Perturbation Definition and 426 Threads

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. 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...
  2. 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...
  3. R

    How Do You Compute Ground-State Energy Using First-Order Perturbation Theory?

    Homework Statement Consider a perturbed hydrogen atom whose Hamiltonian, in atomic units, is: H= -1/2(∆^2) – ½ + b/(r^2) (∆ should be upside down), where b is a positive constant. The Schrodinger equ. for this hamiltonian can be solved exactly for the energy eigenvalues. The...
  4. A

    Understanding Time-Dependent Perturbation Theory's 2nd Order Term

    Time dependent perturbation theory... second order term... For some reason they replace <E_{n}|H_{0}^2|E_{m}> with \Sigma<E_{n}|H_{0}|E_{i}><E_{i}|H_{0}|E_{m}> I know why they are allowed to do this, what I don't understand is how it makes my life better?
  5. 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)}}...
  6. N

    Perturbation vs Bifurcation Theory

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  7. E

    Conceptual problem with perturbation theory

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  8. K

    Perturbation Theory: Solving a/r+br Potential

    A quick question from a high school student, In perturbation theory, what is to be done with the found energy correction? I'm working out the solution to an a/r+br potential and using br as the perturbation. I set up the integral and normalized, but what do I do with the expression that I'm...
  9. E

    Divergent series in perturbation theory of quantum field theory

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  10. E

    Exact Perturbation Theory: Solving H0 and Obtaining Precise Energies

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  11. C

    Perturbation theory using Cohen-Tannoudji

    I'm reading the Cohen-Tannoudji book and I found somthing I don't understand in stationary perturbation theory. the problem the Hamiltonian is split in the known part an the perturbation: H=H_{o}+\lambda \hat{W} H_{o}|\varphi_{p}^{i}\rangle=E_{p}^{o}|\varphi_{p}^{i}\rangle (1) and...
  12. B

    How to Calculate Energy Corrections using Perturbation Theory

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  13. E

    Tough Perturbation Question (At least for me)

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  14. E

    Perturbation Theory: Physically Explained & Applied

    I have been studying Perturbation theory in my Quantum class but my professor has not really explained why physically it comes into play. The book says that perturbation theory is used to help come up with approximate solutions to the Schrodinger Equation. Is this analagous to how we use Fourier...
  15. M

    Degenerate perturbation theory question

    Can anybody explain what Griffiths means when he talks about "good eigenstates" in degenerate time-independent purturbation theory? Mathematically, I know he is just talking about the eigen-vectors of the W matrix (where Wij = <pis_i|H'|psi_j>). But what do the eigen-vectors physically...
  16. J

    Perturbation of hydrogen energy due to nucleus

    Consider the ground state of the hydrogen atom. Estimate the correction \frac{\Delta E}{E_1s} caused by the finite size of the nucleus. Assume that it is a unifromly charged shell with radius b and the potential inside is given by \frac{-e^2}{4\pi \epsilon b} Calculate the first order...
  17. M

    Studying up on an aspect of perturbation theory

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  18. J

    Understanding Perturbation Theory for the Harmonic Oscillator

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  19. A

    Infinite Series from Perturbation Theory

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  20. D

    Perturbation Theory: Calculating Ground State Eigenfunction of Particle in a Box

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  21. R

    Perturbation and Variational Theory

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  22. C

    How Does Delta_k Transform in a Uniformly Moving Frame in Fourier Space?

    can somebody help me to find an expression for the density contrast (in fouruer space; delta_k) in a moving frame. Basically I am trying to figure out how various quantities like power spectrum P(k) etc., will look in a uniformly moving frame .
  23. J

    Understanding Energy Corrections in a Perturbed Square Well Potential

    we have a particle in an infinite one-dimensional square well potential [V(x)=0 for 0<x<L and V(x) is infinite otherwise] and introduce a small potential (perturbation) in the middle of the square well potential. Then the first order energy correction for the ground state is 100 times...
  24. kakarukeys

    How Does Time-Dependent Perturbation Theory Explain Quantum State Evolution?

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  25. M

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  26. E

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