What is Variational method: Definition and 33 Discussions

The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

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  1. Samama Fahim

    I Schrodinger Equation from Ritz Variational Method

    (This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method) 1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
  2. Paulpaulpa

    Spacetime translations and general Lagrangian density for Field Theory

    In Sydney Coleman Lectures on Quantum field Theory (p48), he finds : $$D\mathcal{L} = e^{\mu} \partial _{\mu} \mathcal{L}$$ My calulation, with ##\phi## my field and the variation of the field under space time tranlation ##D\phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}## ...
  3. ExplosivePete

    I Using the Variational Method to get higher sates

    In a typical quantum course we learn how to approximate the ground state of a particular Hamiltonian by making an educated guess at an ansatz with a tunable parameter then calculating the expectation energy for the ansatz. The result will depend on the tunable parameter if done correctly. Then...
  4. T

    A Lagrangian to the Euler-Lagrange equation

    Hello all, I understand the formation of the Lagrangian is: Kinetic Energy minus the potential energy. (I realize one cannot prove this: it is a "principle" and it provides a verifiable equation of motion. Moving on... One inserts the Lagrangian into the form of the "Action" and minimizes it...
  5. J

    I Determining the accuracy of the Variational Method

    The Variational Method allows us to obtain an upper bound on energy of the ground state (and sometimes excited states). Is there any way of determining an upper bound on the error of the energy obtained by the variational method without an analytic or numerical solution to the problem? i.e. Is...
  6. H

    I Derivative = 0 is always minima? (Linear variational method)

    I have a very fundamental question about the linear variational method (Huckel theory). It says in any textbook that the variational method provides energy upper bound to the actual energy of a wavefunction by using test wavefunction. \varepsilon = \frac{\sum_{i,j}^{n}C_{i}C_{j}H_{ij}...
  7. C

    Sturm-Liouville Eigenvalue Problem (Variational Method?)

    Homework Statement Using Sturm-Liouville theory, estimate the lowest eigenvalue ##\lambda_0## of... \frac{d^2y}{dx^2}+\lambda xy = 0 With the boundary conditions, ##y(0)=y(\pi)=0## And explain why your estimate but be strictly greater than ##\lambda_0##Homework Equations ##\frac{d}{dx} \left...
  8. P

    B Guessing trial wave function with variational method

    Hellow i want to ask about guessing the trial wave function at variational method of approximation usually for example at solving harmonic oscillator or hydrogen atom we have conditions for trial wave function but in fact i want to ask generally how could i make the guessing .. some problems...
  9. N

    A Numerical solution to SE - variational method, many electrons

    Hi everyone, I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here. Problem I have can be explained on a very simple...
  10. D

    Deriving geodesic equation using variational principle

    I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got this. $$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
  11. S

    DFPT second order energy variational form

    I am referring to perturbation expansion of density functional Kohn Sham energy expression in Phys. Rev. A 52, 1096. In equation (92) the variational form of the second order energy is listed, but I cannot seem to work out the last 3 terms involving XC energy and density. Particularly, the...
  12. Safinaz

    Question in variational method (QM)

    Homework Statement Hi, in this book " [Nouredine_Zettili]_Quantum_Mechanics_Concepts ", Eq. (9.133) Homework Equations I don't know how the second line had come from the first line:The Attempt at a SolutionI got only two terms such that: $$ < \psi_0| H | \psi_0 > = A^2...
  13. B

    Variational method For Helium Atom

    When using the variational method for the Helium atom, we determine that the lowest possible energy occurs when 1<z<2 where z is the atomic number. My professor elaborated that the number is within this range because there is a probability that the electron may be so close to one of the protons...
  14. R

    Ground state energy from the variational method

    Homework Statement Consider a one-dimentional particle in a box with infinite potential walls at x=0 and x=L. Employ the variational method with the trial wave function ΨT(x) = sin(ax+b) and variational parameters a,b>0 to estimate the ground state energy by minimising the expression E_{T}=...
  15. R

    Choosing the trial wavefunction (variational method)

    Homework Statement I'm going to list two questions as they offer the same problem with more choices, hopefully it will help realize the method (?) used (A) An electron, confined in the two dimensional region 0<x<L and 0<y<L with infinite potential walls, is subject to the potential...
  16. R

    Question about when to use the Variational Method

    Here is the question: We are not given the wavefunction so in this instance would I use the variational method? i.e. should I guess the wavefunction and apply: E_{guess} = \frac{< \Psi_{guess} \mid H \mid \Psi_{guess} > }{< \Psi_{guess} \mid \Psi_{guess} >} I am really unsure about how to...
  17. jfy4

    Lower bounds on energy eigenvalues

    Hi, I'm interested in learning about what would be the compliment to the Variational method. I'm aware that the Variational method allows one to calculate upper bounds, but I'm wondering about methods to calculate lower bounds on energy eigenvalues. And for energies besides the ground state if...
  18. Einj

