Griffiths (1995)

Griffiths' "Introduction to Quantum Mechanics" is probably the most common introductory text used in undergraduate physics programs, so it would be useful to compare. (Equation and table numbers refer to the first edition, 1995.) Griffiths uses the standard (physicists') spherical coordinates, and writes the spherical harmonics as $Y_ℓ^m$.

Spherical harmonics

Equation (4.27) gives the associated Legendre function as

\[P_{ℓ}^{m}(x) = (1-x^2)^{|m|/2} \left(\frac{d}{dx}\right)^{|m|} P_{ℓ}(x),\]

and (4.28) gives the Legendre polynomial as

\[P_{ℓ}(x) = \frac{1}{2^ℓ ℓ!} \left(\frac{d}{dx}\right)^ℓ (x^2-1)^ℓ.\]

Then, (4.32) gives the spherical harmonics as

\[Y_{ℓ}^{m}(θ, ϕ) = ϵ \sqrt{\frac{2ℓ+1}{4π} \frac{(ℓ-|m|)!}{(ℓ+|m|)!}} e^{imϕ} P_{ℓ}^{m}(\cos θ),\]

where $ϵ = (-1)^m$ for $m\geq 0$ and $ϵ = 1$ for $m\leq 0$. Since $P_ℓ^m$ depends only on $|m|$, this places the Condon–Shortley phase entirely in $ϵ$; the result is the standard Condon–Shortley convention, so we expect agreement with ours. In Table 4.2, he explicitly lists the first few spherical harmonics:

\[\begin{aligned} Y_{0}^{0} &= \left(\frac{1}{4π}\right)^{1/2},\\ Y_{1}^{0} &= \left(\frac{3}{4π}\right)^{1/2} \cos θ,\\ Y_{1}^{\pm 1} &= \mp \left(\frac{3}{8π}\right)^{1/2} \sin θ e^{\pm iϕ},\\ Y_{2}^{0} &= \left(\frac{5}{16π}\right)^{1/2} \left(3\cos^2θ - 1\right),\\ Y_{2}^{\pm 1} &= \mp \left(\frac{15}{8π}\right)^{1/2} \sin θ \cos θ e^{\pm iϕ},\\ Y_{2}^{\pm 2} &= \left(\frac{15}{32π}\right)^{1/2} \sin^2θ e^{\pm 2iϕ},\\ Y_{3}^{0} &= \left(\frac{7}{16π}\right)^{1/2} \left(5\cos^3θ - 3\cos θ\right),\\ Y_{3}^{\pm 1} &= \mp \left(\frac{21}{64π}\right)^{1/2} \sin θ \left(5\cos^2θ - 1\right) e^{\pm iϕ},\\ Y_{3}^{\pm 2} &= \left(\frac{105}{32π}\right)^{1/2} \sin^2θ \cos θ e^{\pm 2iϕ},\\ Y_{3}^{\pm 3} &= \mp \left(\frac{35}{64π}\right)^{1/2} \sin^3θ e^{\pm 3iϕ}. \end{aligned}\]

Angular-momentum operators

In Eqs. (4.127)—(4.129), he gives the angular-momentum operators in terms of spherical coordinates:

\[\begin{aligned} L_x &= \frac{\hbar}{i} \left( -\sin ϕ \frac{\partial} {\partial θ} - \cos ϕ \cot θ \frac{\partial} {\partial ϕ} \right), \\ L_y &= \frac{\hbar}{i} \left( \cos ϕ \frac{\partial} {\partial θ} - \sin ϕ \cot θ \frac{\partial} {\partial ϕ} \right), \\ L_z &= -i \hbar \frac{\partial} {\partial ϕ}, \end{aligned}\]

which agree (up to the factor of $\hbar$) with ours, since $\hbar/i = -i\hbar$. Griffiths does not discuss Wigner's $D$ matrices.

