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. E

    Quantum harmonic oscillator with harmonic perturbation

    Homework Statement Let's consider a harmonic oscillator with a harmonic perturbation: H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}. Exact solution is known, but we want to derive it using perturbation theory. More specifically, suppose we want to obtain a series for the ground state...
  2. JK423

    Time-dependent perturbation theory

    I'm studying Sakurai at the moment, time-dependent perturbation theory (TDPT). I'm having a problem in understanding a basic concept here. According to Sakurai we have the following problem: Let a system be described initially by a known hamiltonian H0, being in one of its eigenstates |i>...
  3. E

    Time-dependant perturbation theory & transitions

    Time-dependant perturbation theory & "transitions" I'm studying approximation methods, and something is really bothering me about the standard treatment of time-dependant perturbation theory. In lecture, the prof introduced time-dependant perturbation theory with the following motivation...
  4. B

    Litle help with perturbation theory

    Why when we analyse time dependant perturbation theory, we take that the diagonal elements of matrix <i|W(t)|j> are equal to zero? Why in degenerate perturbation theory we assume that perturbed wavefunctions of degenerate states can be expressed in the base of unperturbed wavefunctions of...
  5. T

    Degenerate perturbation theory help

    So in time-independent degenerate perturbation theory we say that we can construct a set of wavefunctions that diagonalize the perturbation Hamiltonian (H') from the degenerate subspaces of the unperturbed Hamiltonian (Ho). Since the original eigenstates are degenerate, combinations of them are...
  6. O

    What is a perturbation and how does it affect a system in equilibrium?

    Perturbation-the ability of an unperturb system to remain the same when a perturb system is added to it. Can't really understand this?
  7. R

    Quantum Field Theory and Perturbation Approximation

    Pls. answer in the simplest and the most intuitive way. 1. What is the reason our quantum field theory needs perturbative approach. Is it because in the concept of fields, there is an infinite number of freedom in the oscillations of the virtual particles, or is it because the field is...
  8. Y

    Who can solve the probability of perturbation in simple harmonic oscillator

    Pleae help me. a),b),c) was already solved. but question d) is not.
  9. S

    Finding Inner Solutions for Singular Perturbation Problems

    Homework Statement \epsilon\frac{d^{2}u}{dx^{2}} +\frac{du}{dx} + e-x = 0 0<x<1 u(0)=0 u(1)=1 Homework Equations The Attempt at a Solution i want to find the inner solution first i used the substitution x=\epsilon2y i put that in the equation...
  10. Z

    Applying Perturbation to Mercury: Understanding Electron Behavior"

    We can apply perturbation to Helium. ıt has two electrons. But Mercury has lots of electrons. in this case, can we apply perturbation to Mercury? How?
  11. M

    Degenerate Perturbation Theory: Operator commuting with Ho and H'

    Homework Statement Question is: Prove the following: Let A be a Hermitian operator that commutes with H0 and perturbation H'. If two degenerate states have distinct eigenvalues for A, then the matrix element of perturbation between them is zero! The real problem is I don't understand...
  12. C

    Question about a scientific paper - Fluid Mechanics - Perturbation Theory

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  13. A

    Time independent perturbation theory (QM)

    Homework Statement I'm trying to derive the second-order correction of energy in time independent perturbation theory. My professor did it the Landau's way so I'd rather use his notation (without bra and kets). I already derived the first-order correction: E_n^{(1)}=V_{nn}=\int...
  14. K

    A mathematical issue raised from perturbation theory

    Take the usual time-independent perturbation theory in QM for example,H'=H_0+V, a basic assumption is we can expand the new states of H' in terms of the old ones of H_0, most of the textbooks justify this assumption by reasoning that the set of eigenfunctions of Hamiltonian is complete...
  15. R

    Quick Q: First order perturbation theory derivation

    Homework Statement Going over and over the perturbation theory in various textbooks, I feel that I've NEARLY cracked it. However, in following a particular derivation I fail to understand a particular step. Could anyone enlighten me on the following? Multiply |\psi^{1)_{n}>...
  16. Z

    A question about perturbation series inversion

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  17. N

    Can we demontrate the convergence of perturbation quantum field theory?

