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. W

    Stability against small perturbation.

    Hello, I am reading the book, The Quantum Theory of Fields II by Weinberg. In page 426 of this book (about soliton, domain wall stuffs), we have Eq(23.1.5) as the solution that minimizes Eq(23.1.3). The paragraph below Eq(23.1.5), the author said "The advantage of the derivation based on...
  2. U

    Cosine perturbation to potential well

    Homework Statement Part (b): Find the perturbed energy. Homework Equations The Attempt at a Solution I've solved everything, except part (b). I got an answer of 0 for part (b) for all orders, which is kind of strange, as one would expect some perturbation. \Delta E_n = \langle \psi_n...
  3. maverick280857

    Sakurai Degenerate Perturbation Theory: projection operators

    Hi, So, I am working through section 5.2 of Sakurai's book which is "Time Independent Perturbation Theory: The Degenerate Case", and I see a few equations I'm having some trouble reconciling with probably because of notation. These are equations 5.2.3, 5.2.4, 5.2.5 and 5.2.7. First, we...
  4. P

    Perturbation Theory: Finding Eigenfunctions and Energies

    In a text a exercice says that for the Hamiltonian ##H_0 = \frac{p^2}{2m}+V(x)## the eigenfunction and eigen energy are ##\phi_n, E_n##. If we add the perturbation ## \frac{\lambda}{m}p## ¿what is the new eigenfunction? The solution is ## \frac{p^2}{2m} + \frac{\lambda}{m}p+V=...
  5. U

    Perturbation Theory, exchange operator

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  6. U

    Hydrogen - perturbation theory question

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  7. U

    Sudden Perturbation of Potential Well

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  8. U

    Quadratic Stark Effect - Perturbation Theory

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  9. D

    Identical particles/ Time dependent perturbation theory

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  10. Runei

    Perturbation theory (the math)

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  11. G

    Why transition rate independent of time in perturbation theory?

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  12. X

    Perturbation Theory: Calculating Energy Corrections

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  13. X

    Degenerate Perturbation Theory: Two Spin 1/2 Particles

    So I know this might be a lot to read but I am having a very hard time understanding how to use the formulas in degenerate perturbation theory. Here is the problem I am on. Homework Statement A system of two spin-1/2 particles is described by the following Hamiltonian...
  14. L

    Newtonian limit of cosmological perturbation

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  15. F

    One-dimensional linear harmonic oscillator perturbation

    Homework Statement Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state Homework Equations ψn(x) = \frac{α}{√π*2n*n!}1/2 *...
  16. X

    Degenerate Perturbation Theory

    http://farside.ph.utexas.edu/teaching/qm/lectures/node53.html So I was reading this and I don't understand how he goes from 658 to 661 using the completeness relation. In 661 if you use the completeness relaton can you get rid of the I n,l''>s by doing the outer product and ignoring the...
  17. H

    Degenerate perturbation theory (Sakurai's textbook)

    In the theory of degenerate perturbation in Sakurai’s textbook, Modern Quantum Mechanics Chapter 5, the perturbed Hamiltonian is H|l\rangle=(H_0 +\lambda V) |l\rangle =E|l\rangle which is written as 0=(E-H_0-\lambda V) |l\rangle (the formula (5.2.2)). By projecting P_1 from the left (P_1=1-P_0...
  18. H

    Generally applicable development of classical perturbation theory

    Greetings, Does anyone know of some good sources that explain classical perturbation theory, preferably using the Lagrangian formalism? The sources that I have seen more-or-less say, "write L=L_{0}+λδL, where L_{0} is an unperturbed, soluble Lagrangian, δL is the perturbation, and λ is a small...
  19. D

    Degenerate Perturbation Theory (Particle in 3D box)

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  20. H

    Electromagnetic radiation and perturbation

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  21. F

    Time independent perturbation - potential with inner function

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  22. M

    2nd Order Perturbation Coefficients.

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  23. O

    Can a basic knowledge of perturbation theory solve this?

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  24. F

    E-field Perturbation of 2D rotor. Show Y_10 couples ground state?

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  25. Telemachus

    Time dependent perturbation theory, HO subject to electric field

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  26. Telemachus

    Time independent perturbation (Quantum Mechanics)

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  27. B

    First order perturbation for hydrogen

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  28. F

    How do I find the Taylor series for a function with multiple variables?

    Homework Statement I required to make a perturbation expansion in ε of the function: Homework Equations A(X,y,z)=A(x-εsin(wy),y,z). X=x-εsin(wy) The Attempt at a Solution Solution: A(X,y,z)=A0(X,z)+ε[A1(X,z)+∂/∂XA0(X,z)]sin(wy)+o(ε^2) I get the terms A0(X,z) and ∂/∂XA0(X,z)sin(w,y) with the...
  29. B

    Perturbation Theory on Finite Domains

    In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode y'' + P(x)y = 0 using perturbation theory. However he points out that the domain must be finite in order for this to work, I'm wondering how you would phrase a question...
  30. A

    Error in perturbation series question

    Suppose we know the perturbation series E = E(\epsilon) = E_0 + \epsilon E_1 + \epsilon^2 E_2 + \ldots converges, where E_0 is a discrete eigenvalue of H_0 and we are considering a Hamiltonian H = H_0 + \epsilon H_1. Does this mean that we know E - E_0 = O(\epsilon) as...
  31. H

    Sharks can sense the electric field perturbation?

    Sharks use an organ known has the ampullae of Lorenzini to sense electric fields, they can sense fields as weak as 5*10^-9 V/m. Which got me thinking, they should in principle be able to detect vacuum perturbation in the field, interesting?
  32. W

    What does a general PE have to to with a S-O perturbation? Stumped

    Homework Statement Show that for a general potential energy V(r) that the form of the spin-orbit Hamiltonian becomes \hat{H}_{S-O}=\frac{1}{2m_{e}^{2}c^{2}|\hat{\textbf{r}}|}\frac{d\hat{V}}{dr} \hat{\textbf{L}}\cdot\hat{\textbf{S}} Suggestion: Start with \textbf{B} = -( \textbf{v} /c)...
  33. T

    Degenerate perturbation theory degeneracy not lifted by perturbation

    Hi, I have an equation of the form (-i \lambda \frac{d}{dr}\sigma_z+\Delta(r)\sigma_x) g =(\epsilon + \frac{\mu \hbar^2}{2mr^2}) g where \sigma refers to the Pauli matrices, g is a two component complex vector and the term on the right hand side of the equation is small compared to the other...
  34. O

    Time independant perturbation problem

    Example from Schaum's Quantum Mechanics. Picture of the example is attached. What I don't understand is part (c). What are those wavefunctions ##\mid \psi^{(0)}_{1,2} \rangle## and ##\mid \psi^{(0)}_{2,1} \rangle##? How do I find these wavefunctions, if the unperturbed wavefunction is...
  35. O

    Time independant perturbation - Difficulty understanding derivation

    Hamiltonian is in the form ##H = H_0 + \lambda W##, where ##\lambda \ll 1## and ##W## is the perturbation. Assume the eigenstates ##\mid \psi(\lambda) \rangle## and engenenergies ##E(\lambda)## can be expanded in a power series of ##\lambda##. $$\mid \phi(\lambda) \rangle = \mid 0 \rangle +...
  36. C

    Finite Hilbert Space v.s Infinite Hilbert Space in Perturbation Theory

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  37. S

    Perturbation Techniques and Theory for Nonlinear Systems

    Homework Statement Given the equation \ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta} Determine a first-order uniform expansion for small but finite theta. Homework Equations Other than the equation above, none so far as I am aware. The Attempt at a...
  38. C

