fix(HJ): A.09 — Radian convention fix for action variables and Bohr-Sommerfeld

concepts/advanced/rigid-rotator-hj.tex

@@ -86,24 +86,24 @@ A useful substitution is $u = \cos\theta$, for which $\dd\theta = -\dd u/\sqrt{1
 \int \frac{\sqrt{L^2(1-u^2) - L_z^2}}{\sqrt{1-u^2}}\cdot \left(-\frac{\dd u}{\sqrt{1-u^2}}\right)
 = -\int \frac{\sqrt{(L^2-L_z^2) - L^2 u^2}}{1-u^2}\;\dd u.
 \]
-This has the structure of an elliptic integral in general. The denominator $(1-u^2)$ together with the quadratic radicand $\sqrt{(L^2-L_z^2) - L^2 u^2}$ prevents a simple elementary antiderivative. However, for the purpose of computing the action variable --- a closed-loop integral between turning points --- the explicit antiderivative is not needed. The turning points in $u$-space occur at $u = \pm L_z/L$, where the radicand vanishes. A second substitution $u = (L_z/L)\cos\psi$ converts the definite integral to a standard trigonometric form that evaluates in closed fashion, yielding $J_\theta = 2\pi(|L| - |L_z|)$ without computing an indefinite integral.}
+This has the structure of an elliptic integral in general. The denominator $(1-u^2)$ together with the quadratic radicand $\sqrt{(L^2-L_z^2) - L^2 u^2}$ prevents a simple elementary antiderivative. However, for the purpose of computing the action variable --- a closed-loop integral between turning points --- the explicit antiderivative is not needed. The turning points in $u$-space occur at $u = \pm L_z/L$, where the radicand vanishes. A second substitution $u = (L_z/L)\cos\psi$ converts the definite integral to a standard trigonometric form that evaluates in closed fashion, yielding $J_\theta = |L| - |L_z|$ without computing an indefinite integral.}
 
-\nt{Action-angle variables for the rigid rotator}{The two independent action variables are computed by integrating the conjugate momenta over their respective cycles. For the azimuthal coordinate $\phi$, which is $2\pi$-periodic:
+\nt{Action-angle variables for the rigid rotator}{The two independent action variables are computed by integrating the conjugate momenta over their respective cycles, each divided by $2\pi$. For the azimuthal coordinate $\phi$, which is $2\pi$-periodic:
 \[
-J_\phi = \oint p_\phi\,\dd\phi = \int_{0}^{2\pi} L_z\,\dd\phi = 2\pi L_z.
+J_\phi = \frac{1}{2\pi}\oint p_\phi\,\dd\phi = \frac{1}{2\pi}\int_{0}^{2\pi} L_z\,\dd\phi = L_z.
 \]
-Physically, $J_\phi = 2\pi L_z$ measures the vertical spin: the amount of angular momentum aligned with the symmetry axis. A larger $|L_z|$ means the motion stays closer to the equator circle. For the polar coordinate $\theta$, which oscillates between turning points, the full cycle traverses the range twice:
+Physically, $J_\phi = L_z$ equals the vertical spin: the amount of angular momentum aligned with the symmetry axis. A larger $|L_z|$ means the motion stays closer to the equator circle. For the polar coordinate $\theta$, which oscillates between turning points, the full cycle traverses the range twice:
 \[
-J_\theta = \oint p_\theta\,\dd\theta
+J_\theta = \frac{1}{2\pi}\oint p_\theta\,\dd\theta
-= 2\int_{\theta_{\min}}^{\theta_{\max}}\sqrt{L^2 - \frac{L_z^2}{\sin^2\theta}}\;\dd\theta
+= \frac{1}{\pi}\int_{\theta_{\min}}^{\theta_{\max}}\sqrt{L^2 - \frac{L_z^2}{\sin^2\theta}}\;\dd\theta
-= 2\pi\bigl(|L| - |L_z|\bigr).
+= |L| - |L_z|.
 \]
-Physically, $J_\theta = 2\pi\bigl(|L| - |L_z|\bigr)$ measures the tilt from the equator: the ``missing'' angular momentum that keeps the trajectory from being perfectly planar. When $|L| = |L_z|$ the tilt vanishes, $J_\theta = 0$, and the motion is confined to the equatorial circle. When $L_z = 0$ there is no preferred axis and $J_\theta = 2\pi |L|$, the full angular momentum goes into the polar oscillation. Inverting gives $L_z = J_\phi/(2\pi)$ and $|L| = (J_\theta + |J_\phi|)/(2\pi)$. The Hamiltonian in action variables is
+Physically, $J_\theta = |L| - |L_z|$ measures the tilt from the equator: the ``missing'' angular momentum that keeps the trajectory from being perfectly planar. When $|L| = |L_z|$ the tilt vanishes, $J_\theta = 0$, and the motion is confined to the equatorial circle. When $L_z = 0$ there is no preferred axis and $J_\theta = |L|$, the full angular momentum goes into the polar oscillation. Inverting gives $L_z = J_\phi$ and $|L| = J_\theta + |J_\phi|$. The Hamiltonian in action variables is
 \[
-H(J_\theta,J_\phi) = \frac{\bigl(J_\theta + |J_\phi|\bigr)^2}{8\pi^2 I}.
+H(J_\theta,J_\phi) = \frac{\bigl(J_\theta + |J_\phi|\bigr)^2}{2I}.
 \]}
 
