--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.15.2 kernelspec: display_name: Python 3 language: python name: python3 --- ```{code-cell} ipython3 :tags: [hide-cell] import mmf_setup;mmf_setup.nbinit() import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL) %matplotlib inline import numpy as np, matplotlib.pyplot as plt ``` (sec:Deuteron)= Deuteron ======== :::{margin} Heuristically, the nuclear interaction can be described by the exchange of [pions][pion], which gives an energy scale of $m_\pi c^2 \approx 135\;\rm{MeV}$ and a length-scale of $\hbar c/m_{\pi} \approx 0.56\;\rm{fm}$. Whereas exchange of massless particles like the photon give rise to a Coulomb-like potential $V(r) \propto 1/r$, the exchange of massive particles yields a [Yukawa potential][]: \begin{gather*} V_{m}(r) = -g^2 \frac{e^{-\alpha m_{\pi} r /\hbar c}}{r}. \end{gather*} ::: ```{code-cell} :tags: [hide-input, margin] r = np.linspace(0.1, 3) V = -np.exp(-r)/r fig, ax = plt.subplots(figsize=(4,3)) ax.plot(r, V, label="Yukawa") ax.plot(r, -1/r, label="Coulomb") ax.legend() ax.set(xlabel=r"$r$ [$\hbar c /\alpha m_{\pi}$]", ylabel=r"$V(r)/g^2$"); ``` Here we say a little more about Example 8.3.3 in {cite}`Arfken:2013`. The key physical idea is that the nuclear potential is short-ranged. Technically this means that it falls off faster than the Coulomb potential $V \sim 1/r$, but Specifically, we show that in the short-range limit, the potential has a peculiar structure. The naïve choice of $V_0\delta^3(\vect{r})$ is too singular, but we need a combination of two parameters to get the correct short range boundary condition. [pion]: [Yukawa potential]: ### Example: Top-Hat Potential To see this, consider an explicit realization of a top-hap potential $V(r) = -V_0\Theta(r_0-r)$ with depth $-V_0$ and range $r_0$. The bound-states $E = -E_b$ of the radial wavefunction have the following solutions :::{margin} Here we use \begin{gather*} \frac{\hbar^2k^2}{2m} - V_0 = \frac{-\hbar^2 \kappa^2}{2m} = -E_b, \end{gather*} and the fact that both $u(r)$ and $u'(r)$ must be continuous at $r_0$. We also note that $kr_0 > \pi/2$ so let $kr_0 = \tfrac{\pi}{2} + \alpha$ and use $\cot(\tfrac{1}{2}\pi + \alpha) = -\tan(\alpha)$. ::: \begin{gather*} u(r) = \begin{cases} B\sin(k r) & r < r_0,\\ Ae^{-\kappa r} & r>r_0, \end{cases} \qquad \hbar \kappa = \sqrt{2mE_b}, \qquad \hbar k = \sqrt{2m(V_0 - E_b)},\\ B\sin(kr_0) = Ae^{-\kappa r_0}, \qquad kB\cos(kr_0) = -\kappa Ae^{-\kappa r_0}. \end{gather*} Combining these, we have the transcendental equation. \begin{gather*} \hbar k\cot(kr_0) = -\hbar k\tan(k r_0 - \tfrac{\pi}{2}) = -\hbar \kappa = -\sqrt{2mE_b},\\ kr_0 = \frac{\pi}{2} + \cot^{-1}\sqrt{\frac{V_0}{E_b}-1}. \end{gather*} Taking $V_0 \rightarrow \infty$, we can expand this to obtain the limiting behaviour in the