# What is Effective potential: Definition and 40 Discussions

The effective potential (also known as effective potential energy) combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets (both Newtonian and relativistic) and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.

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1. ### I Callan-Symanzik equation for Effective Potential

Hey all, I am looking equations (13.24),(13.25) in Peskin & Schroeder's QFT book and I am confused about how they change from the Callan-Symanzik equation for the Effective Action to the Effective Potential. I thought the relation for constant ##\phi_{cl}## was ##\Gamma[\phi_{cl}] = -(VT)\cdot...
2. ### I Maximum impact parameter given effective potential

This problem is from David Morin's classical mechanics textbook: I am having trouble with Part b. Here is the textbook's answer: I do not understand why large particle energies lead to capture. I would think that smaller energies would lead to capture because the particle wouldn't have enough...
3. ### B L = 0 in the equation for effective potential energy?

Hi, What would happen if I set L in this equation to zero? I can have an L that is zero right?
4. ### A Obtaining the Orbit equation (Effective Potential) from the Newtonian metric

We can write the Newtonian metric in the form of $$ds^2 = -(1 - 2M/r)dt^2 + (1+2M/r)[dr^2 + r^2d\Omega^2]$$ In order to obtain the orbit equation I have written the constant of motion, $$e = (1 - 2M/r)(\frac{dt}{d\tau})$$ and $$l = r^2sin^2(\theta)(\frac{d\phi}{d\tau})$$ I can divide the...
5. ### I Effective Potential & Curved Spacetime: Angular Momentum Effects

I was looking at this chart and I didn't understand how increased angular momentum of the test particle curves the spacetime around the center mass. If that is how it's interpreted. Now the way it looks like is that the curvature is dependent on the angular momentum of the test particle.
6. ### Effective potential in a central field

Hi, I am confused by a point which should be relatively simple. When we consider classical motion of a particle in a central field U(r), we write the total energy E = T + U, where T is the kinetic energy. The kinetic energy contains initially r, r' and φ' (where ' denotes the time derivative)...
7. ### Expression for effective potential energy

<< Mentor Note -- Poster has been reminded to use the Template when starting new schoolwork threads >> Two particles of identical mass m interact with each other via central potential energy Vcentral(r) = -V0(1-|r|/a), if 0 <= |r| <= a 0, if a < |r| Constants are V0 > 0...
8. ### Effective potential problem

Homework Statement A satellite with mass of m is circling a star. The radius of the circle is R. At some moment the mass splits to 2 equal masses (the tangential velocity of the masses doesn't change). As a result of the split the kinetic energy in the system is multiplied by k (k>1). What will...
9. ### A Two questions on Feynman diagram and Green's function

First, is it suitable to solve a Green's function by one-order self-energy, since it only consider partial high order perturbation, so it's unclear that this calculation corresponding to which order perturbation. In other word, if one wants to use self-energy to get Green's function, he should...
10. ### GR- effective potential, maximum attainable radius

Homework Statement part d) from the following question please Homework EquationsThe Attempt at a Solution [/B] sol attached So I see that the idea is that it is a maximum so to set ##\dot{r}=0## and then the maximum value is dependent on some values of J and K, to get the equation. But...
11. ### GR- Energy conservation, effective potential graph sketch

Homework Statement I would like to ask about parts c) and d) , the graph sketching bits. 2. Homework Equations ##V(r) = ( \frac{J^2}{r^2}+\epsilon)(r-\frac{1}{r}) ## where the value of ## \epsilon ## is set according to whether time-like or null etc. The Attempt at a Solution Q1)for...
12. ### A Vacuum in QFT: Fock space or effective potential?

There seem to be two ways of defning what a vacuum is in QFT: 1. It is state $|0\rangle$ such that $$a_k|0\rangle = 0$$ for all anihilation operators $$a_k$$, with creation operators $$a_k^{\dagger}$$. Thus, it is defined in Fock space. 2. It is state $$|0\rangle$$ such that derivative...
13. ### A Is the Effective Potential the same Quantum Potential?

