What is Perturbation: Definition and 422 Discussions

In mathematics, physics, and chemistry, perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter



ϵ


{\displaystyle \epsilon }
. The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of



ϵ


{\displaystyle \epsilon }
usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.
Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The field in general remains actively and heavily researched across multiple disciplines.

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  1. J

    Perturbation Theory (Non-Degenerate)

    If I have V(x)=\frac{1}{2}m\omega^{2}x^{2} (1+ \frac{x^{2}}{L^{2}}) How do I start to solve for the hamiltonian Ho, the ground state wave function ?? Calculate for the energy of the quantum ground state using first order perturbation theory?
  2. N

    Trace reverse perturbation

    From the tensor, ##\bar{h}^{ij}=h^{ij}-1/2\eta^{ij}h## Where, h=##h^i_i##, Prove that ##\bar{h}=-h##, Where, ##\bar{h}=\bar{h}^i_i##
  3. S

    Treating Mass as a perturbation

    Hello again, I also have another question, somewhat related to my previous, on the topic of the Klein-Gordon equation but treating the mass as a perturbation. The feynman diagram shows the particular interaction: I believe the cross is the point of interaction via the perturbation...
  4. S

    Time Dependent Perturbation Theory - Klein Gordon Equation

    Hey, I'm struggling to understand a number of things to do with this derivation of the scattering amplitude using time dependent perturbation theory for spinless particles. We assume we have some perturbation 'V' such that : \left ( \frac{\partial^2 }{\partial t^2}-\triangledown ^2 +...
  5. D

    MHB Can this perturbation problem be solved using a multi-scale approach?

    $$ \frac{d^2x}{dt^2} + x + \epsilon\frac{dx}{dt}\left[1 - \left(\frac{dx}{dt}\right)^2 + \beta\left(\frac{dx}{dt}\right)^4\right] = 0,\quad\quad\epsilon\ll 1, $$ Is there a smart way to do this problem? It will take forever to do directly.
  6. D

    MHB Exploring Multi-Scale Perturbation Techniques

    This is in relation to multi-scale perturbation techniques. $x''+\epsilon f(x,x') + x = 0$ $f(x,x') = (x^2-1)x'$ \begin{alignat*}{3} f\left(x_0 + \epsilon x_1 + \cdots , \left(\frac{\partial}{\partial t} + \epsilon\frac{\partial}{\partial T}\right)(x_0 + \epsilon x_1 + \cdots)\right) & = &...
  7. J

    Energy Levels According To Second-Order Perturbation Theory

    Homework Statement If E1≠E2≠E3, what are the new energy levels according to the second-order perturbation theory? Homework Equations H' = α(0 1 0) (1 0 1) (0 1 0) ψ1= (1) (0) (0) ψ2= (0)...
  8. Hepth

    Chiral Perturbation Theory : Some quick questions

    I just want to make sure that I am doing some things correctly. I'll be using http://www.physics.umd.edu/courses/Phys741/xji/chapter5.pdf from about 5.64 on. The kinetic term : \frac{f^2}{4} Tr[D_{\mu} \Sigma D^{\mu} \Sigma^{\dagger}] Now if I want to expand this out, as \Sigma =e^{i...
  9. D

    Perturbation theory infinite well

    in the infinite well with small potential shown in the attachment. I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x. E=h^2/8mL^2 +∫ ψkψ dx ψ=√(2/L) sin⁡(∏/L x) when integrating ∫...
  10. D

    Perturbation theory infinite well

    in the infinite well with small potential shown in the attachment. I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x. E=h^2/8mL^2 +∫ ψkψ dx ψ=√(2/L) sin⁡(∏/L x) when integrating ∫...
  11. P

    MHB Solving for Roots of a Cubic Equation Using Perturbation Theory

    Question: obtain 2-term expansions for the roots of x^3+x^2-w=0 , 0<w<<1. I assumed an expansion of the form x=a+bw+... and from this obtained a=-1, b=1 as one solution. How do I work out the form of the other 2 expansions? Thanks.
  12. E

    Deuteron Ground State / Perturbation Theory

    Homework Statement The deuteron ground state is made up of l = 0 and l = 2 states; a)Show this mixture cannot be an eigenstate of a central potential Hamiltonian b)Using first-order time independent perturbation theory, argue the potential must contain a term proportional to some combination of...
  13. A

