What is Variational principle: Definition and 47 Discussions
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
In the Jellium model, it is customary to evaluate the exchange term of the Hartree-Fock equation for plane waves ##\varphi_{\mathbf{k}_i}## as a correction to the energy of the non-interacting electron gas obtaining $$\hat{U}^{ex} \varphi_{\mathbf{k}_i}=-e^2 \left( \int \dfrac{\mathrm{d}^3k}{2...
Let us consider an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. The solution of the field ##c## is given by the expression ##f(a,b)##. On taking into account the relations obtained from the solutions for ##a## and ##b##, we find that ##f(a,b)=0##. If the...
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
I'm solving problem number 5 from https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/mit8_05f13_ps2/.
(a) Here I got:
$$
\beta = \frac{\hbar^{\frac{1}{3}}}{(\alpha m)^\frac{1}{6}}
$$
and:
$$
E = \left ( \frac{\alpha \hbar^4}{m^2} \right )^\frac{1}{3}e
$$
(b) Using Scilab I...
The metric is $$ds^2 = \frac{dr^2 + r^2 d\theta ^2}{r^2-a^2} - \frac{r^2 dr^2}{(r^2-a^2)^2}$$
I need to prove the geodesic is: $$a^2 (\frac{dr}{d \theta})^2 + a^2 r^2 = K r^4$$
My method was to variate the action ##\int\frac{(\frac{dr}{d\theta})^2 + r^2 }{r^2-a^2} - \frac{r^2...
(This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method)
1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
I've got that length of a curve on the surface is:
$$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$
So the function to extremise is:
$$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$
Where...
First I picked an arbitrary state ##|ϕ⟩=C_1|φ_1⟩+C_2|φ_2⟩+C_3|φ_3⟩## and went to use equation 1. Realizing my answer was a mess of constants and not getting me closer to a ground state energy, I abandoned that approach and went with equation two.
I proceeded to calculate the following matrix...
This question is actually about relativity and quantum field theories. I have the impression that we just use the variational principle, and given the right lagrangian, they lead to equations that we know, are correct. That seems to me a good reason for "believing" that the variationa principle...
Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't...
Einstein's field equations (EFEs) describe the pointwise relation between the geometry of the spacetime and possible sources described by an energy-momentum tensor ##T^{ab}##. As well known, such equations can be derived from a variational principle applied to the following action: $$S=\int\...
An isolated mechanical system can be represented by a point in a high-dimensional configuration space. This point evolves along a line. The variational principle of Jacobi says that, among many imagined trajectories between two points, only the SHORTEST is real and is associated with situations...
Landau's nonrelativistic quantum mechanics has a "derivation" of Schroedinger's equation using what he calls "the variational principle". Apparently such a principle implies that:
$$\delta \int \psi^{\ast} (\hat{H} - E) \psi dq = 0$$
From here I can see that varying ##\psi## and...
Homework Statement
Okay, in Carrol's Intro to Spacetime and Geometry, Chapter 4, Eq. 4.63 to 4.65 require a derivation of a difference between Christoffel Symbol. I did the calculation and found my answer to be somewhat correct in form, but the indices doesn't match up
Homework Equations
So...
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.
$$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
Background: I am an upper level undergraduate physics student who just completed a course in classical mechanics, concluding with Lagrangian Mechanics and Hamilton's Variational Principle.
My professor gave a lecture on the material, and his explanation struck me as a truism.
Essentially, he...
Hi, i have been struggling to find some good resources on variational principle , I have got an instructor in advanced quantum course who just have one rule for teaching students- "dig the Internet and I don't teach you anything".. So I digged a lot and came up with a lot reading but I need...
These questions applies to both spatially homogenous cosmological models, and multidimensional Kaluza-Klein theories:
Suppose we have a manifold M, of dimension m, for which there is a transitive group of isometries acting on some n-dimensional homogeneous subspace N of M. Thus there exists a...
Homework Statement
A particle of mass m is in a potential of V(x) = Kx4 and the wave function is given as ψ(x)= e^-(ax2) use the variational principle to estimate the ground state energy.
Part B:
The true ground state energy wave function for this potential is a symmetric function of x...
Hello there,
I am struggling in proving the following.
The principle of Minimum energy for an elastic body (no body forces, no applied tractions) says that the equilibrium state minimizes
$$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$
among all vectorial functions u satisfying the...
1. Hey,
I have to find Maxwells equations using the variational principle and the electromagnetic action:
S=-\intop d^{4}x\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
by using
\frac{\delta s}{\delta A_{\mu(x)}}=0
therefore \partial_{\mu}F^{\mu\nu}=0
3. I have had a go at the...
I have the following differential equation which I obtained from Euler-Lagrange
variational principle
\frac{\partial}{\partial x}\left(\frac{1}{\sqrt{y}}\frac{dy}{dx}\right)=0
I also have two boundary conditions: y\left(0\right)=y_{1} and
y\left(D\right)=y_{2} where D, y_{1} and y_{2} are...
Homework Statement
Use the following trial function:
\Psi=e^{-(\alpha)r}
to estimate the ground state energy of the central potential:
V(r)=(\frac{1}{2})m(\omega^{2})r^{2}
The Attempt at a Solution
Normalizing the trial wave function (separating the radial and spherical part)...
I was experimenting with some physics and the mathematics started to get a bit tougher than what I'm used to. I had a professor who looked at what I'm doing, offered to guide me, and told me to do some research on the variational principle.
At the moment, I am in Calculus II. I did a couple...
Homework Statement
It's not exactly a homework question. I can find Christoffel Symbols using general definition of Christoffel symbol. But, when I try to find Christoffel Symbols using variational principle, I end up getting zero.
