Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Lagrangian multiplier and ground state search

+ 0 like - 0 dislike
1055 views

Hi I am trying to understand the paper of Schollwoeck (arxiv version: https://arxiv.org/pdf/1008.3477.pdf)

There on p.64 formular 203 he states:

In order to solve this problem, we introduce a Lagrangian multiplier λ, and extremize

\[\langle\psi|H|\psi\rangle - \lambda \langle\psi|\psi\rangle \]

If I remember correctly this however translates into an optimization problem where one wants to min/maximize \(\langle \psi | H | \psi \rangle\) subject to the constraint \(\langle \psi | \psi \rangle = 0\) .

See for example https://en.wikipedia.org/wiki/Lagrange_multiplier.

My question now is, how can I understand the constraint? If it would be something like \(\lambda(\langle \psi | \psi \rangle -1 )\) then I could read it as the normalization constraint but in this form I don't know how to make sense of it. May some brighter person please enlighten me?

asked Apr 3, 2018 in Computational Physics by anonymous [ no revision ]

You must first consider $|\psi\rangle$ as unknown variable, find the solution to the extremum problem, and only after that apply the constraint, whatever it is.

The stationary points are the same whether you add your term or the term in Schollwoeck; he is just less careful than you in writing the details. The difference is just a constant $\lambda$.

@ArnoldNeumaier what do you mean with stationary points?
I only know Lagrange multiplier in context of optimization of functions over real valued vector spaces.  So what does it actually mean if I want to find the minimal \(| \psi \rangle\)? Does my intuition translate to Hilbert spaces? Since the optimal solution point in a real valued vector space can look very different for different constraints, even if the difference is just a constant. Or are there some properties of the hamiltonian which were implicitly used in the argument?

The infinite-dimensional case works the same, at the physical level of rigor. You minimize a quadratic subject to a norm constraint, by forming the Lagrangian and finding its stationary points (zero variation = zero gradient). The constant does not affect the variation.

@ArnoldNeumaier For a real valued Lagrangian I have to calculate the gradient wrt. x but also wrt. \(\lambda\) . This is were the constant will appear and matter. Then I'll have to solve a system of equations where the constant does not simply vanishes. What is the analog in the infinite-dim. case? What is the search term I have to use to  find more details on this subject?

No. Independent of the dimension, the stationary points are at the gradient of the Lagrangian with respect to the original variables only. In addition, the original constraint must be imposed. The derivation of the technique is the same in all dimensions.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysics$\varnothing$verflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...