Is the Effective Potential the same thing we called in QM as "Quantum Potential" ?
14. ### Effective Potential in Lagrangian & Hamiltonian Mechanics

In Lagrangian and Hamiltonian mechanics it's common to define part of the kinetic energy as the "effective potential energy" but i am unclear on which expression we define this from, if we look at the lagrangian and identify the part of the kinetic energy that's dependant only on the the...
15. ### Effective Potential Energy from Central Force Motion

Hi, I'm studying undergrad mechanics, Central force motion, Marion's book in specific, Here, the Potential Energy is defined weird way (in my opinion though) (μ is reduced mass) So potential Energy becomes Called "Effective Potential Energy" But, I can't agree with calling it potential just...
16. ### 2 Body Radial Equation, Effective Potential derivation

According to my textbook, in the derivation for the effective potential U_{eff}, starting with the Lagrangian L = \frac{1}{2}\mu(\dot r^2 +r^2\dot\phi^2) -V(r), substituting into Lagrange's equation gives \mu\ddot r = -\frac{\partial V}{\partial r} + \frac{l^2}{\mu r^3} =...
17. ### Help with contour plots of effective potential in R3BP.

Hello everyone! I'm currently trying to plot the effective potential for Sun-Jupiter system, to show the lagrangian points in this system. I've converted to a system of units where G=1, m_sun+m_jupiter=1 and R=1, whereby I get the following equation describing the effective potential of a third...
18. ### Einstein's Field Equations: Effective Potential Functions

I have seen written out in various places (including this forum) the effective potential function that comes from the solutions to the Schwarszschild Geodesic. But I haven't been able to find the effective potential functions for other solutions to Einstein's field equations. Are there...
19. ### Effective potential energy minimum

Effective potential energy is defined by U^*(\rho)=\frac{L^2}{2m\rho^2}+U(\rho) in many problems I found that particle will have stable circular orbit if U^*(\rho) has minimum. 1. Why is that a case? Why circle? Why not ellipse for example? 2. Is this condition equivalent with...
20. ### Effective potential energy of a charge in a magnetic field

Hi, I'm studying the Lagrangian and its applications in electromagnetism. I stumbled across this inconsistency: The force of a charge moving through a magnetic field is ## F_b = q v \times B ## If we define B to be in the ## \hat{z} ## direction, this equation can be written as ## F_b = q (...
21. ### Using Effective Potential to Describe motion of an object

Hey guy, this is one of the question in my uni work (I'm not sure whether my answer is correct or not, please have a check). The attachment consists of what the effective potential graph look like. "It starts from a low point where it move downward to the lowest point. It then move...
22. ### Why do we need to consider the effective potential in central force problems?

When dealing with central forces, we can find the energy of a particle under the influence of such a force to be E=0.5m(dr/dt)2+(L2/2mr2)+U(r). We define the effective potential as the sum of the last two terms. In my previous experience of dealing with energy considerations to predict motion...
23. ### Converting effective potential to dimensionless function

Homework Statement Convert the effective potential to a dimensionless function by scaling to the radius of a circular orbit. Homework Equations U_{eff}(r) = - \frac{k}{r}+ \frac{L^2}{2mr^2} \frac{U_{eff}}{U_0} = -\frac{2}{r / r_0} + \frac {1}{(r/r_0)^2} r_0= \frac{L^2}{mk}, U_0=...
24. ### Potential Energy using effective potential

Homework Statement For a given angular momentum L, find the potential energy function U(r) that leads to a spiral path of the form r=r0θk. Choose the total energy E to be zero. Hint: obtain an expression for r' that is only a function of r, not θ, and then use the energy equation with...
25. ### Effective potential of a particle sliding on an inclined rod

Homework Statement Hey guys, Here is the question: A pointlike mass m can slide along a rigid rod of length l and negligible mass. One extremity of the rod is fixed at the origin O of an inertial system (x,y,z), and the rod forms a constant angle α with the z-axis. The rod rotates...
26. ### Effective potential help

An electron (of mass me and charge −e) in a hydrogen atom is located at position vector r relative to the proton (of mass mp and charge +e) constituting the nucleus. It is attracted to the proton by the electrostatic force F =-\frac{e^2}{4\epsilon_0r^2}\hat{r}, where e_0 is a constant (the...
27. ### Orbits of Particle Around a Black Hole using Effective Potential

Homework Statement The possible orbit of a particle moving around a black hole can be described using the effective potential UGR(r) (in effect, potential energy per unit mass): UGR(r) = -GM/r + l2/2m2r2 - Rsl2/2m2r3 where the symbols have their usual meaning and in particular...
28. ### Effective Potential with Power-Law Spiral