Implementing formulas

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

module Griffiths

We'll use some predefined utilities to make the code look more like the equations, including ∂ⁿ, which computes derivatives symbolically so that we can transcribe Eqs. (4.27) and (4.28) literally.

import ..ConventionsUtilities: 𝒾, ❗, ∂ⁿ

Equation (4.28):

function P(ℓ, x)    1 / (2^ℓ * (ℓ)❗) * ∂ⁿ(x -> (x^2 - 1)^ℓ, ℓ)(x)end

Equation (4.27):

function P(ℓ, m, x)    (1 - x^2)^(abs(m)/2) * ∂ⁿ(x -> P(ℓ, x), abs(m))(x)end

Equation (4.32). 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}    ϵ = m ≥ 0 ? (-1)^m : 1    ϵ * √T((2ℓ+1) * (ℓ-abs(m))❗ / (4big(π) * (ℓ+abs(m))❗)) * exp(𝒾 * m * ϕ) * T(P(ℓ, m, cos(θ)))end

Table 4.2:

Y₀⁰(θ, ϕ) = √(1/(4π))Y₁⁰(θ, ϕ) = √(3/(4π)) * cos(θ)Y₁⁺¹(θ, ϕ) = -√(3/(8π)) * sin(θ) * exp(+𝒾*ϕ)Y₁⁻¹(θ, ϕ) = +√(3/(8π)) * sin(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)Y₂⁻¹(θ, ϕ) = +√(15/(8π)) * sin(θ) * cos(θ) * exp(-𝒾*ϕ)Y₂⁺²(θ, ϕ) = √(15/(32π)) * sin(θ)^2 * exp(+2𝒾*ϕ)Y₂⁻²(θ, ϕ) = √(15/(32π)) * sin(θ)^2 * exp(-2𝒾*ϕ)Y₃⁰(θ, ϕ) = √(7/(16π)) * (5cos(θ)^3 - 3cos(θ))Y₃⁺¹(θ, ϕ) = -√(21/(64π)) * sin(θ) * (5cos(θ)^2 - 1) * exp(+𝒾*ϕ)Y₃⁻¹(θ, ϕ) = +√(21/(64π)) * sin(θ) * (5cos(θ)^2 - 1) * exp(-𝒾*ϕ)Y₃⁺²(θ, ϕ) = √(105/(32π)) * sin(θ)^2 * cos(θ) * exp(+2𝒾*ϕ)Y₃⁻²(θ, ϕ) = √(105/(32π)) * sin(θ)^2 * cos(θ) * exp(-2𝒾*ϕ)Y₃⁺³(θ, ϕ) = -√(35/(64π)) * sin(θ)^3 * exp(+3𝒾*ϕ)Y₃⁻³(θ, ϕ) = +√(35/(64π)) * sin(θ)^3 * exp(-3𝒾*ϕ)

end  # module Griffiths

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 = 6

because the symbolic derivatives in the formulas become expensive to compute at higher orders, and this will be sufficient to sort out any sign or normalization differences, which are the most likely source of error.

First, Griffiths' Table 4.2 agrees with his general formula (4.32):

for (θ, ϕ) ∈ θϕrange(rng, Float64, 7)    @test Griffiths.Y₀⁰(θ, ϕ) ≈ Griffiths.Y(0, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₁⁰(θ, ϕ) ≈ Griffiths.Y(1, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₁⁺¹(θ, ϕ) ≈ Griffiths.Y(1, 1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₁⁻¹(θ, ϕ) ≈ Griffiths.Y(1, -1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₂⁰(θ, ϕ) ≈ Griffiths.Y(2, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₂⁺¹(θ, ϕ) ≈ Griffiths.Y(2, 1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₂⁻¹(θ, ϕ) ≈ Griffiths.Y(2, -1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₂⁺²(θ, ϕ) ≈ Griffiths.Y(2, 2, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₂⁻²(θ, ϕ) ≈ Griffiths.Y(2, -2, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁰(θ, ϕ) ≈ Griffiths.Y(3, 0, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁺¹(θ, ϕ) ≈ Griffiths.Y(3, 1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁻¹(θ, ϕ) ≈ Griffiths.Y(3, -1, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁺²(θ, ϕ) ≈ Griffiths.Y(3, 2, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁻²(θ, ϕ) ≈ Griffiths.Y(3, -2, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁺³(θ, ϕ) ≈ Griffiths.Y(3, 3, θ, ϕ) atol=ϵₐ rtol=ϵᵣ    @test Griffiths.Y₃⁻³(θ, ϕ) ≈ Griffiths.Y(3, -3, θ, ϕ) atol=ϵₐ rtol=ϵᵣend

Now, the general formula agrees with ours:

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

This successful test shows that the spherical harmonics defined by Griffiths agree with the spherical harmonics defined by the SphericalFunctions package.

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