    Please teach me this: Can we demontrate the convergence of perturbation series of quantum field theory(Feymann diagrams) after making the renormalizing procedure? If we can't demontrate that,why we still consider the perturbative method using in quantum field theory being useful and believable...
  18. J

    Time dependent perturbation theory

    Hi I was wondering if someone could help me out. I have been studying TDPT and was wondering how it applies to atomic physics or if someone could give me a example that would be great.
  19. C

    Breakdown of perturbation expansion

    Homework Statement consider a perturbation to the simple harmonic oscillator problem Lambda* (x)^4 question a) show tht the first order correction to n-th eigenstate is proportional to (1+2n+2n^2) b) argue that no matter how small lambda is ,the perturbation expansion will break down for...
  20. J

    Perturbation theory and asymptotics

    I need to find the roots of the transcendental function, f(x;a)=x^2-3ax-1-a+exp(-x/a)=0; I've done many problems like this before and am fairly sure this is just a regular perturbation problem. The difficulty I'm having is with the exponential term. Could anyone give me an idea of how...
  21. B

    First-order time-dependent perturbation theory on a Hydrogen atom

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  22. F

    Secular Matrix of Perturbation

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  23. F

    Perturbation Theory - Shift of Ground State

    Homework Statement Use leading order perturbation theory to calculate the ground state shift of hydrogen due to perturbation: \hat{V} Homework Equations 1. Leading terms in expansion of energy: E=mc^{2}+\frac{p^{2}}{2m}-\frac{p^{4}}{8m^{3}c^{2}}+... 2. \hat{H}=\hat{H}_{0}+\hat{V} where...
  24. N

    About mixed integral for perturbation in hydrogen atom

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  25. C

    Energy eigenfunctions in time-independent perturbation theory

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  26. A

    Question on Time-independent perturbation theory: I am confused

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  27. B

    Why do degenerate energy levels shift or split due to perturbation?

    Homework Statement Consider the first excited state of the Hydrogen atom. The principle quantum number is given by n = 2 and so it is four-fold degenerate. Consider now a weak perturbation in the form of V = λxy, where x and y are the Cartesian coordinates of the electron with respect to the...
  28. A

    Question about time limits of time-dependent perturbation theory

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  29. K

    Recommand a nice qm textbook on perturbation and scattering

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  30. C

    How small for perturbation theory to be valid?

    Homework Statement A particle of mass m is in the ground state in the harmonic oscillator potential V(x) = \frac{1}{2}Kx^{2} A small perturbation \beta x^{6} is added to this potential. How small must \beta be in order for perturbation theory to be valid? Homework Equations...
  31. P

    Time dependent perturbation theory for density matrix

    Does anyone kown how to apply time dependent perturbation theory to densities matricies (I'm interested in first order)? Thanks.
  32. 1

    Understanding Perturbation Theory and Spin in Quantum Mechanics

    In my quest to learn quantum mechanics I've become a little confused and I have a question. It is "In The Feynman Lectures on Physics Volume 3, is most of the work, eg. ammonia molecule done using perturbation theory or is this method something else, not perturbation theory?" I started to...
  33. A

    Perturbation theory and total derivatives

    Hi I was just reading about that total derivatives in the Lagrangian does not give any contributions in perturbation theory but that they can play role in non perturbative regimes. But there was no statement WHY that is so? Does anyone have an idea and reading advices? I have the most...
  34. J

    Where Can I Find Feynman Rules for Chiral Perturbation Theory?

    Hello, I was wondering whether anyone knows the Feynman rules for Chiral Perturbation theory? I am trying to calculate K->PiPi and have obtained the relevant diagrams but cannot proceed without the relevant feynman rules.
  35. P

    Mastering Perturbation Theory for Nuclear Engineering Students

    Perturbation Theory Help! Hello physicsforums.com, The last two weeks of my nuclear engineering course covered a mathematical topic known as 'perturbation theory'. It was offered as a 'method to solve anything' with; the problem is, however, that nobody in my class understands it. Basic...
  36. kreil

    Infinite square well perturbation

    Homework Statement A particle of mass m in the infinite square well is subjected to the perturbation H'=Vo, 0<x<L/2, H'=0 else. (a) use first order perturbation theory to calculate the energies of the particle (b) what are the first order corrected wave functions? (c) if the particle is an...
  37. M

    Perturbation Theory & the Ground State in a 1-D Potential Box

    A particle is in the ground state in a one-dimensional box given by the potential v(x)= 0 for 0<x<a v(x)= inifinity other wise A small perturbation V = V(0)x/a is now introduced. Show, correct to first order in perturbation theory, hat the energy change in the ground state is V(0)/2...
  38. V