Homework Statement Given L, find the V(r) that leads to a spiral path of the form r = r0(theta)k. Choose E to be zero. The Attempt at a Solution I know that I need to find r' as a function WITHOUT theta, but I'm not sure how to do that. I know how to find r' as a function of...
29. ### Deriving the effective potential due to screening

Homework Statement I'm supposed to show that the Lindhard dielectric functions gives a contribution to the effective potential of a metals as U_{eff}( \vec{r} ) \propto \frac{cos( 2 k_{F}r)}{r^{3}} in the limit of r\rightarrow\infty for d = 3 (3 dimensions) Homework Equations...
30. ### Potential of 2 Orbiting Objects in Rotating Frame

I would like to express the potential of 2 orbiting objects in the rotating frame, but I'm not quite doing it right. I am a physics major but since I took AP, my mechanics is quite bad. Here's what I'm doing, please tell me what am I missing. First, I consider two objects denoted with 1 and 2...
31. ### Effective potential and geodesics in G.R.

The geodesics around a spherical mass (Schwarzschild solution) in G.R. can be described by \frac{1}{2}\left(\frac{dr}{d\lambda}\right)^2 + V(r) = \mathcal{E} where V(r) is the effective potential \frac{1}{2}\epsilon - \epsilon\frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GML^2}{r^3} and...
32. ### Eff. Pot. Homework: Struggling with Algebra

Homework Statement I am trying to do problem 5, I seem to be having a hard time with the algebra. http://img24.imageshack.us/img24/4224/landau.th.png [Broken] Homework Equations The Attempt at a Solution To find maximum value of the effective potential we just do: dU/dr = 0. I get (which I...
33. ### Second derivative of effective potential

Determine the value of r in terms of l, k, and m for which the following function has a minimum. V(r) = -(k/r) + (l^2/(2mr^2)) where l, k, and m are positive constants. Prove that this is a minimum by showing that the second derivative of V(r) at the minimum is positive. I have no...
34. ### Question on Effective Potential

I am trying to draw one of those nice plots of effective potential [Vr] against radius [r], which suggest the position of stable and quasi-stable orbits. Have no trouble getting the classical curve showing a minimum coinciding with a radius corresponding to a given angular momentum [L], but...
35. ### Effective potential and stability

I am asked to check the stability at theta=0 of the pendulum system shown in the attachment. I set up the energy for the system and found it to be E=\frac{1}{2}m(L\sin(\theta))^{2}+\frac{1}{2}m(L\frac{d\theta}{dt})^{2}-mgL\cos(\theta) which is in agreement with the books answer "A guide to...
36. ### Effective potential concept

Hi, I'm looking for some information about effective potential, but I haven't found any (Wikipedia, Googled...). I was just willing to get a rough understanding of the concept, and understand what it is. Could you explain/link to good info please? Thank you very much.
37. ### Kepler orbital trajectories and 'effective potential'

Ok so the total energy of a body following a given trajectory around a much larger body (eg. Earth and sun), is described by : E(total) = (1/2)mv^2 + U (where U = grav. potential energy) E(total) = (1/2)mv^2 - (GMm)/r (1/2)mv^2 can then be expanded to give : E(total)...
38. ### Effective Potential and Angular Momentum

I have the following equation for potential energy. Actually it's for the effective potential energy. 'V'(r) = - \int F(r) dr - \int \frac{L^{2}}{m r^{3}}dr 'V'(r) = V(r) + \frac{L^{2}}{2 m r^{2}} Where does the second term on the right come from? What does it have to do with potential...
39. ### Effective Potential Method for Solar System Orbits

Hey everyone! I have an exam question, but I don't know how to approach it. The question is, "Discuss orbits of bodies in the Solar System using the effective potential method." I thought about every planet having a certain amount of kinetic and potential energy, showing how they...
40. ### Effective Potential: Krammer, Coulomb, Approximation Questions

Hi, May I ask you some questions on effective potential ? I've heard about Krammer potential ? Exactly, what is it ? Coulomb potential can be replaced by effective potential in some classical theory situation ? Now, my project is about the approximation of effective potential in short range...