Sakurai (1994)

Sakurai [19] is the standard graduate textbook on quantum mechanics, and its Chapter 3 is one of the clearest treatments of rotations. (Page and equation numbers below refer to the Revised Edition of 1994.) Sakurai's conventions are the ones adopted on our conventions pages, so this page mostly serves to make that agreement explicit and to test it.

Rotation operators

Sakurai is careful to state that a rotation "affects the physical system itself, [...] while the coordinate axes remain unchanged" (p. 154), and defines the rotation operator by

\[|α\rangle_R = 𝒟(R) |α\rangle,\]

"where $|α\rangle_R$ and $|α\rangle$ stand for the kets of the rotated and original system, respectively" (p. 156). For an infinitesimal rotation, Eq. (3.1.15) gives

\[𝒟(\hat{𝐧}, dϕ) = 1 - i \left( \frac{𝐉 \cdot \hat{𝐧}}{\hbar} \right) dϕ,\]

which is exactly our $U(e^{ϵ𝐮/2}) = 1 - iϵ L_𝐮$. His Euler angles (Sec. 3.3, p. 173) are defined as ours, and the matrix elements of the rotation operator define the $𝒟$ matrix [Eq. (3.5.42), p. 192]

\[𝒟^{(j)}_{m',m}(R) = \langle j, m' | \exp\left( \frac{-i 𝐉 \cdot \hat{𝐧} ϕ}{\hbar} \right) | j, m \rangle,\]

which obeys the representation property [Eq. (3.5.46)]

\[𝒟^{(j)}_{m'',m}(R_1 R_2) = \sum_{m'} 𝒟^{(j)}_{m'',m'}(R_1)\, 𝒟^{(j)}_{m',m}(R_2).\]

In terms of Euler angles [Eqs. (3.5.50)–(3.5.51), p. 194],

\[𝒟^{(j)}_{m',m}(α, β, γ) = \langle j, m' | \exp\left(\frac{-iJ_z α}{\hbar}\right) \exp\left(\frac{-iJ_y β}{\hbar}\right) \exp\left(\frac{-iJ_z γ}{\hbar}\right) | j, m \rangle = e^{-i(m'α + mγ)}\, d^{(j)}_{m',m}(β), \qquad d^{(j)}_{m',m}(β) = \langle j, m' | \exp\left(\frac{-iJ_y β}{\hbar}\right) | j, m \rangle,\]

and Wigner's formula for $d$ is given as [Eq. (3.8.33), p. 223]

\[d^{(j)}_{m',m}(β) = \sum_k (-1)^{k-m+m'} \frac{\sqrt{(j+m)!\,(j-m)!\,(j+m')!\,(j-m')!}} {(j+m-k)!\,k!\,(j-k-m')!\,(k-m+m')!} \left(\cos\frac{β}{2}\right)^{2j-2k+m-m'} \left(\sin\frac{β}{2}\right)^{2k-m+m'}.\]

Spherical harmonics

Sakurai writes the spherical harmonics as $Y_ℓ^m(θ, ϕ)$ (note the upper index $m$) and relates them to the $𝒟$ matrix by [Eq. (3.6.51), p. 203]

\[𝒟^{(ℓ)}_{m,0}(α, β, γ=0) = \sqrt{\frac{4π}{2ℓ+1}}\; \left. Y_ℓ^{m\ast}(θ, ϕ) \right|_{θ=β,\, ϕ=α},\]

which is our relation. Appendix A gives the explicit forms [Eq. (A.5.7), p. 451]

\[\begin{aligned} Y_0^0 &= \frac{1}{\sqrt{4π}}, & Y_1^{\pm 1} &= \mp\sqrt{\frac{3}{8π}} \sin θ\, e^{\pm iϕ}, & Y_1^0 &= \sqrt{\frac{3}{4π}} \cos θ, \\ Y_2^{\pm 2} &= \sqrt{\frac{15}{32π}} \sin^2 θ\, e^{\pm 2iϕ}, & Y_2^{\pm 1} &= \mp\sqrt{\frac{15}{8π}} \sin θ \cos θ\, e^{\pm iϕ}, & Y_2^0 &= \sqrt{\frac{5}{16π}} \left(3\cos^2 θ - 1\right), \end{aligned}\]

together with the conjugation relation [Eq. (A.5.7b)]

\[Y_ℓ^{-m}(θ, ϕ) = (-1)^m\, \left[Y_ℓ^m(θ, ϕ)\right]^\ast.\]

Implementing formulas

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

module Sakurai

We'll use some predefined utilities to make the code look more like the equations.

import ..ConventionsUtilities: 𝒾, ❗

Wigner's formula for $d$, Eq. (3.8.33). The sum runs over all $k$ for which the factorials have non-negative arguments, $\max(0, m-m') \leq k \leq \min(j+m, j-m')$.