-\nt{From classical action to quantum numbers}{The Bohr--Sommerfeld quantization rule imposes that each action variable must be an integer multiple of Planck\normalsize{}'s reduced constant: $J = n\hbar$. Applying this to the rigid rotator, the azimuthal action quantizes as $J_\phi = m\hbar\cdot 2\pi = 2\pi m\hbar$, which immediately gives $L_z = m\hbar$ with $m$ an integer. The polar action quantizes as $J_\theta = \ell_\theta\hbar\cdot 2\pi = 2\pi \ell_\theta\hbar$, and since $J_\theta = 2\pi(|L| - |L_z|)$ we obtain $|L| = (\ell_\theta + |m|)\hbar$. Defining the total angular-momentum quantum number $\ell = \ell_\theta + |m|$, the condition $\ell_\theta \ge 0$ becomes $\ell \ge |m|$. The semiclassical energy is $E = L^2/(2I) = \ell^2\hbar^2/(2I)$. The fully quantum-mechanical result from solving the angular Schrödinger equation replaces $\ell^2$ with $\ell(\ell+1)$, giving $E = \ell(\ell+1)\hbar^2/(2I)$. The factor $\ell(\ell+1)$ rather than $\ell^2$ emerges from the non-commutativity of $L_x$, $L_y$, and $L_z$ --- no quantum state can simultaneously have definite values of all three components, so the total angular momentum magnitude always exceeds $|L_z|$ by a half-integer step. The Bohr--Sommerfeld approach captures the ladder of energy levels, the integer structure of quantum numbers, and the constraint $\ell \ge |m|$ that limits how far the spin axis can tilt. However, it cannot produce the $+\ell$ correction inside $\ell(\ell+1)$. For large $\ell$ the semiclassical and quantum results agree well, since $\ell(\ell+1) \approx \ell^2$ when $\ell \gg 1$. This bridge from classical action variables to quantum angular momentum was the key step in old quantum theory and its subsequent replacement by matrix and wave mechanics.}
+\nt{From classical action to quantum numbers}{The Bohr--Sommerfeld quantization rule states that each action variable must be an integer multiple of Planck\normalsize{}'s reduced constant: $J_i = n_i\hbar$. Applying this to the rigid rotator, the azimuthal action quantizes as $J_\phi = m\hbar$ with $m$ an integer, giving directly $L_z = m\hbar$. The polar action quantizes as $J_\theta = \ell_\theta\hbar$ with $\ell_\theta$ a non-negative integer. Since $|L| = J_\theta + |J_\phi|$, we obtain $|L| = (\ell_\theta + |m|)\hbar$. Defining the total angular-momentum quantum number $\ell = \ell_\theta + |m|$, the condition $\ell_\theta \ge 0$ becomes $\ell \ge |m|$. Therefore $L^2 = |L|^2 = \ell^2\hbar^2$, and the semiclassical energy is $E = \ell^2\hbar^2/(2I)$. The fully quantum-mechanical result from solving the angular Schrödinger equation replaces $\ell^2$ with $\ell(\ell+1)$, giving $E = \ell(\ell+1)\hbar^2/(2I)$. The factor $\ell(\ell+1)$ rather than $\ell^2$ emerges from the non-commutativity of $L_x$, $L_y$, and $L_z$ --- no quantum state can simultaneously have definite values of all three components, so the total angular momentum magnitude always exceeds $|L_z|$ by a half-integer step. The Bohr--Sommerfeld approach captures the ladder of energy levels, the integer structure of quantum numbers, and the constraint $\ell \ge |m|$ that limits how far the spin axis can tilt. However, it cannot produce the $+\ell$ correction inside $\ell(\ell+1)$. For large $\ell$ the semiclassical and quantum results agree well, since $\ell(\ell+1) \approx \ell^2$ when $\ell \gg 1$. This bridge from classical action variables to quantum angular momentum was the key step in old quantum theory and its subsequent replacement by matrix and wave mechanics.}
 
 \cor{Equatorial orbit}{
 When the total angular momentum equals the absolute value of its $z$-component, $L = |L_z|$, the square root in the $\theta$-equation vanishes everywhere except at $\sin\theta = 1$. The polar momentum $p_\theta = \der{W_\theta}{\theta}$ is zero, so the motion is confined to the equator $\theta = \pi/2$. From Hamilton\normalsize{}'s equations, the azimuthal velocity is
@@ -119,17 +119,17 @@ representing uniform circular motion at angular speed $\omega = |L_z|/I$. This o
 \ex{Action-angle frequencies for the rigid rotator}{
 Using the Hamiltonian in action variables,
 \[
-H = \frac{\bigl(J_\theta + |J_\phi|\bigr)^2}{8\pi^2 I},
+H = \frac{\bigl(J_\theta + |J_\phi|\bigr)^2}{2I},
 \]
 the two frequencies are
 \[
 \omega_\theta = \pdv{H}{J_\theta}
-= \frac{J_\theta + |J_\phi|}{4\pi^2 I}
+= \frac{J_\theta + |J_\phi|}{I}
-= \frac{|L|}{2\pi I},
+= \frac{|L|}{I},
 \qquad
 \omega_\phi = \pdv{H}{J_\phi}
-= \pm\frac{J_\theta + |J_\phi|}{4\pi^2 I}
+= \pm\frac{J_\theta + |J_\phi|}{I}
-= \pm\frac{|L|}{2\pi I}.
+= \pm\frac{|L|}{I}.
 \]
 Here the sign conventionally matches the sign of $J_\phi$, so a negative $J_\phi$ gives retrograde azimuthal motion at the same frequency magnitude. The two frequencies are equal in magnitude, confirming the degeneracy noted above: every trajectory on the sphere is a closed orbit with rational frequency ratio $1:1$. For the special case $J_\theta = 0$ (equatorial orbit), the polar frequency is defined by continuity and the motion is purely azimuthal.}