short-range limit for fixed binding energy $E_b$: \begin{gather*} kr_0 = \frac{\pi}{2} + \sqrt{\frac{E_b}{V_0}} + \frac{1}{6}\sqrt{\frac{E_b}{V_0}}^3 + O(E_b/V_0)^5/2,\\ r_0 \rightarrow \frac{\pi}{2k} \approx \frac{\pi \hbar}{\sqrt{8mV_0}}\left( 1 + \frac{2}{\pi}\sqrt{\frac{E_b}{V_0}} + O\left(\frac{E_b}{V_0}\right) \right)\\ V_0 \rightarrow \frac{m\hbar^2}{2 r_0^2} \left( \pi^2 + 8\kappa r_0 + O(\kappa r_0)^2 \right). \end{gather*} :::{admonition} Details :class: dropdown Let $x = E_b/V_0 \rightarrow 0$. \begin{gather*} \cot^{-1} \sqrt{1/x - 1} = \sqrt{x} + \frac{x^{3/2}}{6} + O(x^{5/2})\\ \frac{1}{\sqrt{1-x}} = 1 + \frac{x}{2} + O(x^2)\\ \frac{\sqrt{2mV_0}}{\hbar}r_0 = \frac{\frac{\pi}{2} + \cot^{-1}\sqrt{1/x - 1}}{\sqrt{1-x}}\\ = \frac{\pi}{2} + \sqrt{x} + \frac{\pi}{4}x + \frac{2}{3}x^{3/2} + O(x^2)\\ = \frac{\pi}{2}\left(1 + \frac{2}{\pi}\sqrt{x} + \frac{1}{2}x + \frac{4}{3\pi}x^{3/2} + O(x^2)\right),\\ \frac{2}{\pi}\kappa r_0 = y = \sqrt{x} + \frac{2}{\pi}\sqrt{x}^2 + \frac{1}{2}\sqrt{x}^{3} + \frac{4}{3\pi}\sqrt{x}^{4} + O(\sqrt{x}^{5}),\\ \end{gather*} We can then use the [series reversion formula](https://mathworld.wolfram.com/SeriesReversion.html) to compute \begin{gather*} y = \sqrt{x} + a_2 \sqrt{x}^2 + a_3 \sqrt{x}^3 + \dots\\ \sqrt{x} = y + A_2y^2 + A_3y^3 + \cdots\\ \sqrt{x} = y - a_2y^2 + (2a_2^2 - a_3)y^3 + \dots = y - \frac{2}{\pi}y^2 + \left(\frac{8}{\pi^2} - \frac{1}{2}\right)y^3 + \dots. \end{gather*} Then we have \begin{gather*} V_0 = \frac{E_b}{x} = \frac{E_b}{(y+A_2y^2 + A_3y^3)^2} = \frac{E_b}{y^2(1+A_2y + A_3y^2)^2}\\ = \frac{E_b}{y^2}\left(1 - 2 A_2 y + (3A_2 - 2A_3)y^2 + (6A_2A_3 - 4A_2^3)y^3 + \cdots\right)\\ = \frac{E_b}{y^2}\left( 1 + \frac{4}{\pi} y + (1 - \tfrac{6}{\pi} - \tfrac{16}{\pi^2})y^2 + (\tfrac{6}{\pi} - \tfrac{64}{\pi^3})y^3 + \cdots\right)\\ = \frac{m\hbar^2 \kappa^2 \pi^2}{2\kappa^2 r_0^2} \left( 1 + \frac{8}{\pi^2}\kappa r_0 + (1 - \tfrac{6}{\pi} - \tfrac{16}{\pi^2})\tfrac{4}{\pi^2}(\kappa r_0)^2 + (\tfrac{6}{\pi} - \tfrac{64}{\pi^3})\tfrac{8}{\pi^3}(\kappa r_0)^3 + \cdots\right). \end{gather*} **Note: There is something wrong here with the higher order terms. Leading order works out.** ::: :::{margin} The point of Shina Tan's work {cite}`Tan:2005uq` is introduce generalized delta-like "selectors" $\lambda(\vect{r})$ and $l(\vect{r})$ that are zero everywhere except $\vect{r}=0$ and satisfy \begin{gather*} \int \d^3{\vect{r}} \Bigl(a\lambda(\vect{r}) + b l(\vect{r})\Bigr) = a,\\ \int \d^3{\vect{r}} \Bigl(a\tfrac{\lambda(\vect{r})}{4\pi r} + b\tfrac{l(\vect{r})}{4\pi r}\Bigr) = b,\\ \int \d^3{\vect{r}} \Bigl(a\uvect{r}\lambda(\vect{r}) + b \uvect{r}l(\vect{r})\Bigr) = 0. \end{gather*} These can be used to rigorously construct appropriate pseudo-potentials. ::: Note that this is a very peculiar limiting behaviour. The leading order behaviour