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. 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...
  2. 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...
  3. 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}^*##...
  4. 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 =...
  5. 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...
  6. U

    Cosine perturbation to potential well

    Homework Statement Part (b): Find the perturbed energy. Homework Equations The Attempt at a Solution I've solved everything, except part (b). I got an answer of 0 for part (b) for all orders, which is kind of strange, as one would expect some perturbation. \Delta E_n = \langle \psi_n...
  7. maverick280857

    Sakurai Degenerate Perturbation Theory: projection operators

    Hi, So, I am working through section 5.2 of Sakurai's book which is "Time Independent Perturbation Theory: The Degenerate Case", and I see a few equations I'm having some trouble reconciling with probably because of notation. These are equations 5.2.3, 5.2.4, 5.2.5 and 5.2.7. First, we...
  8. P

    Perturbation Theory: Finding Eigenfunctions and Energies

    In a text a exercice says that for the Hamiltonian ##H_0 = \frac{p^2}{2m}+V(x)## the eigenfunction and eigen energy are ##\phi_n, E_n##. If we add the perturbation ## \frac{\lambda}{m}p## ¿what is the new eigenfunction? The solution is ## \frac{p^2}{2m} + \frac{\lambda}{m}p+V=...
  9. U

    Perturbation Theory, exchange operator

    Homework Statement Part (a): Find eigenvalues of X, show general relation of X and show X commutes with KE. Part (b): Give conditions on V1, V2 and VI for X to commute with them. Part (c): Write symmetric and antisymmetric wavefunctions. Find energies JD and JE. Part (d): How are...
  10. U

    Hydrogen - perturbation theory question

    Homework Statement Part (a): Explain origin of each term in Hamiltonian. What does n, l, m mean? Part (b): Identify which matrix elements are non-zero Part (c): Applying small perturbation, find non-zero matrix elements Part (d): Find combinations of n=2 states and calculate change in...
  11. U

    Sudden Perturbation of Potential Well

    Homework Statement Part (a): Particle originally sits in well V(x) = 0 for 0 < x < a, V = ∞ elsewhere. The well suddenly doubles in length to 2a. What's the probability of the particle staying in its ground state? Part (b): What is the duration of time that the change occur, for the...
  12. U

    Quadratic Stark Effect - Perturbation Theory

    Homework Statement Homework Equations The Attempt at a Solution With a parity operator, Px = -x implies x has odd parity while Px = x implies x has even parity. Things that puzzle me 1. Why is ##[H_0,P] = 0## and ##H_1P = -PH_1##? Is it because ##H_1 \propto z## so ##Pz = -z##? Then...
  13. D

    Identical particles/ Time dependent perturbation theory

    Homework Statement Two identical spin-1/2 particles interact with Hamiltonian H0=ω0 S1.S2 where ω0>0. A time dependent perturbation is applied, H'=ω1 (S1z-S2z) θ(t) Exp[-t/τ], where ω1>0 and ω1<<ω0. What are the probabilities that a system starting in the ground state will be excited into each...
  14. Runei

    Perturbation theory (the math)

    My study of Quantum Mechanics have brought me to perturbation theory. I'm here talking about the non-degenerate type. My questions relate to the math behind it, and the power series expansion that we do. H = H^0 + \lambda H' (Eq. 1) Question 1: So in equation 1 I think I understand...
  15. G

    Why transition rate independent of time in perturbation theory?

    After time t, the probability of monochromatic absorption of the ground state |1> to the energy state |n> is given by: |<n|1>|^2=4|U_{n1}|^2\frac{\sin^2((E_n-E_1-\hbar\omega)t/2\hbar)}{(E_n-E_1-\hbar\omega)^2} where U is the transition matrix. The claim is that as t goes to infinity, the...
  16. X

    Perturbation Theory: Calculating Energy Corrections

    Hello Everyone. I am very confused on the following questions and have a few confusions about the problem that I hope someone can clear up for me (explained later). Here is the question. Homework Statement The paramagnetic resonance of a paramagnetic ion in a crystal lattice is described...
  17. X

    Degenerate Perturbation Theory: Two Spin 1/2 Particles

    So I know this might be a lot to read but I am having a very hard time understanding how to use the formulas in degenerate perturbation theory. Here is the problem I am on. Homework Statement A system of two spin-1/2 particles is described by the following Hamiltonian...
  18. L

    Newtonian limit of cosmological perturbation

    Homework Statement Problem in question is problem 5.6 in Dodelson's Modern Cosmology (https://www.amazon.com/dp/0122191412/?tag=pfamazon01-20) Take the Newtonian limit of Einstein's equations. Combine the time-time equation (5.27) with the time-space equations of exercise 5 to obtain the...
  19. F

    One-dimensional linear harmonic oscillator perturbation

    Homework Statement Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state Homework Equations ψn(x) = \frac{α}{√π*2n*n!}1/2 *...
  20. X

    Degenerate Perturbation Theory

    http://farside.ph.utexas.edu/teaching/qm/lectures/node53.html So I was reading this and I don't understand how he goes from 658 to 661 using the completeness relation. In 661 if you use the completeness relaton can you get rid of the I n,l''>s by doing the outer product and ignoring the...
  21. H

