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. H

    Probability in first order time-dependent perturbation theory

    Hi , Can anybody help me to solve this question? A time varying Hamiltonian H(t) induces transitions from state |k> at time t=0 to a state |j> at time t=t' with probability P(k to j(t')). Use first order time-dependent peturbation theory to show that if P(j to k(t')) is the prababilty that...
  2. T

    Two-state perturbation problem (QM)

    Homework Statement Given is a hamiltionan H0: E1 0 0 E2 with E1 and E2 being eigenvalues of two eigenstates phi1 and phi2 A distortion W, with W12 = W21* (complex conjugate): 0 W12 W21 0 Calculate the eigenvalues and eigenstates of H = H0 + W Homework...
  3. V

    Probability of staying in same state after time-dep perturbation?

    Let's say that you've got a time-dependent perturbation to your potential (say, the particle-in-a-box to make things simple). Say you start in energy eigenstate #3. What's the probability that the particle will stay in eigenstate 3 after time T? This is not a homework problem. I'm not...
  4. E

    Spins 1/2 and Time-Dependant Perturbation Theory

    Homework Statement We consider two spins 1/2, \vec{S_{1}} and \vec{S_{2}}, coupled by an interaction of the form H=\alpha(t)\vec{S_{1}}*\vec{S_{2}}. \alpha(t) is a function of time who approches 0 for |t|-->infinity and takes appreciable values only in the interval of [-\tau,\tau] near 0...
  5. T

    Degenerate Perturbation Theory Question

    Hello, This is a question on perturbation theory - which I am trying to apply to the following example. Homework Statement The two-dimensional infinitely deep square well (with sides at x=0,a; y=0,a) is perturbed by the potential V(x)=\alpha(x^{2}+y^{2}). What is the first-order correction...
  6. T

    Quantum perturbation corrections

    Homework Statement Substituting |n> = |n^{0}> + \sum_{m}c_{nm}|m^{0}> into the normalisation condition <n|n> = 1 show that the correction c_{nn}^{(1)} is zero. The Attempt at a Solution <n|n> = 1 => \left( <n^{0}| + \sum_{m}c_{nm}^{*}<m^{0}| \right) \left( |n^{0}> +...
  7. S

    Perturbation spectrum after inflation

    The spectrum of perturbations at the end of inflation, which turns out to be flat and slightly red shifted, is calculated under the assumption that each perturbation mode crosses some initial time in the comoving coordinates with 'minimum energy allowed by the Heisenberg uncertainty principle'...
  8. M

    Convergence Analysis of Perturbation Theory with Divergent Quantities

    Hi, i am stuck at this problem , let be the divergent quantity m= clog(\epsilon) +a_{0}+a_{1}g\epsilon ^{-1}+a_{2}g\epsilon ^{-2} +a_{3}g\epsilon ^{-3}+...+ where epsilon tends to 0 and g is just some coupling constant and c ,a_n are real numbers. then i use the Borel transform of the...
  9. cepheid

    Understanding Griffiths' Perturbation Theory in Quantum Mechanics

    I'm looking at the beginning of of Chapter 6 of the 2nd edition of Griffiths Introduction to Quantum Mechanics. He starts out by writing the hamiltonian for a system we'd like to solve as the sum of a hamiltonian with a known solution and a small perturbation: H^0 + \lambda H^\prime He...
  10. N

    Constant force perturbation of the quantum SHO

    [SOLVED] Constant force perturbation of the quantum SHO Homework Statement We're supposed to consider the Hamiltonian for the simple harmonic oscillator: \hat{H}_{0} = \hat{p}^{2}/2m + m\omega^2\hat{x}^2/2 With a perturbation, so that \hat{H} = \hat{H}_0 + \hat{H}' , where \hat{H}' =...
  11. U

