Comparing conventions

Goldstein-Poole-Safko "Classical Mechanics"

Euler angles

1. \(\phi\) about \(z\)
2. \(\theta\) about \(x'\)
3. \(\psi\) about \(z''\)

Goldstein (p. 151) claims that this convention is widely used in celestial mechanics and applied mechanics, and frequently in molecular and solid-state physics. Goldstein also has an Appendix on the various conventions. I would write Goldstein's rotation as \begin{equation} \label{eq:GoldsteinEulerAngles} e^{\phi \vec{z}/2}\, e^{\theta \vec{x}/2}\, e^{\psi \vec{z}/2}. \end{equation} That's right, he uses \(x\)! I think this means that my coordinate \(\gamma\) would be related to his \(\psi\) according to \(\gamma = \psi - \pi/2\).

Wigner \(\mathfrak{D}\); harmonics

I know of nowhere that Goldstein uses either

Varshalovich et al.

"Quantum theory of angular momentum"

Edmonds (1974)

Euler angles

In Sec. 1.3, Edmonds gives rotations "to be performed successively in the order:"

1. A rotation \(\alpha\) about the \(z\) axis
2. A rotation \(\beta\) about the \(y'\) axis
3. A rotation \(\gamma\) about the \(z''\) axis

He notes that these are positive, right-handed rotations about the relevant axes, and the coordinate system is right-handed. Moreover, this rotation describes the rotation of a rigid body about a fixed point, where \(z\), \(y'\), and \(z''\) move with the body. He points out that the net rotation is equivalent to

1. A rotation \(\gamma\) about the \(z\) axis
2. A rotation \(\beta\) about the \(y\) axis
3. A rotation \(\alpha\) about the \(z\) axis

where the axes are fixed with respect to the inertial frame.

Wigner \(\mathfrak{D}\); harmonics

This is where things get ugly with Edmonds. He seems to switch from an active transformation to a passive one, but he still seems to have made an error in writing down the transformation matrix. Basically, all the angles should have their signs reversed. In this version of the book, Edmonds mentions the paper by Wolf (1969), which sorted through various conventions and pointed out an error in older versions of Edmonds. But I think there's still an error.

Devanathan

"Angular Momentum Techniques in Quantum Mechanics"

\begin{align*} R_{\text{Devanathan}}(\alpha, \beta, \gamma) &= e^{\gamma \vec{z}''/2}\, e^{\beta \vec{y}'/2}\, e^{\alpha \vec{z}/2} \\\\ &= e^{\gamma \vec{z}''/2}\, e^{\beta e^{\alpha \vec{z}/2}\, \vec{y}\, e^{-\alpha \vec{z}/2}/2}\, e^{\alpha \vec{z}/2} \\\\ &= e^{\gamma \vec{z}''/2}\, e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2}\, e^{-\alpha \vec{z}/2}\, e^{\alpha \vec{z}/2} \\\\ &= e^{\gamma \vec{z}''/2}\, e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2} \\\\ &= e^{\gamma e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2} \vec{z}\, e^{-\beta \vec{y}/2}\, e^{-\alpha \vec{z}/2}/2}\, e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2} \\\\ &= e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2}\, e^{\gamma \vec{z}/2}\, e^{-\beta \vec{y}/2}\, e^{-\alpha \vec{z}/2}\, e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2} \\\\ &= e^{\alpha \vec{z}/2}\, e^{\beta \vec{y}/2}\, e^{\gamma \vec{z}/2} \\\\ &= R_{\text{spherical}}(\gamma, \beta, \alpha) \end{align*}

Shankar

Sakurai

Wikipedia

Mathematica

The Euler angles correspond to what I would have considered the inverse rotation.

Sympy

Euler angles

The sympy.physics.quantum.spin.Rotation class uses the \(z''\)-\(y'\)-\(z\) convention (which the documentation refers to as the "passive \(z\)-\(y\)-\(z\)" convention). This basically means that I have to swap the \(\alpha\) and \(\gamma\) arguments to keep mine consistent with sympy: \begin{equation*} R_{\text{sympy}}(\alpha,\beta,\gamma) = R_{\text{spherical}}(\gamma,\beta,\alpha). \end{equation*}

\(\mathfrak{D}\) matrices

Sympy (the symbolic math package for python) implements the \(\mathfrak{D}\) matrices as the function sympy.physics.quantum.spin.WignerD. Note that the documentation swaps the symbols \(m\) and \(m'\) in their documentation, relative to the usual order. Nonetheless, the arguments to the function are in the standard order \((m',m)\). That is, if you call the function using what the rest of this module (and other items on this page) regard as \(m'\) and \(m\), you will get the expected result---up to the interpretation of the Euler angles. Don't be alarmed by the fact that the documentation lists the arguments as \((m,m')\).

alpha,beta,gamma = R.euler_angles()
Dsympy = sympy.physics.quantum.spin.Rotation.D(ell, mp, m, gamma, beta, alpha).doit().evalf(n=32).conjugate()
Dsf = sf.Wigner_D_element(R, ell, mp, m)
abs(Dsympy-Dsf) < 1e-13 # True for ell<29

Wigner

(As translated by J. J. Griffin)

The \(\mathfrak{D}\) matrix is given on page 167, Eq. (15.27).