    Degenerate perturbation theory (Sakurai's textbook)

    In the theory of degenerate perturbation in Sakurai’s textbook, Modern Quantum Mechanics Chapter 5, the perturbed Hamiltonian is H|l\rangle=(H_0 +\lambda V) |l\rangle =E|l\rangle which is written as 0=(E-H_0-\lambda V) |l\rangle (the formula (5.2.2)). By projecting P_1 from the left (P_1=1-P_0...
  22. H

    Generally applicable development of classical perturbation theory

    Greetings, Does anyone know of some good sources that explain classical perturbation theory, preferably using the Lagrangian formalism? The sources that I have seen more-or-less say, "write L=L_{0}+λδL, where L_{0} is an unperturbed, soluble Lagrangian, δL is the perturbation, and λ is a small...
  23. D

    Degenerate Perturbation Theory (Particle in 3D box)

    Homework Statement Consider a particle confined in a cubical box with the sides of length L each. Obtain the general solution to the eigenvalues and the corresponding eigenfunctions. Compute the degeneracy of the first excited state. A perturbation is applied having the form H' = V from 0...
  24. H

    Electromagnetic radiation and perturbation

    Why do we often treat the electromagnetic radiation effects on Hamiltonian of a matter as a perturbation? In the other words, why the effects of radiation is so little that is treated as a perturbation?
  25. F

    Time independent perturbation - potential with inner function

    Homework Statement I need to find the corrected ground state for perturbed harmonic oscillator (1D) with perturbation of the form V(x) = λ sin(κx), κ>0. My problem is I have no idea how to handle a potential that has its operator as an inner function. Homework Equations The perturbed...
  26. M

    2nd Order Perturbation Coefficients.

    I have found an expression for the estimated energy contribution a term |I> will bring to a wavefunction |K> \Delta E = \frac{|\langle I|\hat{H}| K\rangle|^2}{(E_K - \langle I |\hat{H}| I\rangle)} Is there a simple way to extract the coefficient that will be associated with |I>? Even a link...
  27. O

    Can a basic knowledge of perturbation theory solve this?

    Hello all, I have boiled a very long physics problem down to the point that I need to solve the coupled equations \frac{\partial^2 x}{\partial u^2} + xf(u) + yg(u) = 0 \frac{\partial^2 y}{\partial u^2} + yf(u) - xg(u) = 0 We may assume that |f| ,|g| << 1. and that both f and g are...
  28. F

    E-field Perturbation of 2D rotor. Show Y_10 couples ground state?

    Homework Statement Consider a molecule with an electric dipole moment d. The Hamiltonian of a molecule in the external electric field E is: \hat{H} = \frac{\hat{L^2}}{2I} - dE \cos{\theta}, where the polar angle \theta characterises the orientation of the molecule. (We have chosen the field...
  29. Telemachus

    Time dependent perturbation theory, HO subject to electric field

    Hi there. I'm dealing with this problem, which says: At time ##t=0## a constant and uniform electric field ##\vec E## oriented in the ##\vec x## direction is applied over a charged particle with charge ##+q##. This same particle is under the influence of an harmonic potential...
  30. Telemachus

    Time independent perturbation (Quantum Mechanics)

    Hi there. I have to find the energy corrections through the perturbation method, and then give the exact result for the hamiltonian: ##H= \begin{pmatrix} E_A & \epsilon & \epsilon & \epsilon \\ \epsilon & E_B & 0 & 0 \\ \epsilon & 0 & E_B & 0\\ \epsilon & 0 & 0 & E_B \\ \end{pmatrix} ## So...
  31. B

    First order perturbation for hydrogen

    Homework Statement Assume that there is a deviation from Coulomb’s law at very small distances, the Coulomb potential energy between an electron and proton is given by V_{mod}(r)=\begin{cases} -\frac{e^{2}}{4\pi\varepsilon_{0}}\frac{b}{r^{2}} & 0<r\leq b\\...
  32. F

    How do I find the Taylor series for a function with multiple variables?

    Homework Statement I required to make a perturbation expansion in ε of the function: Homework Equations A(X,y,z)=A(x-εsin(wy),y,z). X=x-εsin(wy) The Attempt at a Solution Solution: A(X,y,z)=A0(X,z)+ε[A1(X,z)+∂/∂XA0(X,z)]sin(wy)+o(ε^2) I get the terms A0(X,z) and ∂/∂XA0(X,z)sin(w,y) with the...
  33. B

    Perturbation Theory on Finite Domains

    In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode y'' + P(x)y = 0 using perturbation theory. However he points out that the domain must be finite in order for this to work, I'm wondering how you would phrase a question...
  34. A

    Error in perturbation series question

    Suppose we know the perturbation series E = E(\epsilon) = E_0 + \epsilon E_1 + \epsilon^2 E_2 + \ldots converges, where E_0 is a discrete eigenvalue of H_0 and we are considering a Hamiltonian H = H_0 + \epsilon H_1. Does this mean that we know E - E_0 = O(\epsilon) as...
  35. H

    Sharks can sense the electric field perturbation?