    A hydrogen atom in a weak time-dependent perturbation

    Homework Statement A hydrogen atom, which is in its ground state 1s (i.e. \|1,0,0\rangle), is put into a weak time-dependent external electric field, which points into the z direction: \boldsymbol{E}(t,\boldsymbol{r}) = \frac{C\hat{\text{\boldsymbol{e}}}_{z}}{t^{2}+\tau^{2}}, where C and...
  12. R

    Perturbation Theory energy shift

    Homework Statement I'm trying to calculate the energy shift given an electron in a 1D harmonic potential has a wavefunction \Psi_{0}(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}exp\left(\frac{-m\omega x^{2}}{2\hbar}\right) The shift in E_{0} = \frac{\hbar\omega}{2} = 2eV due to...
  13. 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...
  14. W

    Harm osc, anharmonic perturbation

    Homework Statement Particle bound by V = \frac{1}{2} m \omega^2 x^2 - a x^3 for small x. Show that the mean position of the particle changes with the energy of the eigenstates when a is small, so first order perturbation theory works. Homework Equations For the harmonic oscillator x...
  15. 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}...
  16. 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...
  17. 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...
  18. N

    Perturbation of the simple harmonic oscillator

    [SOLVED] Perturbation of the simple harmonic oscillator Homework Statement An additional term V0e-ax2 is added to the potential of the simple harmonic oscillator (V and a are constants, V is small, a>0). Calculate the first-order correction of the ground state. How does the correction change...
  19. E

    Another problem - time dependent perturbation and transition probabilities

    Homework Statement (the actual question is now as an attachment) Assuming that the perturbation V(x,t)=\betax exp(-\gamma t) is applied at t = 0 to a harmonic oscillator (HO) in the ground state, calculate to the first approximation the transition probabilty to any excited state n\geq1...
  20. 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...
  21. P

    2-D Harmonic Osc. with Perturbation

    2-D Harmonic Oscillator with Perturbation Homework Statement A 2-D harmonic oscillator has an energy Ensubxnsuby and wavefunctions \phinsubxnsuby The first excited states are 2-fold degenerate E_{01}=E_{10}=2\hbar\omega What are the energies and wavefunctions if we add the...
  22. N

    Spin-orbit coupling perturbation

    Homework Statement An electron in a hydrogen atom is in the n = 2, l = 1 state. It experiences a spin-orbit interaction H' = \alpha \mathbf{L} \cdot \mathbf{S}. Calculate the energy level shifts due to the spin-orbit interaction.Homework Equations Degenerate perturbation theory. The Attempt...
  23. 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...
  24. F

    Perturbation effect on harmonic osclillator

    Im confused about taking out the first and second order perturbation effect on a 1d harmonic oscillator.I get to the integration part but don't know where to go from there. for example if a term of ax^3 is added to the hamiltonian of the harmonic oscillator this is how i start.if i want to...
  25. S

    Problem on perturbation theory

    Homework Statement Determine approximately the ground state energy of a helium like atom using first order perturbation theory in the electron-electron interaction. Ignore the spins of the electrons and the Pauli principle. Homework Equations given that \intd\tau1\intd\tau2...
  26. 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...
  27. 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...
  28. 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...
  29. S

    Functional determinant approach to perturbation

    given the functional integral with 'g' small coupling constant \int \mathcal D [\phi]exp(iS_{0}[\phi]+\int d^{4}x \phi ^{k}) so k >2 then could we use a similar 'Functional determinant approach' to this Feynman integral ?? in the sense that the integral above will be equal to...
  30. S

    Haag's Theorem, Perturbation, Existence and QFT.

