What is Perturbation theory: Definition and 263 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. T

    Stationary Perturbation Theory

    Hi together... When reading Sakurai's Modern Quantum Mechanics i found two problems in the chapter "Approximation Methods" in section "Time-Independent Perturbation Theory: Nondegenerate Case" First: The unperturbed Schrödinger equation reads H_0 | n^{(0)}\rangle=E_n^{(0)}...
  2. L

    Non-degenerate and degenerate perturbation theory

    Consider a system of a rigid rotator together with a uniform E-field directing along z-axis. So to calculate the perturbed energy and wavefunction we have to use perturbation theory. But the book said we can use non-degenerate one to calculate the result. I wonder why. It is because the original...
  3. E

    Is WKB or Perturbation Theory More Applicable for Slowly Varying Potentials?

    Dear All, I have recently read about WKB approximation and about perturbation theory. Both methods are applicable in the range of slowly varying potentials. What I have not understood is which is the range of applicability of one of the method compared with the other one. More...
  4. E

    Quantum Mechanics: Degenerate Perturbation Theory on square well

    Homework Statement Hi I am trying to apply degenerate perturbation theory to a three dimensional square well v= 0 for x, y,z interval 0 to a, perturbed by H' = xyz (product) from 0 to a, otherwise infinite. I need to find the correction to energy of the first excited state which I know is...
  5. C

    Limits of Perturbation theory for hydrogenic atoms

    Homework Statement Why can't we use perturbation theory to calculate the effect of the spin orbit interaction in hydrogen like uranium? Homework Equations The Attempt at a Solution Is it something to do with the fact that the perturbation must be small compared to the rest of the...
  6. O

    Particle in a box - perturbation theory

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  7. Q

    QM 1st order perturbation theory

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  8. J

    Quick question about a simple perturbation theory question

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  9. D

    Time-Dependent Perturbation Theory - Two-level System

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  10. N

    Griffiths' explanation of degenerate perturbation theory

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  11. I

    Degenerate time indep. perturbation theory

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  12. I

    Time independent perturbation theory

    Homework Statement Assume that H0 describes a paramagnetic system that couples to a magnetic field via the Zeeman effect, i.e. V = −μB, where μ is the magnetic moment. Note that for the unperturbed paramagnetic system the probability of having an up-spin is equal to that for a down-spin. Show...
  13. M

    Perturbation Theory with Symmetric Rotator

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  14. P

    Perturbation theory energy shift for hydrogen atom

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  15. W

    Time Dependent Perturbation Theory

    Homework Statement Question: If a particle is in the ground state at time t<0, use the 1st order time dependent perturbation theory to calculate the probability that the particle will still be in the ground state at time t. Suppose we turn on the perturbation at time t=0 H(x) = ax...
  16. A

    Time-dependent perturbation theory: hydrogen atom in electric field

    Homework Statement A Hydrogen atom in its ground state (n,l,m) = (1,0,0) is placed in a weak electric fieldE(t) = 0 if t < 0 Eo *e^{\frac{-t}{\tau}} if t > 0E is in the positive z direction What is the probability that it will be found in any of the n=2 states at time t > 0 ? use...
  17. B

    Virtual particles and perturbation theory

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  18. A

    First order perturbation theory, quantum physics

    I have an infinite potential well with length L. The first task was to calculate the eigenvalues and -functions for the energy of a particle in the well. The requirements were \psi(0, L) = 0 and there is no time-dependence. I've calculated: \hat{H}\psi(x) = E\psi(x) E =...
  19. T

    Resources and guides about Perturbation theory

    Hello, I have to learn about the classic Perturbation Theory. I'm looking for guides, textbooks etc about Perturbation Theory. I already know the basis (Poincare method), but I found it hard to find resources for more advanced material on the one hand, that will also teach it from basis on...
  20. T

    Poincare's method in perturbation theory

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  21. T

    Perturbation Theory - Poincare Method

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  22. P

    QM perturbation theory: rigid diatomic molecule

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  23. J

    Understanding Perturbation Theory: Solving for Roots and Coefficients

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  24. O

    Standard perturbation theory - what exactly is meant?

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  25. H

    Probability in first order time-dependent perturbation theory

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  26. E

    Spins 1/2 and Time-Dependant Perturbation Theory

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  27. T

    Degenerate Perturbation Theory Question

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  28. M

    Convergence Analysis of Perturbation Theory with Divergent Quantities

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  29. cepheid

    Understanding Griffiths' Perturbation Theory in Quantum Mechanics

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  30. R

    Perturbation Theory energy shift

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  31. R

    Perturbation Theory transmission probability

    I'm trying to bridge the gap between several expressions describing the insertion of a constant perturbation: a_{f}(t) = \frac{1}{i\hbar} V_{fi} \int^{t}_{0} e^{i(E_{f}-E_{i})t'/\hbar}dt' = \frac{1}{i\hbar}V_{fi}\frac{e^{i(E_{f}-E_{i})t/\hbar} - 1}{i(E_{f}-E_{i})/\hbar} so...
  32. E

    Time-dependent perturbation theory

    [SOLVED] time-dependent perturbation theory Homework Statement My book uses time-dependent perturbation theory to derive the following expression for the transition of \psi_{100} to \psi_{210} in the hydrogen atom in a uniform magnetic field with magnitude \mathcal{E} \frac{131072}{59049}...
  33. E