    Time-Dependent Perturbation Theory and Transition Probabilities

    I'm rather stuck on this problem. I seem to be having issues with the simplest things on this when trying to get started. Homework Statement There is a particle with spin-1/2 and the Hamiltonian H_0 = \omega_0 S_z. The system is perturbed by: H_1 = \omega_1 S_x e^{\frac{-t}{\tau}}...
  14. E

    Need book or webpage recommendation for singular perturbation class

    I need some example driven learning material for my "singular perturbation" class. Someone help me please.
  15. E

    Perturbation techniques examples please?

    As expected, my textbook and teacher are both lacking clear, concise examples for me to work with, so would someone point me to early examples within the context of "perturbation techniques", preferably with WORKED OUT solutions for cross reference? thanks
  16. B

    First order perturbation theory problem

    Ok so I have a classic particle in a box problem. If a particle in a box, the states of which are given by: ψ = (√2/L) * sin(nπx/L) where n=1,2,3... is perturbed by a potential v(x) = γx , how do I calculate the energy shift of the ground state in first order perturbation I'm guessing that...
  17. Z

    How to make relativistic correction to perturbation?

    I want to know about relativistic correction to perturbation. I searched but failed to find any teaching on this topic. Is it true that we just need to replace the non-relativistic Hamiltonian perturbation terms with the relativistic ones while leaving the perturbation formulae unchanged...
  18. M

    Optical theorem and renormalised perturbation theory (c.f. Peskin 10.2)

    Hi all, On p.327 in my second edition of Peskin and Schroeder, I have an expression for the one loop correction to the 4-point amplitude of phi^4 theory: i\mathcal{M}=-i\lambda - \frac{i \lambda^2}{32 \pi^2}\text{[Complicated integral]} Mathematica can do the integral for me, and all that...
  19. B

    Specific question on Goldstein section on Time-independent perturbation theory

    I apologize that this is rather specific, but hopefully enough people have used Goldstein. I have a basic grasp of action-angle variables, and I'm going through the time-independent perturbation theory section in Goldstein (12.4). In this section we seek a transformation from the unperturbed...
  20. B

    Classical Perturbation Theory-Time Dep. vs. Time Indep (Goldstein).

    Classical Perturbation Theory--Time Dep. vs. Time Indep (Goldstein). Hi, I'm going through Goldstein, and I'm a little confused on the distinction between time dependent and time independent perturbation theory. In section 12.2, they do the case of a simple harmonic perturbation on force...
  21. L

    Perturbation theory (qualitative question)

    Homework Statement How does the energy change (negative, positive or no change) in the HOMO-LUMO transition of a conjugated polyene where there are 5 double bonds when a nitrogen is substituted in the center of the chain? The substitution lowers the potential energy in the center of the box...
  22. B

    Gravitational Perturbation - How does it work?

    It is easy to understand for example how Jupiter pulls (perturbation) the orbit of the Earth more elliptic. But after a certain period the orbit will again be more circular. How does that (the opposite) work ?
  23. S

    Naive regular perturbation

    can anyone hlep me with this qustion ? Consider the equation ε x^3 + x^2 - x - 6 = 0 ,ε > 0. (1) 1. Apply a naive regular perturbation of the form x~^{0}_{∞}Ʃ xn εn as ε→0+ do derive a three-term approximation to the solutions of (1). 2. The above perturbation expansion...
  24. alemsalem

    Weak field zeeman effect, why don't we still use degenerate perturbation theory?

    the spin orbit coupling removes the degeneracy but not completely, should we still use the degenerate perturbation theory. is it because of relativistic corrections? Thanks!
  25. M

    Question about Quantum + Thermodynamic Perturbation theory

    The following comes from Landau's Statistical Physics, chapter 32. Using a Hamiltonian \hat{H} = \hat{H}_0 + \hat{V} we get the following expression for the energy levels of a perturbed system, up to second order: E_n = E_0^{(0)} + V_{nn} + \sideset{}{'}{\sum}_m \frac{\lvert...
  26. A