    Hello, I just reading and learning QFT and there is something I've been wondering, hopefully somebody here can help me. Let's say we have an interacting Quantum Field Theory, such as Quantum Electrodynamics if we want to compute an amplitude such as two electrons scattering off each other, then...
  31. V

    Perturbation Methods for 1st order ODE - Find the asymptotic solution

    Homework Statement Apply the regular perturbation method to solve the following ordinary differential equation \left(1\,+\,\epsilon\,y\right)\,\frac{dy}{dx}\,+\,y\,=\,0 subject to x\,=\,1;\,y\,=\,1 Show that the asymptotic solution is of the form...
  32. P

    Free particle with Coulomb Perturbation

    Homework Statement This is a question I have about something stated in a textbook without much explanation. From Richard D. Mattuck's "A guide to Feynman Diagrams in the Many-Body Problem" Appendix A.1 pg 337 "for example consider the Coulomb interaction between two electrons in a metal...
  33. P

    Perturbation Theory: Higher Order Energy + Deviation from True Solution?

    The higher order psi are written with larger epsilon factors in front but the higher order psi are precisely the ones that are meant to more exactly approach the purturbed system however we are decreasing its importance. Is the reason because they can also more easily deviate greater from the...
  34. V

    3rd order energy perturbation correction

    Homework Statement Derive the general expression of 3rd-order perturbation energy for a non-degenerate quantum system. Homework Equations for nth order we have (Ho-Eo)|n>+(H'-E1)|n-1> -E2\n-2>-En|0>=0 (given) also, <0|0>=1, <1|0> = <0|1>=0, <0|2>=<2|0>=-1/2<1|1>...
  35. V

    How Do You Find the Eigenenergies of a Rotor in an Electric Field?

    Homework Statement A plane (x-y plane) rotor has moment of inertia I and electric dipole moment P. A uniform electric field is applied along the x-direction (Ex, so the interaction energy can be written as H'=P*E cosΦ, where phi is the angle between the E-field and the dipole P...
  36. 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...
  37. D

    Primodial Curvature Perturbation Equation

    This equation gives us (delta (rho))/rho (which I understand is the fractional perturbation in the energy density), at the time of "horizon entry" (which I'm unsure about). Does this mean the time that decoupling occured?
  38. H

    Perturbation expansion when solving Kdv equation

    Hi all. I have actually asked this question in my another thread but seems not many people notice that. I am going to repeat the question here and hope to get some replies. The question is about assuming a perturbation series for the KdV equation. u_t + 6uu_x + u_xxx = 0 Why one would...
  39. 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...
  40. quasar987

    Quantum time dependant perturbation HW

    Homework Statement We consider an hydrogen atom in its fundamental state. At t=0, we apply an electric field in the z direction, \mathcal{E}(t)=\mathcal{E}_0e^{-t/\tau} What is the probability that the atom be in the state 2p at t>>\tau?The Attempt at a Solution I thought time dependant...
  41. quasar987

    How does TDPT handle time dependent perturbations?

    Hi, I have a problem with the very nature of time dependant perturbation theory (TDPT). In TDPT, we consider a system of Hamiltonian H(t) = H_0 (for t<0), H(t)=H_0 + kW(t) (for t>0) [where k<<1], where H_0 is, for simplicity, discrete, non-degenerate and time-independent, and given that at...
  42. 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...
  43. L

    Singly Ionized Li, Perturbation and Integral Help

    Homework Statement An approximate value for the ground state energy of a two electron atom can be found by starting with the energy of two noninteracting electrons in 1s states and calculating the first order correction due to the perturbation V= \frac{ {e}^{2} }{ 4 \pi {\epsilon}_{0} } \frac{...
  44. M

    Quantum: Perturbation Theory

    For a particle in a two-dimensional box. The particle is subject to perturbation V=Cxy. What are the eigenenergies and eigenfunctions of the unperturbed system and what is the first-order energy correction?
  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.

    Let's suppose we have a theory with Lagrangian: \mathcal L_{0} + gV(\phi) where the L0 is a quadratic Lagrangian in the fields then we could calculate 'exactly' the functional integral: \int\mathcal D[ \phi ]exp(iS_{0}[\phi]/\hbar+gV(\phi)) where J(x) is a source then we could...
  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. R

    First-order perturbation

    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...
  50. 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?
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