    Perturbation Theory/Harmonic Oscillator

    Homework Statement I am given the hamiltonian, where H^{^}_{0} is that of the harmonic oscillator and the perturbation is (lambda)*(h-bar)*(omega)*[(lowering operator)^2 + (raising operator)^2]. I am asked to find the ground state, second-order approx. energy value. Homework...
  39. N

    Perturbation theory / harmonic oscillator

    Homework Statement An electron is confined by the potential of a linear harmonic oscillator V(x)=1/2kx2 and subjected to a constant electric field E, parallel to the x-axis. a) Determine the variation in the electron’s energy levels caused by the electric field E. b) Show that the second order...
  40. G

    Perturbation theory question (in Quarks & Leptons)

    Hi I'm referring to the book Quarks and Leptons (Halzen, Martin). On pages 79-82 nonrelativistic perturbation theory is investigated (i.e. by using the Schroedinger equation, which is first order in time). On Page 85, however, the transition amplitude (T_fi) is used that has been derived on...
  41. C

    Energy shift using perturbation theory

    Homework Statement I am looking at the relativistic correction to the kinetic energy for a hydrogen atom. I am told that the perturbation is usually written as H = -p^4/(8 m^3 c^2) and need to find the energy shift Homework Equations I know that from the perturbation theory the energy...
  42. M

    Perturbation: First order correction to particle-in-box eigenstates

    First order correction to particle-in-box eigenstates for Dirac perturbation Homework Statement Calculate the first three nonzero terms in the expansion of the correction to the ground state \psi^{1}_{1} for a Dirac delta perturbation of strength alpha at a/2 (box from 0 to a). Homework...
  43. J

    Where can I find perturbation theory formulas up to fourth order?

    Does anybody happen to know where to find the perturbation theory formulas for the energies and states up to fourth order? I have to do a calculation up to this order and don't want to have to derive them if I don't have to (I know that Wikipedia has high order energies, but they only have the...
  44. B

    Time-dependent perturbation theory question not a hard one

    Homework Statement A hydrogen atom is placed in a uniform electric field E(t) given by E(t) = Enaught*exp(-a*t) (where a is a constant) for t >0. The atom is initially in the ground state. What is the probability that, as t→∞ , the atom makes a transition to the 2p state? I know...
  45. F

    Does the sum of perturbation expansions in QED and QCD converge?

    I heard that there are semi-rigorous arguments that the sum of all (renormalized) terms in QED or QCD perturbation expansions do not converge, even though each term in the sum is systematically renormalized and rendered finite. This is suspected to be the case even if the coupling constant is...
  46. H

    Perturbation of Potential (Particle in a Box)

    Homework Statement Assume that the particle in the box is perturbed by a potential V_{1}(x) = x . Calculate the energy shift of the ground state and the first excited state in first-order perturbation theory. Homework Equations Unperturbed wave functions for the particle given by...
  47. H

    Anharmonic Oscillator - Energy Shift Calculation Using 1st Order Perturbation

    Homework Statement V(x) = \frac{1}{2}mw^{2}x^{2} + \lambdax^{4} Using first-order perturbation theory to calculate the energy shift of: 1. The ground state: \psi_{0}(x) = (2\pi\sigma)^{\frac{-1}{4}}\exp(\frac{-x^{2}}{4\sigma}) of the harmonic oscillator, where...
  48. M

    Perturbation Theory description of Positronium

    Hi everyone, I'm trying to work on Problem 5.32 from Sakurai's Modern Quantum Mechanics. In a nutshell, we need to use Perturbation Theory to find some of the energy levels of Positronium. Here's the full problem: http://img714.imageshack.us/img714/6515/sakurai532.gif Problem 3.3 isn't much...
  49. N

    Perturbation Theory: How Does Oscillation Amplitude Vary with Length?

    Homework Statement The length of a pendulum is slowly doubled (l=l_0(1+epsilon*t), 0<=t<=1/epsilon). How does the amplitude q_max of the oscillations vary? Any hints? Homework Equations The Attempt at a Solution
  50. S

    Degenerate perturbation theory

    Homework Statement Hi, i have put the question, my attempt and actual answer in the attached picture. My answer is not quite right; firstly why is the second term a minus lambda, and where does the O(lamdba^2) come from? Homework Equations The Attempt at a Solution
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