    Fine structure constant and perturbation theory

    Hi all, I have a question about perturbation theory and the fine structure constant. Consider an electron moving through the vacuum - this wil induce vacuum polarization, and (if I understand correctly) perturbation theory can be used to analyze the situation. My question is essentially: if...
  27. P

    Potential well with inner step, perturbation theory

    hey, say you have a infinite potential well of length L, in the middle of the well a potential step of potential V and length x. Inside the well is a particle of mass m. why are the first order energy corrections large for even eigenstates compared to odd ones? also, say well...
  28. W

    Convergence radius of a perturbation series

    i see people discussing the convergence radius of a perturbation series in the literature i am really baffled generally, one can only get the first few coefficients of a perturbation series that is, the perturbation series is not known at all how can one determine the convergence...
  29. V

    Time dependent perturbation for harmonic oscillator

    Homework Statement I'm looking at the 1d harmonic oscillator \begin{equation} V(x)=\frac{1}{2}kx^2 \end{equation} with eigenstates n and the time dependent perturbation \begin{equation} H'(t)=qx^3\frac{(\tau^2}{t^2+\tau^2} \end{equation} For t=-∞ the oscillator is in the groundstate...
  30. J

    Singularity theorms and perturbation from exact symmetry

    The singularity theorems apply to situations away from exact symmetry ... away from Schwarzschild solution or Friedmann solutions for example. There are a number of accounts of the singularity theorems but none addressing the problem of proving a 'trapped set' still persists after slight...
  31. H

    Perturbation of 2D Oscillator along one axis

    The problem given is a perturbation on the two dimensional harmonic oscillator where the perturbation is simply: H'=-qfy. It seems that all of the elements of the matrix H' are zero and so constructing a diagonal matrix in the subspace is eluding me. Any ideas?
  32. D

    Metric Perturbation: Finding Info for Einstein's Field Equation

    Hello guys. I was told to prepare a presentation on perturbed Einstein's field equation by my advisor. I got some of the things I needed to start with in the Weinberg's Cosmology book but it was not enough. Can anyone please tell me a book or anything with information on metric perturbation? Thanks
  33. P

    Perturbation Theory: Time-Independent, Non-Degenerate Results

    time-independent, non-degenerate. I am referring to the following text, which I am reading: http://www.pa.msu.edu/~mmoore/TIPT.pdf On page 4, it represents the results of the 2nd order terms. In Eqs. (32), (33) and (34) I don't understand the second equality, i.e. basing on which formula he has...
  34. N

    Degenerate Perturbation Theory

    Hi I am reading about Degenerate Perburbation Theory, and I have come across a question. We all know that the good quantum numbers in DPT are basically the eigenstates of the conserved quantity under the perburbation. As Griffiths he says in his book: "... look around for some hermitian...
  35. R

    How does Perturbation theory account for interactions in QED?

    Many of you stated how ad hoc is QFT as the field is supposed to be non-interacting yet how could they get an incredibly accurate value of calculated magnetic moment of the electron of value 1.0011596522 compared to measured 1.00115965219 with accuracy to better than one part in 10^10, or...
  36. P

    Hydrogen like atom, perturbation theory

    Hi all ! I need some help Homework Statement The nucleus of an hydrogen-like atom is usually treated as a point charge Ze. Using the first order perturbation theory, estimate the error due to this approximation assuming that the nucleus is a sphere of radius R with a uniform charge...
  37. H

    Calculating Transition Probability and Width in a System with Perturbation

    Homework Statement Consider a system with states |i> (with energy Ei) and |f> with energy eF) AND A PPERTURBATION OF THE FORM w(t)=wcosωt i) WHAT IS THE PROBABILITY P(t) FOR A TRANSITION BETWEEN |i> AND |f> ? ii)FOR WHICH VALUE OF ω IS THE PROBABILITY MAXIMAL ?GIVE THE PROBABILITY IN THIS...
  38. V

    Perturbation theory to solve diff eq?