    Variational Method and Bound States

    Homework Statement Consider a potential function V(x) such that: $$ \begin{cases} V(x)\leq 0\text{ for }x\in[-x_0,x_0] \\ V(x)=0 \text{ for }x\not\in[-x_0,x_0] \end{cases} $$ Show, using the variational method that: (a) In the 1-dimensional case \lambda^2V(x) always possesses at...
  19. C

    Variational method, is the wavefunction the best for all

    I understand that the variational method can give me the "best" approximate ground state wavefunction among the class of the function belongs to. It is the "best" wavefunction in a sense that its energy level is closest to the ground state among its own class. Question: Is it also true that...
  20. H

    Variational Method vs Inverse Rayleigh Method with Shifting

    I must fundamentally misunderstand what the variational method is. According to my textbook, it's used to find the minimum eigen energy of an operator (in particular, the time-independent schrodinger equation). This appears to be synonymous to finding the eigenvalues of the matrix representation...
  21. A

    Tricky integral in QM variational method

    Homework Statement Find the variational parameters \beta, \mu for a particle in one in one dimension whose group-state wave function is given as: \varphi(\beta,\mu)=Asin(βx)exp(-\mux^{2}) for x≥0. The wavefunction is zero for x<0. Homework Equations The Hamiltonian is given as...
  22. M

    Variational method approximation for half-space linear potential

    Homework Statement This was a test question I just had, and I'm fairly certain I got it wrong. I'm confused as to what I did wrong, though. We were told that our potential was infinite when x<0, and Cx where x>0. We were asked to approximate the ground state potential using the...
  23. U

    Variational Method - Gaussian Trial Wavefucntion

    Homework Statement Use the variational method with a gaussian trial wavefunction ψ(x) = Ae^{\frac{-a^{2}x^{2}}{2}} to prove that in 1 dimension an attractive potential of the form shown, no matter how shallow, always has at least 1 bound state. *Figure is of a potential V(x) that has a minimum...
  24. C

    Quantum Mechanics Variational Method

    Homework Statement Consider a particle in a box in the interval [-a,a]. Use the trial wavefunction ψT = x(a-x2) to obtain an approximate energy for the first excited state of the box as a function of a. Homework Equations Schrodinger equation, Hamiltonian for atomic units is...
  25. J

    Variational method particle in box approx.

    Homework Statement use the variational method to approximate the ground state energy of the particle in a one-dimentional box using the normalized trial wavefunction ∅(x)=Nx^{k}(a-x)^{k} where k is the parameter. Demonstrate why we choose the positive number rather than the negative...
  26. H

    Variational Method: Best Estimate of Ground State Energy

    Homework Statement In the variational method the ground state energy of a quantum system has been calculated as (ћω_o)f(λ), where f(λ) is a function of an arbitrary parameter λ. If f(λ)= λ² + λ, then what is the best estimate of the ground state energy. The Attempt at a Solution Is...
  27. S

    Charmonium Mass Estimation: Trial Wavefunction for Variational Method

    Charmonium: estimating the mass - trial wavefunction for use in variational method?? Hi, I need any suggestions of trial wavefunctions I can use to find an order of magnitude estimate for the mass of charmonium in the variational method. I am ignoring coulombic effects (and relativistic)...
  28. Y

    Variational Method: Approximating Complex Wave Functions with Real Quantities?

    I recently saw the Rayleigh Ritz variational approach used in spectral graph theory, so I was curious to look it up again in the quantum mechanics context. Anyway, there was a real sticking point quite quickly... When we pick our trial wave function, because we want our overlap integrals...
  29. K

    Get the equation catenary using variational method

    I tried to solve the equation of catenary by variational method the other day. The integral we want to minimize is the potential energy: U = \int_{{x_2}}^{{x_1}} {\rho gy\sqrt {1 + y{'^2}} } dx Then I got stuck at the constraint problem, and in this...
  30. J

    Variational method in a finite square well

    I am trying to prove that there is always one bound state for a finite square well using variational method, and I am stuck. I've tried using e^(-bx^2) as my trial wave function, but I am left with E(b)=(hbar^2)b/2m - V, where V is the depth of the well. In this equation, taking the derivative...
  31. A

    Stuck deriving the Hylleraas variational method

    Hi, first post here. I've been trying (out of personal interest, not homework) to re-derive http://www.springerlink.com/content/mm61h49j78656107/" relatively famous calculation on the ground state of Helium from 1929. And I'm stuck at one point. What Hylleraas did, was to parametricize the...
  32. C

    Variational method for geodesics - I'm stuck

    Homework Statement Hi, I am reading Ray d'Inverno's book, 'Introducing Einstein's Relativity' and there is a particular derivation of the geodesic equation that I get stumped on (chapter 7). It is a variational method and the final equation is df/dx_alpha-d/du{df/dx_alpha_dot}=0 where...
  33. U

    Evaluate the ground state energy using the variational method

    Homework Statement V(x) = k|x|, x \in [-a,a], V(x) = \infty, x \notin [-a,a]. Evaluate the ground state energy using the variational method. Homework Equations a = \infty and \psi = \frac{A}{x^{2}+c^{2}}. The Attempt at a Solution 1 =...
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