    Perturbation Theory for a Hamiltonian

    Hi guys, this is my first time posting, I'm studying physics at uni, in my third year and things are getting a bit tough, so basically my question is in relation to solving problem 1, (i included a picture...) I missed the class and don't really know what I'm doing. Any help would be appreciated.
  39. B

    First order perturbation question

    Homework Statement Suppose we put a delta function bump in the center of the infinite square well: H' = \alpha \delta(x -a/2), where \alpha is constant. a) Find the first order correction to the allowed energies. b) Find the first three non-zero terms in the expansion of the correction...
  40. M

    Momentum perturbation to harmonic oscillator

    Homework Statement the problem and a possible solution(obtained from a book) is attached as a pdf to the post.However Iam unable to understand it.Please download the attachment. Homework Equations equation no (2) in the pdf.Is there any use of space translation operator in here.The Attempt at...
  41. A

    Analyzing perturbation of vectors

    Homework Statement We have A \in R^{mxm} \text{ and } b \in R^{m} \text{ and } b \neq 0 \text{. Show that } Ax = b \text{ and } A(x+ \delta x) = b+ \delta b The Attempt at a Solution I did the first part just by the definition of A being non singular. The second part is tripping...
  42. D

    Simple time-independent non-degenerate quantum perturbation

    I'm reading through this pdf (http://www.pa.msu.edu/~mmoore/TIPT.pdf) on simple quantum perturbation theory and I'm quite confused with equations 32 through 34. They have E_{n}^{(2)} = <n^{(0)}|V|n^{(1)}> = - \sum_{m \neq 0}{\frac{|V_{mn}|^{2}}{E_{mn}}} but I would have done E_{n}^{(2)} =...
  43. D

    MHB Is there a mistake in the satellite's trajectory calculation?

    The L4 position is stable in the Earth Moon and I perturbing a satellite by km in the x direction to see the trajectory over the course of the year. However, the satellite isn't moving. Can anyone see if there is something wrong? I gave the satellite no initial velocity. In[2587]:=...
  44. J

    Equation decoupling in Cosmological Perturbation theory

    Hi, I am recently reading Weinberg's Cosmology, and getting subtle on Ch5, small fluctuation. One of the subtle point is on P.225-226 (same as (F.11) -> (F.13) and (F.14) in appendix F). The equations of motion (5.1.24)-(5.1.26) are decomposed into many parts. For example, (5.1.24) is...
  45. M

    Diagonal Matrix & Perturbation Theory in Quantum Mechanics

    What does it mean for a matrix to be diagonal, especially in Quantum Mechanics, where we get to Perturbation theory (Degeneracy). I don't get it. Please if you can explain in 'simple' language.
  46. B

    First-Order Perturbation Theory Derivation in Griffiths

    Homework Statement On page 251 of Griffiths's quantum book, when deriving a result in first-order perturbation theory, the author makes the claim that <\psi^0|H^0\psi^1> = <H^0\psi^0|\psi^1> where H^0 is the unperturbed Hamiltonian and \psi^0 and \psi^1 are the unperturbed wavefunction and its...
  47. C

    QED perturbation series convergence versus exact solutions

    It is well known due to the famous argument by Dyson that the perturbation series for quantum electrodynamics has zero radius of convergence. Dysons argument essentially goes like that: If the power series in α had a finite (r>0) radius of convergence it also would converge for some small...
  48. E

    A perturbation operator problem

    hi,my friends.I have a perturbation operator problem. v=ezE; why this formula is right?how to deduce it?a is bohr radius. thank you!
  49. Hepth

    Chiral Perturbation Theory : pi0 pi0 Z vertex?

    Not sure if anyone has any experience with chiral perturbation theory, but I'm trying to see what all of the vertices are for interactions with a single Z boson. I've looked at the lagrangian up to order p^4 so far, and it seems that the Z only interacts with charged pions/kaons. I'm using...
  50. J

    Perturbation Theory: Calculating for the correction on the ground state energy

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