From 08c45e7b808fef27ae1c3f6d18efe2a9fc3caf21 Mon Sep 17 00:00:00 2001 From: Krishna Ayyalasomayajula Date: Sat, 2 May 2026 13:07:44 -0500 Subject: [PATCH] =?UTF-8?q?fix(HJ):=20A.13=20=E2=80=94=20Radian=20conventi?= =?UTF-8?q?on=20fix=20for=20action=20variables?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- concepts/advanced/kepler-coulomb-hj.tex | 39 ++++++++++++------------- 1 file changed, 18 insertions(+), 21 deletions(-) diff --git a/concepts/advanced/kepler-coulomb-hj.tex b/concepts/advanced/kepler-coulomb-hj.tex index 82714ca..895561f 100644 --- a/concepts/advanced/kepler-coulomb-hj.tex +++ b/concepts/advanced/kepler-coulomb-hj.tex @@ -50,25 +50,25 @@ These match the gravitational Kepler equations exactly, with $k$ playing the rol The Coulomb HJ equation is structurally identical to the gravitational Kepler problem treated in A.08 (kepler--hj.tex). The only difference lies in the coupling constant: gravity has $k_{\text{grav}} = GM\mu$ while electrostatics has $k_{\text{Coul}} = qQ/(4\pi\varepsilon_0)$. Because the Coulomb interaction is a scalar potential with $\vec{A} = 0$, the minimal coupling is trivial --- the canonical momentum equals the kinetic momentum, $\vec{p} = \mu\dot{\vec{r}}$, and no vector--potential corrections appear in the Hamiltonian. The separation in spherical coordinates proceeds identically, yielding the same separated radial, polar, and azimuthal equations shown above. The three action variables $J_r$, $J_\theta$, and $J_\phi$ retain the exact integral structure from the gravitational case; all results for orbits, action--angle variables, and frequencies carry over with the replacement $GM\mu \to k$.} \nt{From classical action to quantum numbers}{ -In the Hamilton--Jacobi formalism, an action variable is defined as the phase--space integral of a canonical momentum over one complete cycle of its conjugate coordinate: $J = \oint p\,\dd q$. This construction assigns a single number to each degree of freedom that measures the area enclosed by the orbit in phase space. Planck introduced a fundamental constant $h = 6.626\times 10^{-34}\,\mathrm{J\!\cdot\!s}$ to explain blackbody radiation. The reduced Planck constant $\hbar = h/(2\pi) = 1.055\times 10^{-34}\,\mathrm{J\!\cdot\!s}$ is the natural quantum of angular momentum. Bohr and Sommerfeld proposed that classical action variables should be restricted to integer multiples of $h$: +In the Hamilton--Jacobi formalism, an action variable is defined as the phase--space integral of a canonical momentum over one complete cycle of its conjugate coordinate: $J = \frac{1}{2\pi}\oint p\,\dd q$. This construction assigns a single number to each degree of freedom that measures the area enclosed by the orbit in phase space, normalized by $2\pi$. Planck introduced a fundamental constant $h = 6.626\times 10^{-34}\,\mathrm{J\!\cdot\!s}$ to explain blackbody radiation. The reduced Planck constant $\hbar = h/(2\pi) = 1.055\times 10^{-34}\,\mathrm{J\!