---
jupytext:
formats: ipynb,md:myst
text_representation:
extension: .md
format_name: myst
format_version: 0.13
jupytext_version: 1.13.8
kernelspec:
display_name: Python 3 (phys-581)
language: python
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---
```{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:RG-SEQ)=
How to Renormalize The Schrödinger Equation
===========================================
Here we work through the example discussed in {cite:p}`Lepage:1997`, which uses the
example of bound states in a spherically symmetric potential, with Coulomb-like
behaviour for large radii $r\rightarrow \infty$, but with "unknown" corrections at short
distances $r\rightarrow 0$:
\begin{gather*}
\left(\frac{-\hbar^2\nabla^2}{2m} + V(r) - E\right)\Psi(r, \Omega) = 0, \qquad
\lim_{r\rightarrow \infty} V(r) \rightarrow \frac{-\alpha}{r}.
\end{gather*}
To do this, I suggest you complete the following tasks:
1. Get a method to numerically solve the radial Schrödinger equation. You will use this
to compute your "experimental" data. Note: this is a little challenging due to the
nature of the Coulomb potential being singular at the origin, and long-ranged
(falling off slowly $V(r) \sim 1/r$ at long distances). The following notes have
many hints and suggestions, but please try to use them only as needed.
```{toctree}
---
maxdepth: 2
titlesonly:
glob:
---
RG_SEQ/*
```
2. Perform the perturbative analysis discussed in {cite:p}`Lepage:1997` but following
with your own data. If there is anything you don't know how to do, please ask on the
[Hypothes.is document](https://hyp.is/ITZMcI5tEe6EBmPaQth57Q/emailwsu-my.sharepoint.com/personal/m_forbes_wsu_edu/Documents/Courses/Physics%20581%20Standard%20Model/1997_Lepage-HowToRenormalizeTheSchrodingerEquation.pdf?CT=1701230953632&OR=ItemsView) so I can help fill in background in class or online.
## How to Solve the Schrödinger Equation
To easily work through the discussion in {cite:p}`Lepage:1997`, we must be able to
formulate and solve the Schrödinger equation for spherically symmetric potentials.
### Radial Equation
Spherical symmetry allows use to express this as a simple 1D boundary value problem
(BVP) for the radial equation:
:::{margin}
With the usual spherical coordinates $\phi$ and $\theta$, and
angular momentum quantum numbers $l$ and $m$, we have
| $d$ | $\Omega$ | $\lambda$ |
|-----|------------------|-----------|
| $2$ | $\phi$ | $\abs{m}$ |
| $3$ | $(\phi, \theta)$ | $l$ |
for $d=2$ dimensions (i.e. cylindrical coordinates, but no $z$ dependence) and $d=3$
dimensions respectively. The total orbital [angular momentum operator] is related to the
[Laplace-Beltrami operator] by
\begin{gather*}
\op{L}^2 = \hbar^2\op{\Delta}_{S^{d-1}}
\end{gather*}
which has eigenvalues $\hbar^2 m^2$ in $d=2$ (i.e. $\op{L}^2 \equiv \op{L}_z^2$ and
$\hbar^2 l (l - 1)$ in $d=3$.
:::
Our aim is to satisfy the Schrödinger equation for central potentials in $d$-dimensions,
which we can do in the usual way by expressing the wavefunction $\Psi(r, \Omega) =
\psi(r)Y_{\lambda}(\Omega)$ in terms of the radial wavefunction $u(r)$ and
the appropriate generalized spherical harmonics $Y_{\lambda}(\Omega)$:
\begin{gather*}
\left(\frac{-\hbar^2\nabla^2}{2m} + V(r) - E\right)\Psi(r, \Omega) = 0, \\
\left[
\frac{\hbar^2}{2m}
\underbrace{
\left(-\diff[2]{}{r} + \frac{\nu^2 - 1/4}{r^2}\right)
}_{\op{K}} +
V(r)\right]u(r) = E u(r),\\
u(r) = r^{(d-1)/2}\psi(r), \qquad
\nu = \lambda + \frac{d}{2} - 1.
\end{gather*}
:::::{admonition} Details
:class: dropdown
Here $\Omega$ is the generalized solid angle. Rotational invariance implies that
angular momentum is a good quantum number, so all eigenstates can be factored $\Psi(r, \Omega) =
\psi(r)Y_{\lambda}(\Omega)$ where $Y_{\lambda}(\Omega)$ is the generalized spherical
harmonic on the ($d-1$)-dimensional sphere and $\lambda \in \{0, 1, \dots\}$ is the
generalized angular momentum:
\begin{gather*}
\nabla^2 = \frac{1}{r^{d-1}}\diff{}{r}\left(r^{d-1} \diff{}{r}\right)
+ \frac{1}{r^2}\Delta_{S^{d-1}}, \\
\Delta_{S^{d-1}}Y_{\lambda}(\Omega) =
\lambda (\lambda + d - 2)Y_{\lambda}(\Omega),
\end{gather*}
where $\Delta_{S^{d-1}}$ is the [Laplace-Beltrami operator]. Introducing the radial
wavefunction $u(r)$, this becomes quadratic:
\begin{gather*}
u(r) = r^{(d-1)/2}\psi(r), \qquad
\nu = \lambda + \frac{d}{2} - 1,\\
r^{(d-1)/2}\nabla^2 \psi(r) =
\left(\diff[2]{}{r} - \frac{\nu^2 - 1/4}{r^2}\right)u(r).
\end{gather*}
This follows after a little algebra from
\begin{gather*}
\frac{1}{r^{d-1}}\diff{}{r}\left(r^{d-1} \diff{}{r}\right)r^{(1-d)/2}u(r)\\
=
r^{(1-d)/2}
\left(
\diff[2]{}{r}
-
\frac{(d-3)(d-1)}{4}
\right)u(r).
\end{gather*}
:::::
This can be solved quite simply -- but not very accurately -- with finite differences.
Highly accurate solutions can be obtained by shooting, and we present details in
{ref}`sec:RadialSEQ` about how to do this.
## The Essence
The essential idea is low-energy properties, like the bound state energies $E_n$, should
not be highly sensitive to details about the nature of the potential $V(r)$ at short
distances $r \ll 1/\Lambda$ where $\Lambda$ is a large momentum scale.
[Manim Community]:
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