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. Diracobama2181

    Time Dependent Perturbation Problem

    I am assuming this is the interaction picture, so I start with $$|\psi>=c_1(t)|1>+c_2(t)|2>$$. Plugging this into the Schrodinger equation, I get the equations $$i\hbar c_1(t)=<1|H'|2>c_2(t)$$ and $$i\hbar c_2(t)=<1|H'|2>c_1(t)$$. I am assuming H' (the perturbation) is $$H'= − f(t)[...
  2. Diracobama2181

    A Time Dependent Perturbation of Harmonic Oscillator

    An electric field E(t) (such that E(t) → 0 fast enough as t → −∞) is incident on a charged (q) harmonic oscillator (ω) in the x direction, which gives rise to an added ”potential energy” V (x, t) = −qxE(t). This whole problem is one-dimensional. (a) Using first-order time dependent perturbation...
  3. Baibhab Bose

    Infinitesimal Perturbation in a potential well

    If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term. SO how should I approach this? And how the Δ is relevant in this calculation?
  4. Robin04

    Perturbation theory for solving a second-order ODE

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  5. saadhusayn

    Perturbation expansion with path integrals

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  6. M

    I Time independent perturbation theory in atom excitation

    Hello! In Griffiths chapter on Time independent perturbation theory, he has a problem (9.20) in which he asks us to calculate the first order contribution to the electron Hamiltonian in an atom if one takes into account the magnetic dipole/electric quadrupole excitations, beside the electric...
  7. Futurestar33

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  8. A

    I Perturbation to Flat Space Metric: Geodesic Equation

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  9. V

    Gauge invariance in GR perturbation theory

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  10. iVenky

    What happens when injecting a current at different instants in an oscillator?

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  11. N

    A Perturbation solution and the Dirac equation

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  12. BeyondBelief96

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  13. W

    B Time independent perturbation theory

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  14. C

    MHB Regular perturbation nonlinear problem

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  15. D

    I Problem: perturbation of Ricci tensor

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  16. N

    A TD perturbation - state initially in continuous part

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  17. E

    First order perturbation energy correction to H-like atom

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  18. JuanC97

    I Restrictions of 1st Order Perturbation Theory

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  19. S

    I Degenerate Perturbation Theory

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  20. T

    Perturbation of a Magnetic Field

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  21. Warda Anis

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  22. I

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  23. Necmi

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  24. V

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  25. diegzumillo

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  26. S

    Hydrogen transition probability

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  27. Tursinbay

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  28. Fips

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  29. S

    A Calculating the power spectra of scalar perturbation

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  30. F

    I Higher order terms in perturbation theory (QFT)

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  31. D

    Quick question about hydrogen atom perturbation

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  32. D

    Time-dependent perturbation theory

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  33. Leechie

    Perturbation of a Hydrogen atom

    Homework Statement Suppose there is a deviation from Coulomb's law at very small distances, with the mutual Coulomb potential energy between an electron and a proton being given by: $$V_{mod}(r)= \begin{cases} - \frac {e^2} {4 \pi \varepsilon_0} \frac {b} {r^2} & \text {for } 0 \lt r \leq b \\...
  34. Leechie

    Using perturbation to calculate first order correction

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  35. S

    I First order perturbation derivation

    In lectures, I learned that in first order perturbation, \hat{H}_0 term cancels with E_0 term because \hat{H}_0 is Hermitian. What property does Hermitian operators hold that cancels with the unperturbed energy?
  36. acdurbin953

    Time-Dependent Perturbation of a 1D Infinite Square Well

    Homework Statement At t < 0 we have an unperturbed infinite square well. At 0 < t < T, a small perturbation is added to the potential: V(x) + V'(x), where V'(x) is the perturbation. At t > T, the perturbation is removed. Suppose the system is initially in the tenth excited state if the...
  37. V

    Linear perturbation to harmonic oscillator

    Homework Statement Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##. Homework Equations First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...
  38. L

    I Why do we need perturbation theory

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  39. P

    Time-Dependent Perturbation Theory

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  40. G

    I Time dependent perturbation(determination of order)

    I have a question about time dependent perturbation. In time dependent perturbation, unlike time independent perturbation, there is no lamda which is used for comparing order. So, I`m confused how can I determine order. Is there any explanation which use lambda or some other method for...
  41. L

    Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

    Homework Statement How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x) The attempt at a solution H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x) I know how to find the matrix of the normal...
  42. S

    Perturbation Methods + Airfoils

    First of all, thanks for reading. I'd like to ask if some of you here can suggest a reference text concerning perturbation methods on thin airfoil? I'm using the text of Katz and Plotkin as well as Van Dyke but for now, I find it challenging to follow their discussion. I hope that finding a...
  43. F

    I Gauge transformation in cosmological perturbation

    Based on this lecture notes http://www.helsinki.fi/~hkurkisu/CosPer.pdf For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system...
  44. bananabandana

    I Degenerate Perturbation Theory

    I'm struggling to understand degenerate perturbation theory. It's clear that in this case the 'normal' approximation method fails completely seeing as you get a divide by zero. I follow the example for a two state system given in e.g D.J Griffiths "Introduction to Quantum Mechanics" However...
  45. B

    First Order Perturbation Theory - QM

    Homework Statement The ground state energy of the 1D harmonic oscillator with angular frequency ##\omega## is ##E_0 = \frac{\hbar \omega}{2}##. The angular frequency is perturbed by a small amount ##\delta \omega##. Use first order perturbation theory to estimate the ground state energy of the...
  46. A

    A Canonical perturbation for infinite chain

    I've been Dealing with a problem of perturbation of the movement of an infinite chain of harmonic oscillator and I tried to apply the von Zeippel-Poincare formalism of canonical perturbation theory just to see what I get. This was too naive since I quickly stumbled into the problem of defining...
  47. A

    How Do You Calculate First Order Correction in a Perturbed Infinite Square Well?

    Homework Statement I have the particle in the infinite square well and need to calculate the first order correction energy and the wave function. L is the width and the potential is: 1/2 mw2x2 in the -L/2 < x < L/2 and infinity in x <= -L/2 and x>=L/2 Homework Equations H'=H-H0[/B] The...
  48. D

    I Time dependent perturbation theory

    Hi. I have been looking at some notes for time dependent perturbation theory. The equation for the transition probability involves the matrix element < f | H | i > where f is the final state , i is the initial state and H is the perturbation switched on at t=0. If H is a constant , ie. just a...
  49. F

    Perturbation matrix: free electron model on a square lattice

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  50. G

    Time-Independent Perturbation Theory

    Homework Statement I am working on a physics project for which I need to use perturbation theory to calculate the first- and second-order corrections to the eigenvalues and eigenvectors of a perturbed matrix. The unperturbed matrix is real and symmetric, and the eigenvalues and eigenvectors are...
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