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Diffstat (limited to 'content/know/concept/ritz-method')
| -rw-r--r-- | content/know/concept/ritz-method/index.pdc | 11 |
1 files changed, 11 insertions, 0 deletions
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. |