Talk:Variational method (quantum mechanics)

Comments

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)?Nimur 19:27, 3 May 2006 (UTC)

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.

Also, it would be nice to talk about the multiparameter variational method (i.e. minimizing wrt more than one paramter). Batamtig 18:22, 30 November 2006 (UTC)

Merge Suggested

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. Nimur 10:01, 4 May 2006 (UTC)

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.137.222.31.145 13:14, 30 May 2006 (UTC)

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. Stevvers 03:33, 5 November 2006 (UTC)

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)

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.

The result of the proposal was moved. --RegentsPark (sticks and stones) 02:05, 10 October 2009 (UTC)

Variational method (quantum mechanics) → Variational method — The page "Variational method" is just a redirect to here, so, in the spirit of keeping article titles as simple as possible, I propose dropping the "(quantum mechanics)" from the title of this article. Djr32 (talk) 18:55, 3 October 2009 (UTC)

Survey

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 Wikipedia's naming conventions.

• Support Agree with nominator. YeshuaDavid • Talk • 21:23, 4 October 2009 (UTC)

Discussion

Any additional comments:

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)

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Variational method → Variational method (quantum mechanics) – I think that the move in 2009 was, unfortunately, a clear mistake. The problem is that Variational methods certainly means the general methods of Calculus of variations. This article is just one example of these methods (perhaps not even the sole example even within quantum mechanics). So Variational method should redirect to Calculus of variations and we need to rename this article. Variational method (quantum mechanics) might not be the very best name, but for simplicity it is easiest to move it back there. Dingo1729 (talk) 16:23, 23 March 2013 (UTC)

• Support move. Seems obvious. Sławomir Biały (talk) 01:27, 25 March 2013 (UTC)
  • 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)
• Move it back I agree. "Variational method" would usually be interpreted as calculus of variations, i.e. differentiating a functional with respect to its argument, a function, and setting that to zero. JRSpriggs (talk) 01:46, 25 March 2013 (UTC)
• Move. "Variational method" is more usually found in calculus of variations. Deltahedron (talk) 07:16, 25 March 2013 (UTC)
• Comment: Turing into a disambig page? -- Taku (talk) 10:26, 25 March 2013 (UTC)
• Rename and disambiguate per Taku, move the Variational method (disambiguation) page to the prime location -- 65.92.180.137 (talk) 23:50, 25 March 2013 (UTC)
  • Comment Variational methods should also point to the disambiguation page. -- 65.92.180.137 (talk) 23:53, 25 March 2013 (UTC)
• Support doing something to de-emphasize the quantum mechanics meaning in favor of the dab or mathematics meaning. —BarrelProof (talk) 18:07, 26 March 2013 (UTC)
• 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)

It would be very helpful if you could reference a variational method which is not the Calculus of variations. Certainly variational Bayesian methods, the Variational method (quantum mechanics) and everything linked from Variational method (disambiguation) are all Calculus of variations. Dingo1729 (talk) 13:31, 27 March 2013 (UTC)

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)

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)

• Support (more or less) My suggestion: Variational method should be a brief article, based on the current disambiguation page, stating that "variational method" can refer to either (1) the use of the variational principle to reformulate problems in terms of the calculus of variations (e.g., variational methods in general relativity) or (2) an approximation method that finds the "best" approximate solution by optimizing within some restricted class of ansätze (e.g., Variational method (quantum mechanics), Variational Bayesian methods).—Stevvers (talk) 09:32, 28 March 2013 (UTC)

• Support: Obviously the inference to QM reflects the article content and removes possible conflation with the calculus of variations. M∧Ŝc²ħεИ_τlk 06:11, 30 March 2013 (UTC)

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)

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)