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. D

    Time dependant perturbation theory

    Hi, I am basically trying to put a wavefunction into the Time Dependant Schrodinger Eqn, as shown in my lecture notes, but i don't understand one of the steps taken... |\right \Psi (t)\rangle=\sum c_n (t) |\right u_n\rangle e^-(\frac{E_n t}{\hbar}) into i\hbar \frac{\delta}{\delta t}|\right...
  2. J

    First order perturbation theory

    The potential of an electron in the field of a nucleus is: -Ze^2/(4 Pi Epsilon0 r) r > r0 -Ze^2/(4 Pi Epsilon0 r0) r <= r0 where r0 is the fixed radius of the nucleus. What is the pertubation on the normal hydrogenic Hamiltonian? Calculate the change in energy of the 1s state to the first...
  3. E

    Perturbation in Infinite Square well

    Homework Statement Calculate the 1st order probability an electron in the ground state of an infinite sqaure well (width 1) will be found in the first excited state t seconds after the pertubation H=sin(PI*x) is switched on. Homework Equations Transition frequency is omega_12 The Attempt at a...
  4. MathematicalPhysicist

    Another 2 questions in perturbation theory.

    Homework Statement 1. A particle of mass M is in a square well, subject to the potential: V(x)= V0\theta(x-a/2) for x in (0,a) and diverges elsewhere, where theta is heaviside step function. In perturbation theory, find O(V0^2) correction to the energy and O(V0)to the eigenstate. 2. A...
  5. R

    Understanding Curvature Perturbation in Cosmology

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  6. MathematicalPhysicist

    Calculating Electron Energy Shift with Perturbation Theory

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  7. N

    Time-dependent perturbation theory

    Hi all Please look at this link (Search for the phrase "The quantum state at each instant can be expressed as a linear combination of the eigenbasis"): http://en.wikipedia.org/wiki/Perturbation_theory_%28quantum_mechanics%29 If we write the wavefunction for the perturbed system as a...
  8. A

    How Does Degenerate Perturbation Theory Apply to Matrix Elements?

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  9. G

    Measurement collapse as a perturbation?

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  10. Z

    Perturbation Theory: 2D Harmonic Oscillator & Energy Levels

    Homework Statement 1. Considered the 2D harmonic oscillator potential, V(x,y) = m\omega^{2}x^{2}/2+m\omega^{2}y^{2}/2+ \lambda xy and showed that the energy eigenvalues could be found exactly. Now, treat this as a perturbation theory problem with perturbing Hamiltonian, H^{'}=\lambda xy...
  11. J

    Perturbation theory in 3D potential

    Homework Statement Consider a quantum particle of mass m in a 3-D harnonic potential with frequency \omega and it experiences a perturbation H_{1}=az^{2} a. Determine the effect of H_{1} on the 1st exicted level of the system ( at the 1st order perturbation) b. what happen to L^{2} and...
  12. L

    Perturbation Theory: Square Well & Energy Levels

    a particle moves in one dimension in the potential V(x)=\infty \forall |x|>a, V(x)=V_0 \cos{\frac{\pi x}{2a}} \forall |x| \leq a now the unperturbed state that i use is just a standard infinite square well. anyway the solution says that perturbation theory is only valid provided that...
  13. G

    Asymptotic behaviour after perturbation

    Is it possible to write down a statement about the asymptotic behaviour of the wave function after a small perturbation has been switched on? So I have and initial wave function \psi_0 and the Hamiltonian H=\begin{cases} H_0 & t<0\\ H_0+V+R & t\geq 0 \end{cases} where V is a small perturbation...
  14. T

    Stationary Perturbation Theory

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  15. H

    Exploring Degenerate Perturbation Theory for Finding Zero Order Eigenstates

    Hi, I know that for degenerate states, we need to apply degenerate perturbation theory by looking at the perturbative hamiltonian in the subspace of the degenerate states. What then if the states still degenerate after we cast them the generate subspace. Is there a way to find the zero...
  16. Q