\cdot\!s}$ is the natural quantum of angular momentum. Bohr and Sommerfeld proposed that classical action variables should be restricted to integer multiples of $\hbar$: \[ -J_i = n_i h = n_i\cdot 2\pi\hbar, +J_i = n_i\hbar, \] -where $n_i$ is a positive integer called a quantum number. The physical idea is that phase space is granular at atomic scales: only classical orbits whose action variables satisfy this condition survive in the quantum theory. Why quantize action? The action variable sets the scale of an orbit in phase space, and imposing $J = nh$ is equivalent to requiring an integer number of de Broglie wavelengths to fit along the orbit, producing a standing wave. The simplest action variable to evaluate is the azimuthal action. For any central potential, the canonical momentum $p_\phi = \pdv{\mcS}{\phi} = L_z$ is a constant of motion. Because $L_z$ does not vary with $\phi$, the line integral reduces to a product: +where $n_i$ is a positive integer called a quantum number. The physical idea is that phase space is granular at atomic scales: only classical orbits whose action variables satisfy this condition survive in the quantum theory. Why quantize action? The action variable sets the scale of an orbit in phase space, and imposing $J = n\hbar$ is equivalent to requiring an integer number of de Broglie wavelengths to fit along the orbit, producing a standing wave. The simplest action variable to evaluate is the azimuthal action. For any central potential, the canonical momentum $p_\phi = \pdv{\mcS}{\phi} = L_z$ is a constant of motion. Because $L_z$ does not vary with $\phi$, the integral evaluates directly: \[ -J_\phi = \oint p_\phi\,\dd\phi -= \int_{0}^{2\pi} L_z\,\dd\phi -= L_z\int_{0}^{2\pi}\dd\phi -= 2\pi L_z. +J_\phi = \frac{1}{2\pi}\oint p_\phi\,\dd\phi += \frac{1}{2\pi}\int_{0}^{2\pi} L_z\,\dd\phi += \frac{1}{2\pi}L_z\int_{0}^{2\pi}\dd\phi += L_z. \] -Bohr--Sommerfeld quantization $J_\phi = n_\phi h = n_\phi\cdot 2\pi\hbar$ then immediately gives $L_z = n_\phi\hbar$. The $z$-component of angular momentum is quantized in units of $\hbar$, exactly as full quantum mechanics predicts.} +Bohr--Sommerfeld quantization $J_\phi = n_\phi\hbar$ then immediately gives $L_z = n_\phi\hbar$. The $z$-component of angular momentum is quantized in units of $\hbar$, exactly as full quantum mechanics predicts.} \nt{Action--angle quantization and the hydrogen spectrum}{ -For the $1/r$ potential the three action variables are $J_\phi = 2\pi L_z$, $J_\theta = 2\pi(L - |L_z|)$, and $J_r = 2\pi(-L + k\sqrt{\mu/(2|E|)})$. The polar and radial actions $J_\theta$ and $J_r$ have the same integral structure as those derived for the gravitational Kepler problem in A.08, with the only change being the coupling constant $k$. Their sum eliminates the angular--momentum dependence: +For the $1/r$ potential the three action variables are $J_\phi = L_z$, $J_\theta = L - |L_z|$, and $J_r = -L + k\sqrt{\mu/(2|E|)}$. The polar and radial actions $J_\theta$ and $J_r$ have the same integral structure as those derived for the gravitational Kepler problem in A.08, with the only change being the coupling constant $k$. Their sum eliminates the angular--momentum dependence: \[ -J_{\mathrm{tot}} = J_r + J_\theta + J_\phi = 2\pi k\sqrt{\frac{\mu}{2|E|}}. +J_{\mathrm{tot}} = J_r + J_\theta + J_\phi = k\sqrt{\frac{\mu}{2|E|}}. \] -Bohr--Sommerfeld quantization requires $J_{\mathrm{tot}}/2\pi = n\hbar$, where $n$ is the principal quantum number. Setting $k\sqrt{\mu/(2|E|)} = n\hbar$ and solving for energy: +Bohr--Sommerfeld quantization requires $J_{\mathrm{tot}} = n\hbar$, where $n$ is the principal quantum number. Setting $k\sqrt{\mu/(2|E|)} = n\hbar$ and solving for energy: \[ |E| = \frac{\mu k^2}{2n^2\hbar^2}, \qquad @@ -84,8 +84,8 @@ For an electron bound to a proton, the electrostatic coupling constant is $k = e \begin{enumerate}[label=(\alph*)] \item For a bound orbit with semimajor axis $a_0 = 0.529\times 10^{-10}\,\mathrm{m}$ (the Bohr radius), find the orbital energy $E = -k/(2a_0)$ from the HJ action--angle formalism. Express the result in both joules and electron volts. -\item Find the angular momentum $L = \sqrt{\mu k a_0}$ for this circular orbit and compute the total action $J_{\mathrm{tot}} = 2\pi L$. Compare the energy found in part (a) to the quantum $n=1$ energy of $-13.6\,\mathrm{eV} = -2.18\times 10^{-18}\,\mathrm{J}$. -\item Using the Bohr--Sommerfeld quantization $J_{\mathrm{tot}} = n h$ with $n=1$, verify that the quantized energy $E_1 = -\mu k^2/(2\hbar^2)$ matches $-13.6\,\mathrm{eV}$. +\item Find the angular momentum $L = \sqrt{\mu k a_0}$ for this circular orbit and compute the total action $J_{\mathrm{tot}} = L$. Compare the energy found in part (a) to the quantum $n=1$ energy of $-13.6\,\mathrm{eV} = -2.18\times 10^{-18}\,\mathrm{J}$. +\item Using the Bohr--Sommerfeld quantization $J_{\mathrm{tot}} = n\hbar$ with $n=1$, verify that the quantized energy $E_1 = -\mu k^2/(2\hbar^2)$ matches $-13.6\,\mathrm{eV}$. \end{enumerate}} \sol \textbf{Part (a).} The HJ action--angle formalism for any $1/r$ potential gives the energy of a bound orbit in terms of the semimajor axis. The binding energy follows from the virial relation $2T + V = 0$ for a $1/r$ potential, giving @@ -132,20 +132,17 @@ L = \sqrt{1.111\times 10^{-68}}\,\mathrm{J\!\cdot\!s} \] This equals the reduced Planck constant $\hbar = 1.055\times 10^{-34}\,\mathrm{J\!\cdot\!s}$. The total action is \[ -J_{\mathrm{tot}} = 2\pi L = 2\pi\hbar = h = 6.63\times 10^{-34}\,\mathrm{J\!\cdot\!s}. +J_{\mathrm{tot}} = L = \hbar = 1.055\times 10^{-34}\,\mathrm{J\!\cdot\!s}. \] -The total action equals Planck\normalsize{}'s constant $h$. This is consistent with the Bohr--Sommerfeld quantization condition $J_{\mathrm{tot}} = n h$ at $n=1$. +The total action equals the reduced Planck constant $\hbar$. This is consistent with the Bohr--Sommerfeld quantization condition $J_{\mathrm{tot}} = n\hbar$ at $n=1$. Comparing energies: part (a) yielded $E = -2.18\times 10^{-18}\,\mathrm{J} = -13.6\,\mathrm{eV}$, which is exactly the stated quantum $n=1$ energy. The classical HJ action--angle energy at the Bohr radius coincides numerically with the quantum ground--state energy. \textbf{Part (c).} The Bohr--Sommerfeld quantization condition reads \[ -J_{\mathrm{tot}} = n h = n\cdot 2\pi\hbar. -\] -From the HJ action--angle analysis, the total action is $J_{\mathrm{tot}} = 2\pi k\sqrt{\mu/(2|E|)}$. Equate the two expressions: -\[ -2\pi k\sqrt{\frac{\mu}{2|E|}} = 2\pi n\hbar, +J_{\mathrm{tot}} = n\hbar. \] +From the HJ action--angle analysis, the total action is $J_{\mathrm{tot}} = k\sqrt{\mu/(2|E|)}$. Equate the two expressions: \[ k\sqrt{\frac{\mu}{2|E|}} = n\hbar. \] @@ -194,7 +191,7 @@ E = -2.18\times 10^{-18}\,\mathrm{J} = -13.6\,\mathrm{eV}, L = 1.055\times 10^{-34}\,\mathrm{J\!\cdot\!s} = \hbar, \] \[ -J_{\mathrm{tot}} = h = 6.63\times 10^{-34}\,\mathrm{J\!\cdot\!s}, +J_{\mathrm{tot}} = \hbar = 1.055\times 10^{-34}\,\mathrm{J\!\cdot\!s}, \qquad E_1 = -\frac{\mu k^2}{2\hbar^2} = -13.6\,\mathrm{eV}. \]