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.
(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...
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}}## ...
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...
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...
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...
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}...
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...
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...
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...
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.
$$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
I am referring to perturbation expansion of density functional Kohn Sham energy expression in
Phys. Rev. A 52, 1096.
In equation (92) the variational form of the second order energy is listed, but I cannot seem to work out the last 3 terms involving XC energy and density. Particularly, the...
Homework Statement
Hi, in this book " [Nouredine_Zettili]_Quantum_Mechanics_Concepts ", Eq. (9.133)
Homework Equations
I don't know how the second line
had come from the first line:The Attempt at a SolutionI got only two terms such that:
$$ < \psi_0| H | \psi_0 > = A^2...
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...
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}=...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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 =...