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I think there is a "=" sign missing in the second equation in section Ansatz!
This seems written from a Mathematical Physics standpoint, instead of from a Quantum Mechanics angle. Any suggestions for converging this article (for example, application to a sample problem)?
Nimur19:27, 3 May 2006 (UTC)reply
One thing is that most physics textbooks present the variational method with non-normalized ansatz functions, unlike the normalized fn's presented here. This makes writing an ansatz simpler, but introduces a slight modification of the formulae. I agree about the samnple problems, and there are a variety of such problems in many texts.
There is a very similar
article about the theory behind this technique. Either we should merge them or we should integrate them into a more coherent framework.
Nimur10:01, 4 May 2006 (UTC)reply
I found this article very useful as a sepperate entity. Articles are easy to get lost in when they are this technical and they try and disscuss to much at once. Would be best to just surgest in this article that it may be usefull to refer to the other.
137.222.31.14513:14, 30 May 2006 (UTC)reply
It may be that you've mistakenly referred to the wrong article, Nimur, but this doesn't seem to be at all related to 'variational pertubation theory' as described on that page.
Stevvers03:33, 5 November 2006 (UTC)reply
I agree with Steevers that this article is not very related to the article suggested by Nimur about 'variational perturbation theory. This article should be expanded, but not merged with the suggested article.
Just google searching for the variational method brings up a link (2nd after wikipedia) that is far more useful in application and with respect to quantum mechanics and the application of the variational method. As I am new to this, I will just post the link to the page here
http://vergil.chemistry.gatech.edu/notes/quantrev/node28.html From the knowledge I have gained in my quantum classes, this is a very comprehensive guide and touches upon strengths and weaknesses of actually applying the method to a QM system. Also, the website uses a more physical chemistry approach, which I find more valuable than the one currently posted.
Vints1 (
talk)
22:15, 15 December 2008 (UTC)reply
Requested move (2009)
The following is a closed discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.
Feel free to state your position on the renaming proposal by beginning a new line in this section with*'''Support'''or*'''Oppose''', then sign your comment with~~~~. Since
polling is not a substitute for discussion, please explain your reasons, taking into account
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The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.
Requested move back to where it was
The following discussion is an archived discussion of a
requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a
move review. No further edits should be made to this section.
The result of the move request was: moved. Clear consensus that this article requires disambiguation, but I can't say there's a consensus on whether the dab page should be moved to
Variational method or whether it should be a redirect to
Calculus of variations. My original plan was to default to wherever
Variational method had originally pointed, but unfortunately there was no original to default to. As such, I'm going to move the dab page to the 'primary' location because it strikes me that when there's no consensus on if there's a primary topic we are better off to disambiguate. There should of course be no prejudice against starting a new RM to debate whether
Variational method should redirect to
Calculus of variations or be a dab page.
Jenks24 (
talk)
11:57, 4 April 2013 (UTC)reply
Comment. I would like to argue against a disambiguation page. It's not that there are many different things called "variational method", rather there is a single "variational method" (at least as far as the calculus of variations is a single subject) that is applied to many different problems. The appropriate way to address this is with a single article that summarizes and links to the various ways this method is applied to areas of science.
Sławomir Biały (
talk)
21:18, 26 March 2013 (UTC)reply
Comment I think it's important to distinguish the
calculus of variations, which deals with the mathematical problem of extremizing a functional, from the
variational method, which is an approximation method. The latter is used in various contexts, including quantum mechanics, classical statistical mechanics and
variational Bayesian methods, and relies on the
variational principle. In practice, applying the variational method often (but certainly not always) involves minimizing a functional, for which one uses the methods of the calculus of variations. My preferred solution would be to have the article
variational method explain this in general terms, with sections on the applications to quantum mechanics and to Bayesian statistics (with a link in the latter to
variational Bayesian methods). —
Stevvers (
talk)
13:13, 27 March 2013 (UTC)reply
Calculus of variations is (currently) defined as "a field of mathematical analysis that deals with maximizing or minimizing functionals", and is therefore a particular type of
mathematical optimization where the "set of available alternatives" is a set of functions. Taking for example
the variational method applied to the helium atom, an approximation to the ground-state energy is found by minimizing with respect to a single real parameter (Z)—this is an application of single-variable calculus, with no functionals involved. (As stated in
Calculus of variations#Eigenvalue problems, eigenvalue problems can generally be formulated variationally, so the exact solution of the Schrödinger equation can be found by minimizing with respect to the wavefunction. But that is not what the article on the variational method in quantum mechanics deals with.)—
Stevvers (
talk)
09:08, 28 March 2013 (UTC)reply
Direct methods have for a long time been regarded as part of the general subject of the calculus of variations (see any textbook on the subject written in the last 50 years). This includes, for instance,
finite element analysis, the
Ritz method, the
Galerkin method, and so forth. All of these direct methods optimize in a finite-dimensional subspace. They are still part of the calculus of variations, and there are precise estimates on the quality of these approximations to the actual variational problem. I would need a great deal of convincing that anything called "variational method" is not already subsumed under the general subject of calculus of variations.
Sławomir Biały (
talk)
23:30, 28 March 2013 (UTC)reply
The above discussion is preserved as an archive of a
requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a
move review. No further edits should be made to this section.
Error in statement of orthonormality of states
It seems the sum over $\lambda_1,\lambda_2$ in the statement of the orthonormality of the set of states $|\phi_{\lambda}\rangle$ is misplaced. Remove? Change to simply $\langle \phi_{\lambda_1}|\phi_{\lambda_2}\rangle = \delta_{\lambda_1,\lambda_2}$. — Preceding
unsigned comment added by
176.22.14.195 (
talk)
18:14, 20 May 2015 (UTC)reply
Sign error in effective charge treatment of helium atom
In the expression for $\langle H \rangle$ there seems to be a sign error (in fact, the expression is maximum at $Z_{eff} = 27/16$ rather than minimum). — Preceding
unsigned comment added by
176.22.14.195 (
talk)
18:27, 20 May 2015 (UTC)reply