From f5105dc7b183fd540006fb4f21039d8b2d126621 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 7 Oct 2022 19:43:33 +0200
Subject: Expand knowledge base

---
 content/know/concept/ritz-method/index.pdc | 11 +++++++++++
 1 file changed, 11 insertions(+)

(limited to 'content/know/concept/ritz-method/index.pdc')

diff --git a/content/know/concept/ritz-method/index.pdc b/content/know/concept/ritz-method/index.pdc
index 320771a..1fe0d31 100644
--- a/content/know/concept/ritz-method/index.pdc
+++ b/content/know/concept/ritz-method/index.pdc
@@ -229,6 +229,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
+In the context of quantum mechanics, this is not surprising,
+since any superposition of multiple states
+is guaranteed to have a higher energy than the ground state.
+
 Note that the convergence to $\lambda_0$ goes as $|c_n|^2$,
 while $u$ converges to $u_0$ as $|c_n|$ by definition,
 so even a fairly bad guess $u$ will give a decent estimate for $\lambda_0$.
@@ -348,10 +352,17 @@ in a limited basis would yield a matrix $\overline{H}$ giving rough eigenvalues.
 The point of this discussion is to rigorously show
 the validity of this approach.
 
+If we only care about the ground state,
+then we already know $\lambda$ from $R[u]$,
+so all we need to do is solve the above matrix equation for $a_n$.
+Keep in mind that $\overline{M}$ is singular,
+and $a_n$ are only defined up to a constant factor.
+
 Nowadays, there exist many other methods to calculate eigenvalues
 of complicated operators $\hat{H}$,
 but an attractive feature of the Ritz method is that it is single-step,
 whereas its competitors tend to be iterative.
+That said, the Ritz method cannot recover from a poorly chosen basis.
 
 
 
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