    A few questions about perturbation and string theory

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  17. 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...
  18. K

    Perturbation Technique for Temperature Distribution Along Fin

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  19. E

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

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  20. E

    Quantum Mechanics: Degenerate Perturbation Theory on square well

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  21. Angelos K

    Derivation of (linearized) perturbation equations

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  22. L

    Perturbation approximation of the period of a pendulum

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  23. C

    Limits of Perturbation theory for hydrogenic atoms

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  24. F

    Components of metric perturbation in TT gauge

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  25. O

    Particle in a box - perturbation theory

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  26. Q

    QM 1st order perturbation theory

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  27. E

    How Does Perturbation Theory Apply to Quantum Mechanics Eigenvalue Problems?

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  28. J

    Quick question about a simple perturbation theory question

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  29. D

    Time-Dependent Perturbation Theory - Two-level System

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  30. T

    Timescale to propagate a perturbation through a system?

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  31. N

    Griffiths' explanation of degenerate perturbation theory

    Hi all. I'm reading about time-independent perturbation theory for degenerate states in Griffiths' Introduction to QM. I have a question on the things he writes in chapter 6.2, page 269. What does he mean by the so-called "good" linear combinations? I hope you can shed some light on...
  32. I

    Degenerate time indep. perturbation theory

    why in time independent degenerate perturbation we diagonalize the matrix of the perturbation part of the hamilitonian and not the original hamiltonian?
  33. 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...
  34. M

    Perturbation Theory with Symmetric Rotator

    Homework Statement Given the Hamiltonian and perturbation below, what are the energy shifts for the states with l=1 Given H_{0}=(L^2)/(2I) H_{1}=E_{1}cos\vartheta Homework Equations L= r x P The Attempt at a Solution in order to find the first order correction to the energy...
  35. P

    Perturbation theory energy shift for hydrogen atom

    I'm trying to follow some working by lecturer; Treating delK (previously found in first bit of question), show that the energy En of the usual hydrogenic state [nlm> is shifted by some expression given. basically we start with \[ \frac{1}{2m_{0}c^{2}} \left\langle...
  36. 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...
  37. A

    Simple time-independent perturbation problem. QM

    Homework Statement "Suppose we put a delta-function in the center of the infinite square well: {H^{'}} = \alpha\delta(x-a/2) where a is a constant. Find the first order correction to the allowed energies. Explain why the energies are not peturbed for even n" Homework Equations The...
  38. 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...
  39. B

    Virtual particles and perturbation theory

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  40. 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 =...
  41. 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...
  42. N

    Hydrogen atom - small perturbation

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  43. T

    Poincare's method in perturbation theory

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  44. T

    Perturbation Theory - Poincare Method

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  45. P

    QM perturbation theory: rigid diatomic molecule

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  46. J

    Understanding Perturbation Theory: Solving for Roots and Coefficients

    From the following attachments I understand how the roots of the equation and the perturbation coefficients were found. What I don't get is the solid line in the graph that is allegedly the plot of two of the three roots versus epsilon. Can somebody clear this up for me? Also, how would I...
  47. O

    Standard perturbation theory - what exactly is meant?

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  48. V

    Simple degenerate perturbation problem.

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  49. J

    Sudden perturbation approximation for oscillator

    Homework Statement An oscillator is in the ground state of H = H^0 + H^1 , where the time-independent perturbation H^1 is the linear potential (-fx). It at t = 0, H^1 is abruptly turned off, determine the probability that the system is in the nth state of H^0 . Homework...
  50. J

    Sudden Perturbation Approximation Question

    Homework Statement In a beta decay H3 -> He3+, use the sudden perturbation approximation to determine the probability of that an electron initially in the 1s state of H3 will end up in the |n=16,l=3,m=0> state of He3+ Homework Equations |<n'l'm'|nlm>|^2 The Attempt at a Solution...
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