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

The $𝒟$ matrix in terms of Euler angles, Eq. (3.5.50):

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

The spherical harmonics, obtained by solving Eq. (3.6.51) for $Y_ℓ^m$:

function Y(ℓ, m, θ::T, ϕ::T) where {T<:Real}    conj(√((2ℓ+1) / (4T(π))) * 𝒟(ℓ, m, 0, ϕ, θ, zero(T)))end

The explicit formulas of Eq. (A.5.7):

Y₀⁰(θ, ϕ) = 1 / √(4π)Y₁⁻¹(θ, ϕ) = +√(3/(8π)) * sin(θ) * exp(-𝒾*ϕ)Y₁⁰(θ, ϕ) = √(3/(4π)) * cos(θ)Y₁⁺¹(θ, ϕ) = -√(3/(8π)) * sin(θ) * exp(+𝒾*ϕ)Y₂⁻²(θ, ϕ) = √(15/(32π)) * sin(θ)^2 * exp(-2𝒾*ϕ)Y₂⁻¹(θ, ϕ) = +√(15/(8π)) * sin(θ) * cos(θ) * exp(-𝒾*ϕ)Y₂⁰(θ, ϕ) = √(5/(16π)) * (3cos(θ)^2 - 1)Y₂⁺¹(θ, ϕ) = -√(15/(8π)) * sin(θ) * cos(θ) * exp(+𝒾*ϕ)Y₂⁺²(θ, ϕ) = √(15/(32π)) * sin(θ)^2 * exp(+2𝒾*ϕ)

end  # module Sakurai

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 = 5

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 a modest grid of Euler angles.

αβγs = αβγrange(rng, Float64, 5)

First, Sakurai's own explicit spherical harmonics and conjugation relation are consistent with his general expression via $𝒟$:

for (θ, ϕ) ∈ θϕrange(rng)    @test Sakurai.Y₀⁰(θ, ϕ) ≈ Sakurai.Y(0, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₁⁻¹(θ, ϕ) ≈ Sakurai.Y(1, -1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₁⁰(θ, ϕ) ≈ Sakurai.Y(1, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₁⁺¹(θ, ϕ) ≈ Sakurai.Y(1, 1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₂⁻²(θ, ϕ) ≈ Sakurai.Y(2, -2, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₂⁻¹(θ, ϕ) ≈ Sakurai.Y(2, -1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₂⁰(θ, ϕ) ≈ Sakurai.Y(2, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₂⁺¹(θ, ϕ) ≈ Sakurai.Y(2, 1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Sakurai.Y₂⁺²(θ, ϕ) ≈ Sakurai.Y(2, 2, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    for ℓ ∈ 0:ℓₘₐₓ, m ∈ -ℓ:-1        @test Sakurai.Y(ℓ, m, θ, ϕ) ≈ (-1)^m * conj(Sakurai.Y(ℓ, -m, θ, ϕ)) atol=ϵₐ rtol=ϵᵣ    endend

Now the spherical harmonics agree with ours:

for (θ, ϕ) ∈ θϕrange(rng)    for (ℓ, Yˡ) ∈ SphericalFunctions.YlmCalculator(θ, ϕ, ℓₘₐₓ), m ∈ -ℓ:ℓ        @test Sakurai.Y(ℓ, m, θ, ϕ) ≈ Yˡ[m] atol=ϵₐ rtol=ϵᵣ    endend

The $d$ matrix agrees with ours:

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

And the $𝒟$ matrix agrees with ours — with no complex conjugation, transposition, or reordering of the Euler angles:

for (α, β, γ) ∈ αβγs    for (j, 𝔇ʲ) ∈ SphericalFunctions.DCalculator(α, β, γ, ℓₘₐₓ), m′ ∈ -j:j, m ∈ -j:j        @test Sakurai.𝒟(j, m′, m, α, β, γ) ≈ 𝔇ʲ[m′, m] atol=ϵₐ rtol=ϵᵣ    endend

These successful tests show that Sakurai's spherical harmonics and Wigner $d$ and $𝒟$ matrices agree with the corresponding functions defined by the SphericalFunctions package.

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