has constant $V_0 \propto 1/r_0^2$ which is **independent of the binding energy** $E_b = \hbar^2\kappa^2/2m$. This is a two-dimensional delta-function: more singular than a single delta-function, but not as singular as the 3D delta-function discussed in {cite}`Lepage:1997`. Thus, in order to specify the universal low-energy behaviour of short-range potentials, one needs a special type of pseudo-potential that can be constructed using selectors {cite}`Tan:2005uq`. We can understand the effect here in two steps: 1. The second order term has constant $V_0 r_0$: this is a delta-function in 1D, and can give the kink in the wavefunction that develops as we take $r_0 \rightarrow 0$. However, representing the potential for $u(r)$ as $c\delta(r)$ by itself should have no effect since the radial wavefunction $u(0) = 0$ vanishes. 2. Thus, we need something **even stronger** to "kick" $u(0)$ to a finite value before the delta-function above can be used to specify the magnitude of the kink, and therefore fix the energy. This is the leading order piece with $V_0 r_0^2$ that is independent of $\kappa$ and $E_b$. This is why Tan needs a combination of two selectors {cite}`Tan:2005uq`. ```{code-cell} :tags: [hide-input] hbar = 1 m = 0.5 Eb = 1 kappa = np.sqrt(2*m*Eb) A = np.sqrt(kappa**3/np.pi) R = 2.0 r = np.linspace(0, R, 1000) u0 = A*np.exp(-kappa*r) fig, axs = plt.subplots(2, 1, sharex=True, height_ratios=(2, 1), gridspec_kw=dict(hspace=0)) ax, axr = axs V0_Es = [5, 10, 20, 40] for V0_E in V0_Es: V0 = V0_E * Eb k = np.sqrt(2*m*(V0 - Eb)) r0 = (np.pi/2 + np.arctan(1/np.sqrt(V0/Eb - 1)))/k r0_ = np.pi*hbar /np.sqrt(8*m*V0)*(1 + 2/np.pi*np.sqrt(Eb/V0)) # Limiting approximation y = V0/(m*hbar**2*np.pi**2/2/r0**2) c1 = 8/np.pi**2 c2 = (1- 6/np.pi - 16/np.pi**2)*4/np.pi**2 # This is wrong, it should be about 0.24103 #print(y, # (y - 1)/(c1*kappa*r0), # (y - 1 - c1*kappa*r0)/((kappa*r0)**2)) B = A*np.exp(-kappa*r0)/np.sin(k*r0) u = np.where(r [Bohr radius]: [Jacobi elliptic functions]: [Jupyter Book with Sphinx]: [Jupyter Book]: [Jupyter]: "Jupyter" [Jupytext]: "Jupyter Notebooks as Markdown Documents, Julia, Python or R Scripts" [Laplace-Beltrami operator]: [Liouville's Theorem]: [Manim Community]: [Markdown]: [MyST Cheatsheet]: [MyST]: "MyST - Markedly Structured Text" [MySt-NB]: [Rutherford scattering]: [Sphinx]: [angular momentum operator]: [glue]: [hydrogenic atoms]: [orthogonal polynomials]: ## See Also * [Physics 555: Bound States in the 1D Schrödinger Equation]( https://physics-555-quantum-technologies.readthedocs.io/en/latest/Notes/Shooting.html) [hydrogen atom]: [reduced Bohr radius]: [generalized Laguerre polynomial]: [spherical harmonics]: [finite difference methods]: [Dirichlet boundary conditions]: [Numerov's method]: [harmonic oscillator]: [spherical Bessel functions]: [optical theorem]: