What is Perturbation: Definition and 421 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. ShayanJ

    A Degenerate perturbation theory -- Sakurai

    I'm reading section 5.2 "Time-Independent Perturbation Theory: The Degenerate Case" of the book "Modern Quantum Mechanics" by Sakurai and Napolitano and I have trouble with some parts of the calculations. At firsts he explains that there is a g-dimensional subspace(which he calls D) of...
  2. S

    Change in ground state energy due to perturbation

    Homework Statement Consider a quantum particle of mass m in one dimension in an infinite potential well , i.e V(x) = 0 for -a/2 < x < a/2 , and V(x) =∞ for |x| ≥ a/2 . A small perturbation V'(x) =2ε|x|/a , is added. The change in the ground state energy to O(ε) is: Homework Equations The...
  3. dwdoyle

    Degenerate Perturbation Theory and Matrix elements

    Homework Statement I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive...
  4. gasar8

    Hamiltonian and first order perturbation

    Homework Statement [/B] Particle is moving in 2D harmonic potential with Hamiltonian: H_0 = \frac{1}{2m} (p_x^2+p_y^2)+ \frac{1}{2}m \omega^2 (x^2+4y^2) a) Find eigenvalues, eigenfunctions and degeneracy of ground, first and second excited state. b) How does \Delta H = \lambda x^2y split...
  5. J

    How Do You Determine Rotational Invariance in a Two-Fermion System?

    Homework Statement Homework EquationsThe Attempt at a Solution I suppose to determine if a hamiltonian is rotational invariant, we check if [H(1),L^2], however, I am not sure how to do it if the hamiltonian is operate on a two particle wave function. Is it just to evaluate [S1z Z2 +S2z Z1...
  6. S

    Perturbation theory, Intermediate states, Virtual particles

    The following is taken from page 13 of Peskin and Schroeder. Any relativistic process cannot be assumed to be explained in terms of a single particle, since ##E=mc^{2}## allows for the creation of particle-antiparticle pairs. Even when there is not enough energy for pair creation, multiparticle...
  7. P

    Time independent perturbation theory

    In my course notes for atomic physics, looking at time independent perturbation for the non-degenerate case, we have the following: http://i.imgur.com/ao4ughk.png However I am confused about the equation 5.1.6. We know that < phi n | phi m > = 0 for n =/= m, so shouldn't this mean that < phi n...
  8. S

    Deriving Lorentz transformations using perturbation theory

    Homework Statement Derive the transformations ##x \rightarrow \frac{x+vt}{\sqrt{1-v^{2}}}## and ##t \rightarrow \frac{t+vx}{\sqrt{1-v^{2}}}## in perturbation theory. Start with the Galilean transformation ##x \rightarrow x+vt##. Add a transformation ##t \rightarrow t + \delta t## and solve for...
  9. S

    Harmonic oscillator perturbation

    Homework Statement Consider the one-dimensional harmonic oscillator of frequency ω0: H0 = 1/2m p2 + m/2 ω02 x2 Let the oscillator be in its ground state at t = 0, and be subject to the perturbation Vˆ = 1/2 mω2xˆ2 cos( ωt )at t > 0. (a) Identify the single excited eigenstate of H0 for...
  10. I

    Time-dependent Perturbation theory -maths problems

    Hey guys, I signed up here because I needed some information on some quantum physics problems. My question is related to quantum physics, and more precisely the derivation of time dependent perturbation theory. First of all, I am not able to understand all the maths structures and formulas...
  11. ShayanJ

    Functional time-dependent perturbation theory

    Today, in my advanced particle physics class, the professor reminded the time-dependent perturbation theory in NRQM and derived the formula: ##\displaystyle \frac{da_m(t)}{dt}=-i \sum_n e^{-i(E_n-E_m)} \int_{\mathbb R^3}d^3 x \phi^*_m (\vec x) V(\vec x,t) \phi_n(\vec x)##. Then he said that...
  12. R

    Multiple-scale analysis for 2D Hamiltonian?

    I came across a technique called "multiple-scale analysis" https://en.wikipedia.org/wiki/Multiple-scale_analysis where the equation of motion involves a small parameter and it is possible to obtain an approximate solution in the time scale of $$\epsilon t$$. I am wondering if it is possible to...
  13. B

    [check exercise] Perturbation theory

    I have solved this exercise, but I'm not sure that it is good. Please, can you check it? A lot of thanks! 1. Homework Statement The hamiltonian is ##H_0=\epsilon |1><1|+5/2 \epsilon (|2><2|+|3><3|)## The perturbation is given by ##\Delta(|2><3|+|3><2|)## Discuss the degeneration of H0. Using...
  14. B

    [quantum mechanics] Perturbation theory in a degenerate case

    Homework Statement I'm trying to understand how we can find - at the first order - the energy-shift and the eigenstates in a degenerate case. My notes aren't clear, so I have searched in the Sakurai, but the notation is different, I have read other notes but their notation is different...
  15. H

    Confused about perturbation theory with path integrals

    Homework Statement Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you...
  16. M

    New Energy Levels for Degenerate Perturbation Theory

    Homework Statement The e-states of H^0 are phi_1 = (1, 0, 0) , phi_2 = (0,1,0), phi_3 = (0,0,1) *all columns with e-values E_1, E_2 and E_3 respectively. Each are subject to the perturbation H' = beta (0 1 0 1 0 1 0 1 0) where beta is a positive constant...
  17. A. Neumaier

    Insights Causal Perturbation Theory - Comments

    A. Neumaier submitted a new PF Insights post Causal Perturbation Theory Continue reading the Original PF Insights Post.
  18. Runei

    Time-Dependent Perturbation Theory & Completeness

    Hello! I just want to make sure that I have understood the following argument the correct way: For a given quantum system we take the hamiltonian to be a time-independent (and soluble) part, and a time-dependent part. ## \hat{H} = \hat{H_0} + H'(t) ## Now, the solutions to the unperturbed...
  19. gonadas91

    Perturbation theory in strong interaction regime

    In QFT, we can expand the propagator and obtain the diagrammatic expansion to build up the Green's function. If we have a hamiltonian of the type H = H_{0}+V, where V is the perturbation, we can build up the Feynman diagrams,and if we could build up all of them to infinite order, we would...
  20. A

    How do you plot eigenfunctions of perturbed HO?

    Homework Statement Find eigenvalues and eigenvectors of a perturbed harmonic oscillator (H=H0+lambda*q4 numerically using different numerical methods and plot perturbed eigenfunctions. I wrote a code in c++ which returns a row of eigenvalues of the perturbed matrix H and a matrix of...
  21. C

    Applicability of perturbation theory

    Consider some system in some initial state ##|k^{(0)}\rangle##. The probability that such a state makes a transition to some other state ##|m^{(0)}\rangle## can be computed to various orders in time dependent perturbation theory. E.g the total first order probability that the system has made a...
  22. C

    Time dependent perturbation theory of the harmonic oscilator

    Homework Statement A 1-d harmonic oscillator of charge ##q## is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form ##E(t) = \frac{K }{\sqrt{\pi} \tau} \exp (−(t/\tau)^2) ##. Assuming that when ##t = -\infty##, the...
  23. quantatanu0

    Cut-off Regularization of Chiral Perturbation Theory

    I was trying to learn renormalization in the context of ChPT using momentum-space cut-off regularization procedure at one-loop order using order of p^2 Lagrangian. So, 1. There are counter terms in ChPT of order of p^4 when calculating in one-loop order using Lagrangian of order p^2 . 2...
  24. W

    QM: Work done due a time dependent perturbation

    Homework Statement A quantum particle of mass ##m## is bound in the ground state of the one-dimensional parabolic potential well ##\frac{K_0x^2}{2}## until time ##t=0##. Between time moments of ##t=0## and ##t=T## the stiffness of the spring is ramped-up as ##K(t) = K_0...
  25. C

    Degenerate perturbation theory for harmonic oscillator

    Homework Statement [/B] The isotropic harmonic oscillator in 2 dimensions is described by the Hamiltonian $$\hat H_0 = \sum_i \left\{\frac{\hat{p_i}^2}{ 2m} + \frac{1}{2} m\omega^2 \hat{q_i}^2 \right\} ,$$ for ##i = 1, 2 ## and has energy eigenvalues ##E_n = (n + 1)\hbar \omega \equiv (n_1 +...
  26. L

    How's diffraction pattern modied by phonon perturbation?

    Hi there, I have a problem on phonon perturbation's effect on diffraction pattern. Assume atomic planes parallel to (100) of bcc lattice is periodically perturbed by phonon. How will diffraction pattern be modified as a result of such perturbation? Will we see any diffraction peaks in addition...
  27. Ahmad Kishki

    Perturbation theory in qm self study

    i am currently self studying qm, and i am trying to plan ahead since i am relatively over with griffiths part1 (which is the theory part) and i was wondering if i should go ahead to part 2 (applications) or should i just keep this for later and attempt to stregnthen my basics in qm from another...
  28. K

    Variance and 2 point function in perturbation theory

    When we try to find the statistical correlation of some perturbation between two positions, we always calculate the quantum 2-point function. Are these two concepts really the same? Also, people say vacuum fluctuation is gaussian. For normalized fields, we always use Bunch-Davies initial...
  29. K

    Question regarding time independent perturbation theory

    Let's say we've a system which can be described by the Hamiltonian: $$H_0 = \dfrac{p^2}{2m} + V(x)$$ Now suppose we introduce a perturbation given by: $$H_1 = \lambda x^2$$ Our total hamiltonian: $$H = H_0 + H_1 = \dfrac{p^2}{2m} + V(x) + \lambda x^2 $$ Normally, the perturbation doesn't...
  30. B

    Time independent perturbation theory

    Homework Statement The following text on the time independent perturbation theory is given in a textbook: \hat{H} = \hat{H}_0 + \alpha \hat{H'} We expand its eigenstates \mid n \rangle in the convenient basis of \mid n \rangle^{(0)} \mid n \rangle = \sum_m c_{nm} \mid m \rangle^{(0)}...
  31. ShayanJ

    Perturbation proportional to momentum

    Homework Statement I'm struggling with the problem below: If E_n^{(0)} \ \ (n\in \mathbb N) are the energy eigenvalues of a system with Hamiltonian H_0=\frac{p^2}{2m}+V(x) , what are the exact energy eigenvalues of the system if the Hamiltonian is changed to H=H_0+\frac{\lambda}{m} p \ ...
  32. R

    First-order perturbation for a simple harmonic potential well

    Homework Statement The ground state of the wavefunction for an electron in a simple one-dimensional harmonic potential well is \Psi _{0}(x)= \left ( \frac{m\omega }{\pi \hbar} \right )^{1/4} exp(-\frac{m\omega x^{2}}{2\hbar}) By employing first-order perturbation theory calculate the energy...
  33. fluidistic

    Sudden perturbation, particle in a box, ground state

    Hi guys! Suppose there's a particle in a box, initially in its ground state. Suppose that one chooses a system of coordinates such that the potential V(x) is 0 from 0 to L. Suppose that one suddenly perturbate the system at a particular time so that V(x) becomes 0 from 0 to 2L. I've calculated...
  34. Z

    Solution distribution of degenerate perturbation secular EQ

    When two states |k> and |k'> degenerate, a perturbation H' would lead to an energy split of <k|H'|k'>. As the number of degenrate states increases, the order of the secular equation rises correspondingly (and the equation could hardly be solved ?) My question is: is there any knowledge of the...
  35. S

    Validity of perturbation theory

    I was wondering why perturbation theory works in quantum mechanics. My lecturer said that no one really bothered why it worked anyway, until they found it gave problems in QFT and came back to non-relativistic quantum mechanics and found why it worked in this domain. Can anybody explain?
  36. Z

    First order correction of wavefunction in degenerate perturbation

    In first order correction of wavefunction, |ψ(1)n>=∑ψ(0)m<ψ(0)m|V|ψ(0)n>/(E(0)n−E(0)m) when any two of the original states degenerate, we replace the two states with their corresponding "good states" to get a new set of "undisturbed" states (ψ(0)m), AND then we determine the first order...
  37. P

    Spin-orbit Interaction & Degenerate Perturbation Theory

    Hello! This is my first time posting, so please correct me if I have done anything incorrectly. There's something that I don't understand about the spin-orbit interaction. First of all I know that [\hat{S} \cdot \hat{L}, \hat{L_z}] \ne 0 [\hat{S} \cdot \hat{L}, \hat{S_z}] \ne 0 so this means...
  38. ddd123

    Is Schiff's Quantum Mechanics wrong? Degenerate stationary perturbation theory.

    Homework Statement However incorrect the text seems to me, I suspect there's something I'm missing, since it's a renowned text: Schiff - Quantum Mechanics 3rd edition 1968. The topic is degenerate stationary perturbation theory. In this example there's only two eigenfunctions associated with...
  39. K

    Temperature at horizon entry

    Homework Statement How do I calculate the temperature at which a galactic scale perturbation enters the horizon? This would be for radiation domination. Homework Equations \left( \frac{\delta \rho}{\rho} \right)_{\lambda_0} (t) = \left( \frac{a(t)}{a_{eq}} \right) \left( \frac{\delta...
  40. T

    Degenerate Perturbation Theory

    Homework Statement We have spin-1 particle in zero magnetic field. Eigenstates and eigenvalue of operator \hat S_z is - \hbar |-1> , 0 |0> and \hbar |+1> . Calculate the first order of splitting which results from the application of a weak magnetic field in the x direction. Homework...
  41. DataGG

    Perturbation theory, second-order correction - When does the sum stop?

    I've no idea if I should be posting this here or in the general forums. This is not really an exercise as much as an example. I'm not understanding something though: 1. Homework Statement Using perturbation theory, find the exact expression for the energy given by the hamiltonian...
  42. K

    Weinberg 3.4 -- Derive the perturbation expansion....

    Homework Statement Basically I wanted to see if anyone would be willing to give me the solution to the 4th problem of the Weinberg textbook on quantum field theory. The exact question in the book is "Derive the perturbation expansion (3.5.8) directly from the expansion (3.5.3) of old-fashioned...
  43. Q

    Asymptotic perturbation theory

    Having just watched Prof Carl Bender's excellent 15 lecture course in mathematical physics on YouTube, the following question arose: The approach was to work in one space dimension and to solve the schrodinger equation for more general potentials than the harmonic oscillator using asymptotic...
  44. G

    Origin of infrared divergences in perturbation theory

    If you have a momentum integral over the product of propagators of the form \frac{1}{k_o^2-E_k^2+i\epsilon} , why are there divergences associated with setting m=0? Factoring you get: \frac{1}{k_o^2-E_k^2+i\epsilon}=\frac{1}{(k_o-E_k+i\epsilon) (k_o+E_k-i\epsilon)} . This expression has...
  45. S

    Perturbation Theory - First Order Approximation

    If: ##\hat{H} \psi (x) = E \psi (x)## where E is the eigenvalue of the *disturbed* eigenfunction ##\psi (x)## and ##E_n## are the eigenvalues of the *undisturbed* Hamiltonian ##\hat{H_0}## and the *disturbed* Hamiltonian is of the form: ##\hat{H} = \hat{H_0} +{\epsilon} \hat{V}...
  46. P

    Particle on a ring with perturbation

    So I'm trying to solve old qualifying exam problems, one of which is a particle on a ring with a constant electric field perturbation. The un-perturbed problem is straightforward, and we then add a constant electric field in the x-direction (the ring lies in the xy-plane) of magnitude E...
  47. carllacan

    Perturbation of a degenerate isotropic 2D harmonic oscillator

    Homework Statement A two-dimensional isotropic harmonic oscillator of mass μ has an energy of 2hω. It experiments a perturbation V = xy. What are its energies and eigenkets to first order? Homework Equations The energy operator / Hamiltonian: H = -h²/2μ(Px² + Py²) + μω(x² + y²) The...
  48. carllacan

    Time-dependent delta-function perturbation

    Homework Statement We have a system whose state can always be expressed as the sum of two states ##\Psi_a## and ##\Psi_b##. the system undergoes a perturbation of the form ##H'=U\delta(t)##, where ##\delta## is the delta-function in time and ##U_{aa} = U_{bb} = 0## and ## U_{ab} = U_{ba}^*##...
  49. carllacan

    Conmutative Hermitian operator in degenerate perturbation theory

    Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...
  50. 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...
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