    Hi all, I have a tricky problem in pertubation theory. I have a function: f(\vec{r}) = P(\vec{r}) + \left( B(\vec{r}) + b(\vec{r}) \right)^2 where b(\vec{r}) is a small perturbation and is equal to 0 when P(\vec{r}) = 0 Now, to solve the equation \nabla f(\vec{r}) = 0 for b(r) is...
  39. A

    Question on time-independent perturbation theory

    Hi all. I have been thinking about a very simple question, and I am a little confused. We know from time-independent perturbation theory that if the system is perturbed by the external perturbation λV which is much smaller compared to the unperturbed hamiltonian H0, we can write the ground state...
  40. C

    Perturbation to free surface in bispherical coordinates

    Hi there!I'm working on a physics problem where there is a liquid droplet (not necessarily spherical) on a plane. Transforming from cylindrical coordinates to bispherical: (r,\phi,z)\mapsto (\xi,\eta,\varphi;a) such that r={\frac {a\sin \left( \eta \right) }{\cosh \left( \eta \right) +\cos...
  41. N

    Solving for Constants in Perturbed Simple Harmonic Oscillator with HF Potential

    Homework Statement The potential of a simple harmonic oscillator of HF has the following form \frac{1}{2}kx^2 + bx^3 + cx^4 The first part of the problem involved finding expressions for the first-order energy corrections for the first three states, which I found below. Basically the x3 term...
  42. T

    2nd order perturbation calculation for a system involving spins

    Hello! I am answering a problem which involves spins in the hamiltonian. The hamiltonian is given by H = B(a1Sz^(1) + a2Sz^(2)) + λS^(1)dotS^(2). The Sz^(1) and Sz^(2) refers to the Sz of the 1st and 2nd spins respectively. B is the magnetic field and the others are just constants. The...
  43. C

    Question on perturbation theory

    Hey everyone, I'm studying quantum mechanics from Griffiths (Introduction to Quantum Mechanics, 2nd edition), and I'm puzzling over his derivation of the nth order corrections to the energies and corresponding eigenstates for a perturbed Hamiltonian. The steps that are outlined in Griffiths...
  44. T

    Taylor series at a point for which the function isn't defined (perturbation)

    Homework Statement This problem arises from the following ODE: \epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta where 0 < x < 1, 0 < \epsilon \ll 1 Find the exact solution and expand it in a Taylor series for small \epsilon Homework Equations I guess knowing the Taylor...
  45. P

    Diagonalizing a matrix using perturbation theory.

    Homework Statement Consider the following Hamiltonian. H=\begin{pmatrix} 20 & 1 & 0 \\1 & 20 & 2 \\0 & 2 & 30 \end{pmatrix} Diagonalize this matrix using perturbation theory. Obtain eigenvectors (to first order) and eigenvalues (to second order). Ho=\begin{pmatrix} 20 & 0 & 0 \\0 & 20 & 0...
  46. H

    Perturbation with equations of motion for air resistance

    Homework Statement "A ball is tossed upwards with speed V_0. Air resistance is -mkv^2 and there's gravity too. Find the the time it takes the ball to reach the maximum height. Do not solve the equation of motion exactly. Use the perturbation method on the equation of motion. Solve the equation...
  47. A

    Perturbation of a uniform electrostatic field by a dielectric cube

    Hi, Is there any way to analytically calculate the perturbation of a uniform electrostatic field by a dielectric cube. I know a solution exists for dielectric spheres but I haven't been able to come across the solution, when dealing with a cube. Ohh.. and I'm assuming the simplest case...
  48. E

    Perturbation theory - need a couple of articles

    I'm studying a perturbation theory (behaviour of its series) and have found two articles which might be of particular interest. Unfortunately, all my three institutions do not have subscription to these journals (articles are too old). I'm kindly asking for your help. These are the articles I'm...
  49. M

    Degenerate Perturbation Theory Wavefunction Correction

    Hi, If we have a non degenerate solution to a Hamiltonian and we perturb it with a perturbation V, we get the new solution by |\psi_{n}^{(1)}> = \sum \frac{<\psi_{m}^{(0)}|V|\psi_{n}^{(0)}>}{E_n^{(0)} - E_m^{(0)}}\psi_m^{(0)} where we sum over all m such that m\neq n. When we do the same...
  50. C

    Homotopy Analysis Method (or Homotopy Perturbation Method)?

    Homotopy Analysis Method (or Homotopy Perturbation Method)?? How effective is this Homotopy Analysis Method (HAM) in solving coupled non-linear PDE? I see some papers, but they seem to be cross-referencing a small group of people most of the time. This sounds strange for a method that is so...
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