′), γ)L_x(f) = (α, β, γ) -> -𝒾 * ħ * (-cos(α)*cot(β) * ∂α(f)(α, β, γ) - sin(α) * ∂β(f)(α, β, γ) + cos(α)/sin(β) * ∂γ(f)(α, β, γ))L_y(f) = (α, β, γ) -> -𝒾 * ħ * (-sin(α)*cot(β) * ∂α(f)(α, β, γ) + cos(α) * ∂β(f)(α, β, γ) + sin(α)/sin(β) * ∂γ(f)(α, β, γ))L_z(f) = (α, β, γ) -> -𝒾 * ħ * ∂α(f)(α, β, γ)

end  # module Edmonds

Tests

We can now test the functions against the equivalent functions from the SphericalFunctions package. We will need to test approximate floating-point equality, so we set absolute and relative tolerances (respectively) in terms of the machine epsilon:

ϵₐ = 100eps()ϵᵣ = 1000eps()

We only test up to

ℓₘₐₓ = 4

because the formulas are slow, and this will be sufficient to sort out any sign or normalization differences, which are the most likely source of error. For the same reason we use modest grids of points.

αβγs = αβγrange(rng, Float64, 5)

First, the spherical harmonics agree with ours. The formula involves $\sin^m θ$ with $m$ possibly negative, so we avoid the poles. Close to them, the formula divides a nearly vanishing derivative by a power of the small quantity $\sin θ$, which amplifies its rounding errors enormously — in Float64 arithmetic at $θ ≈ 10^{-8}$, to an absolute error of about $2 × 10^{-8}$ — so we evaluate it in BigFloat arithmetic.

for (θ, ϕ) ∈ θϕrange(rng, Float64, 7; avoid_poles=ϵₐ/40)    for (ℓ, Yˡ) ∈ SphericalFunctions.YlmCalculator(θ, ϕ, ℓₘₐₓ), m ∈ -ℓ:ℓ        @test Edmonds.Y(ℓ, m, big(θ), big(ϕ)) ≈ Yˡ[m] atol=ϵₐ rtol=ϵᵣ    endend

Edmonds' $d$ is the transpose of ours:

for β ∈ βrange(rng)    for (j, dʲ) ∈ SphericalFunctions.dCalculator(β, ℓₘₐₓ), m′ ∈ -j:j, m ∈ -j:j        @test Edmonds.d(j, m′, m, β) ≈ dʲ[m, m′] atol=ϵₐ rtol=ϵᵣ    endend

And Edmonds' $𝒟$ is our $𝔇$ of the inverse rotation — equivalently, the Hermitian conjugate of ours:

for (α, β, γ) ∈ αβγs    𝔇 = SphericalFunctions.DCalculator(α, β, γ, ℓₘₐₓ)    𝔇⁻¹ = SphericalFunctions.DCalculator(-γ, -β, -α, ℓₘₐₓ)  # the inverse rotation    for ((j, 𝔇ʲ), (_, 𝔇ʲ⁻¹)) ∈ zip(𝔇, 𝔇⁻¹), m′ ∈ -j:j, m ∈ -j:j        @test Edmonds.𝒟(j, m′, m, α, β, γ) ≈ 𝔇ʲ⁻¹[m′, m] atol=ϵₐ rtol=ϵᵣ        @test Edmonds.𝒟(j, m′, m, α, β, γ) ≈ conj(𝔇ʲ[m, m′]) atol=ϵₐ rtol=ϵᵣ    endend

Finally, his operators (2.2.2) agree with ours if they act on our $𝔇$ as our summary page says ours do: $L_z 𝔇^{(j)}_{m',m} = -m' 𝔇^{(j)}_{m',m}$ and $(L_x \pm i L_y) 𝔇^{(j)}_{m',m} = -\sqrt{(j \pm m')(j \mp m' + 1)}\, 𝔇^{(j)}_{m' \mp 1,m}$, where the lowered or raised function vanishes at the edges. We apply them to our $𝔇$ in the form $\overline{𝒟^{(j)}_{m m'}}$ just established, which automatic differentiation can evaluate. The operators involve $1/\sin β$, so we use a few generic Euler-angle triples away from the poles.

𝔇(j, m′, m, α, β, γ) = conj(Edmonds.𝒟(j, m, m′, α, β, γ))for (α, β, γ) ∈ [(0.7, 1.1, 2.3), (2.9, 0.4, 5.1), (4.0, 2.2, 0.3), (1.3, 2.9, 4.7)]    for (j, m′, m) ∈ ℓm′mrange(3)        f = (α, β, γ) -> 𝔇(j, m′, m, α, β, γ)        L₊f = Edmonds.L_x(f)(α, β, γ) + im * Edmonds.L_y(f)(α, β, γ)        L₋f = Edmonds.L_x(f)(α, β, γ) - im * Edmonds.L_y(f)(α, β, γ)        @test Edmonds.L_z(f)(α, β, γ) ≈ -m′ * f(α, β, γ) atol=ϵₐ rtol=ϵᵣ        @test L₊f ≈ (m′ > -j ? -√((j+m′) * (j-m′+1)) * 𝔇(j, m′-1, m, α, β, γ) : 0) atol=ϵₐ rtol=ϵᵣ        @test L₋f ≈ (m′ < j ? -√((j-m′) * (j+m′+1)) * 𝔇(j, m′+1, m, α, β, γ) : 0) atol=ϵₐ rtol=ϵᵣ    endend

These successful tests show that Edmonds' spherical harmonics and angular-momentum operators agree with ours, while his rotation matrices are those of the inverse rotation: the Hermitian conjugates of the $𝔇$ matrices defined by the SphericalFunctions package.

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