I have started with the space-time metric in a weak...
Homework Statement
http://img4.imageshack.us/img4/224/32665300.png
The Attempt at a Solution
http://img684.imageshack.us/img684/2920/scan0003xo.jpg
I've uploaded my work so far since its much faster than typing and I'm stuck on the last line trying to solve the integral.
The first...
Homework Statement
The problem statement is a bit length so I have attached a picture of the problem instead. The issue I am having pertains to part (b).
Homework Equations
The Attempt at a Solution
The main issue I am having is with what my Hamiltonian should look like when I do...
Consider the variational principle used to obtain that in the vacuum the Einstein tensor vanish.
So we set the lagrangian density as L(g,\partial g)=R
and asks for the condition
0 = \delta S =\delta\int{d^4 x \sqrt{-g}L}
proceeding with the calculus I finally have to vary R such that...
I have a question regarding the variational principle in quantum mechanics.
Usually we have a Hamiltonian H and we construct a state |ψ> using some trial states. Then we minimize E = <ψ|H|ψ> and get an upper bound for the ground state energy. In many cases the state |ψ> is then used to...
It is very well known that the result of varying some functional gives a differential equation which solutions minimizes the given functional. What about the other way around? Can one find a functional that is minimized given a differential equation? Is there a procedure for this?
The reason...
I am trying to prove the variational principle on 1st excited state, but have some questions here.
The theory states like this: If <\psi|\psi_{gs}>=0, then <H>\geq E_{fe}, where 'gs' stands for 'grand state' and 'fe' for 'first excited state'.
Proof: Let ground state denoted by 1, and...
Homework Statement
Show that the Lorentz force law follows from the following variational principle:
S=\frac{m}{2}\int\eta_{\mu\nu}u^\mu u^\nu ds-q\int A_\mu u^\mu ds
Homework Equations
Definition of Field Strength Tensor
Integration by Parts
Chain Rule & Product Rule for Derivatives
The...
My notes give the variational principle for a geodesic in GR:
c\tau_{AB} = c\int_A^B d\tau = c\int_A^B \frac{d\tau}{dp}dp = \int_A^B Ldp
and then apply the Euler-Lagrange equations. By choosing p to be an "affine parameter" where \frac{d^2 p}{d\tau^2} the Euler-Lagrange equations are...
Quantum Field Theory -- variational principle
In non-relativistic quantum mechanics, the ground state energy (and wavefunction) can be found via the variational principle, where you take a function of the n particle positions and try to minimize the expectation value of that function with the...
Obtain a variational estimate of the ground state energy of the hydrogen atom by taking as a trial function \psi_T(r) = \text{exp } \left( - \alpha r^2 \right)
How does your result compare with teh exact result?
You may assume that
\int_0^\infty \text{exp } \left( - b r^2 \right) dr =...
Homework Statement
Use trial wavefunction exp(-bx^2) to get an upper limit for the groundstate energy of the 1-d harmonic oscillator
The Attempt at a Solution
This is always going to give an integral of x^2*exp(-x^2). How do you do it? :/
Homework Statement
Consider the variation principle for the space-time functional of the variables \eta, \phi
A( \eta, \phi) = \int \int \phi \partial _t \eta -\frac{1}{2}g \eta ^2 -\frac{1}{2} h ( \partial_x \phi ) ^2\ \mbox{d}x \mbox{d}t Derive the two coupled equations for the critical...
I once tried to come up with a variational principle that would lead to Emden's equation. I think this is instructive. Start with the mass
M = - 4 \pi a^{3} \rho_{c} \xi^{2} \Theta'
rewrite this as
M / 4 \pi a^{3} \rho_{c} + \xi^{2} \Theta' = 0
but just let
X = M / 4 \pi a^{3}...
A text I am reading has used the variational principle not only to find the ground state of a system, but also to find some higher order states. (Specifically, it was used to derive the Roothaan equations, which are ultimately related to the LCAO method of orbital calculations.) I don't see how...
Homework Statement
Find the best bound state on Egs for the one-dimensional harmonic oscillator using a trial wave function of the form
\psi(x) = \frac{A}{x^2 + b^2}
where A is determined by normalization and b is an adjustable parameter.Homework Equations
The variational principle...
Homework Statement
If I'm given a potential say A(x/a-m) m an integer, (this is the sawtooth wave)
What kind of trial function should I use to approximate this?
Homework Equations
The Attempt at a Solution
I do recall this function arising in Fourier series. Should I actually...
[SOLVED] QM variational principle
Homework Statement
In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx^4, where K is a constant, which of the following wave functions would be a better trial wave function:
1) \psi(x) =...
I am in search for some part of physics that could not be derived from a variational principle.
For the small part of physics I know (CM, Schrödinger), everything can be derived from a variational principle. I would like to know if this is a deep fingerprint of physics or a general...
In classical mechanics, for conservative systems, it well knows that the differential laws of motion can be derived from a variational principle called "least action principle".
I know also that some non-conservative systems can be derived from a variational principle: the damped harmonic...
I need someone who can briefly (in easy way) explain me the variational principle or tell me where (in checked source) I can find this. I will be very greatful.
What is this used for?..i don,t see any utility on using it..:frown: :frown: for Commuting and Anti-commuting operators we would have:
\delta{<A|B>}=i<A|\delta{S_{AB}}|B>
but i don,t see that it provides a way to obtain Schroedinguer equation or the propagator for the theorie...what is...
I was asked to do an assigment for a Chemical Physics class on the Ritz variational principle (used to calculate an approximation of an observable). We are working a simple potential, the one dimensional particle in the box (v=0 for 0<x<L, V= infinite elsewhere) and only considering the ground...