    Time-independent perturbation theory

    Homework Statement In each of my QM books, they always say something like "we can write the perturbed energies and wavefunctions as" E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots |n\rangle = |n^{(0)}\rangle + \lambda |n^{(1)}\rangle + \lambda^2 |n^{(2)}\rangle + \cdots...
  34. J

    Introductory perturbation theory

    I've been reading a paper at the following link: www.cims.nyu.edu/~eve2/reg_pert.pdf I have several questions: In the first example they use the method to approximate the roots for x^2 - 1 = "epsilon" x I was under the impression - wrongly perhaps - that f(x) had to have...
  35. O

    Time-dep perturbation theory

    while I`m reading the griffiths` textbook.. got my curiosity from "Typically, the diagonal matrix elements of H` vanish" i.e. <a|H`|a>=0 in general.. If V(x) does not have an angular dependence.. the selection rule implies <a|H`|a>=0 (since Δl=0)..yes.. but what if it does...
  36. N

    How Does a Magnetic Field in the X-Direction Affect Electron Energy Levels?

    Homework Statement An electron is inside a magnetic field oriented in the z-direction. No measurement of the electron has been made. A magnetic field in the x-direction is now switched on. Calculate the first-order change in the energy levels as a result of this perturbation. The Attempt...
  37. S

    Problem on perturbation theory

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  38. H

    GRE Question (QM, Perturbation theory?)

    Homework Statement Initially, you have a one dimensional square well potential with infinitely high potential fixed at x = 0 and x = a. In the lowest energy state, the wave function is proportional to sin (kx). If the potential is altered slightly by introducing a small bulge(symmetric about...
  39. C

    Time independent perturbation theory

    H=H0 + lambda * W lambda << 1 must hold and the matrix elements of W are comparable in magnitude to those of H0. More precisely, the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0. I don't understand what is the meaning of " the matrix...
  40. C

    Time independent perturbation theory

    H=H0 + lambda * W lambda << 1 must hold and the matrix elements of W are comparable in magnitude to those of H0. More precisely, the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0. I don't understand what is the meaning of " the matrix...
  41. T

    WKB and perturbation theory.

    for a Hamiltonian H=H_0 + \epsilon V(x) my question is (for small epsilon) can WKB and perturbative approach give very different solutions ?? to energies eigenvalues and so on the index '0' means that is the Hamiltonian of a free particle. problem arises perhaps in calculation of...
  42. G

    Degenerate perturbation theory question

    This's a question from Griffiths, about degenerate pertrubation theory: For \alpha=0, \beta=1 for instance, eq. 6.23 doesn't tell anything at all! What does it mean "determined up to normalization"?. Equations 6.21 and 6.23 involve 3 unknowns (\alpha, \beta, E^1), and Griffiths solved them...
  43. A

    Challenge/discuss/help? introductory quantum mechanics and perturbation theory

    We discussed this problem in class to some extent, and I'd just like to post it here so that I can continue the discussion on the conceptual physics of it as well as the algebra. I believe a lot can be learned from this problem. "When an atom is placed in a uniform external electric field...
  44. M

    Quantum: Perturbation Theory

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  45. quasar987

    Probably easy perturbation theory question (quantum)

    Note that the post is long but only because I wanted to make the content cristal clear. The same post could easily have been 10 lines long. Homework Statement A spinless particle of charge q is in a spherically symetric potentiel V(r). The energy levels depend on l but not on m_l. The system...
  46. K

    Perturbation theory and Path integrals.

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  47. S

    Understand Quantum-Confined Stark Effect in Nanoparticles

    Please help me try to understand this problem. It deals with the quantum-confined Stark effect in nanoparticles. For odd n, n = 1, 3, 5, ... \psi_{n}(x) = \sqrt{\frac{2}{a}} \cos (\frac{n \pi x}{a}) and for even n = 2, 4, 6, ... \psi_{n}(x) = \sqrt{\frac{2}{a}} \sin (\frac{n \pi x}{a}) and...
  48. H

    Perturbation Theory Homework: Find Eigenvalues to 2nd Order

    Homework Statement I'm given that a harmonic oscillator is in a uniform gravitational field so that the potential energy is given by: V(x)=\frac{1}{2}m\omega^2x^2 - mgx, where the second term can be treated as a perturbation. I need to show that the first order correction to the energy of a...
  49. E

    Time-Independent Perturbation Theory

    Hi, I'm working out the 2nd Edition of Quantum Mechanics by Bransden & Joachain and I'm a little puzzled by the sign of the last term in equation 8.30 on page 380, which reads... a_{nl}^{(2)} = \frac{1}{E_n^{(0)} - E_l^{(0)}}\sum_{k{\neq}n} \frac{H_{lk}^{'}H_{kn}^{'}}{E_n^{(0)} - E_l^{(0)}}...
  50. E

    Conceptual problem with perturbation theory

    -Ok..Let,s be the Hamiltonian H=H_0 +W in one dimension where W is a "weak" term so we can apply perturbation theory. -The "problem" comes when we need to calculate the eigenvalues and eigenfunction of H0 of course we set the system in an "imaginary potential well of width L" so we have the...
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