    Sharks use an organ known has the ampullae of Lorenzini to sense electric fields, they can sense fields as weak as 5*10^-9 V/m. Which got me thinking, they should in principle be able to detect vacuum perturbation in the field, interesting?
  36. W

    What does a general PE have to to with a S-O perturbation? Stumped

    Homework Statement Show that for a general potential energy V(r) that the form of the spin-orbit Hamiltonian becomes \hat{H}_{S-O}=\frac{1}{2m_{e}^{2}c^{2}|\hat{\textbf{r}}|}\frac{d\hat{V}}{dr} \hat{\textbf{L}}\cdot\hat{\textbf{S}} Suggestion: Start with \textbf{B} = -( \textbf{v} /c)...
  37. T

    Degenerate perturbation theory degeneracy not lifted by perturbation

    Hi, I have an equation of the form (-i \lambda \frac{d}{dr}\sigma_z+\Delta(r)\sigma_x) g =(\epsilon + \frac{\mu \hbar^2}{2mr^2}) g where \sigma refers to the Pauli matrices, g is a two component complex vector and the term on the right hand side of the equation is small compared to the other...
  38. O

    Time independant perturbation problem

    Example from Schaum's Quantum Mechanics. Picture of the example is attached. What I don't understand is part (c). What are those wavefunctions ##\mid \psi^{(0)}_{1,2} \rangle## and ##\mid \psi^{(0)}_{2,1} \rangle##? How do I find these wavefunctions, if the unperturbed wavefunction is...
  39. O

    Time independant perturbation - Difficulty understanding derivation

    Hamiltonian is in the form ##H = H_0 + \lambda W##, where ##\lambda \ll 1## and ##W## is the perturbation. Assume the eigenstates ##\mid \psi(\lambda) \rangle## and engenenergies ##E(\lambda)## can be expanded in a power series of ##\lambda##. $$\mid \phi(\lambda) \rangle = \mid 0 \rangle +...
  40. C

    Finite Hilbert Space v.s Infinite Hilbert Space in Perturbation Theory

    Hi all, I have a question about the concept of complete set when I apply the perturbation theory in two situations -Finite Hilbert Space and Infinite Hilbert Space. Consider a Hamiltonian H=H0+H', where H0 is the unperturbed Hamiltonian and H' is the perturbed Hamiltonian. Let ψ_n be the...
  41. S

    Perturbation Techniques and Theory for Nonlinear Systems

    Homework Statement Given the equation \ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta} Determine a first-order uniform expansion for small but finite theta. Homework Equations Other than the equation above, none so far as I am aware. The Attempt at a...
  42. C

    Perturbation Theory for a Hamiltonian

    Hi guys, this is my first time posting, I'm studying physics at uni, in my third year and things are getting a bit tough, so basically my question is in relation to solving problem 1, (i included a picture...) I missed the class and don't really know what I'm doing. Any help would be appreciated.
  43. B

    First order perturbation question

    Homework Statement Suppose we put a delta function bump in the center of the infinite square well: H' = \alpha \delta(x -a/2), where \alpha is constant. a) Find the first order correction to the allowed energies. b) Find the first three non-zero terms in the expansion of the correction...
  44. M

    Momentum perturbation to harmonic oscillator

    Homework Statement the problem and a possible solution(obtained from a book) is attached as a pdf to the post.However Iam unable to understand it.Please download the attachment. Homework Equations equation no (2) in the pdf.Is there any use of space translation operator in here.The Attempt at...
  45. A

    Analyzing perturbation of vectors

    Homework Statement We have A \in R^{mxm} \text{ and } b \in R^{m} \text{ and } b \neq 0 \text{. Show that } Ax = b \text{ and } A(x+ \delta x) = b+ \delta b The Attempt at a Solution I did the first part just by the definition of A being non singular. The second part is tripping...
  46. D

    Simple time-independent non-degenerate quantum perturbation

    I'm reading through this pdf (http://www.pa.msu.edu/~mmoore/TIPT.pdf) on simple quantum perturbation theory and I'm quite confused with equations 32 through 34. They have E_{n}^{(2)} = <n^{(0)}|V|n^{(1)}> = - \sum_{m \neq 0}{\frac{|V_{mn}|^{2}}{E_{mn}}} but I would have done E_{n}^{(2)} =...
  47. D

    MHB Is there a mistake in the satellite's trajectory calculation?

    The L4 position is stable in the Earth Moon and I perturbing a satellite by km in the x direction to see the trajectory over the course of the year. However, the satellite isn't moving. Can anyone see if there is something wrong? I gave the satellite no initial velocity. In[2587]:=...
  48. J

    Equation decoupling in Cosmological Perturbation theory

    Hi, I am recently reading Weinberg's Cosmology, and getting subtle on Ch5, small fluctuation. One of the subtle point is on P.225-226 (same as (F.11) -> (F.13) and (F.14) in appendix F). The equations of motion (5.1.24)-(5.1.26) are decomposed into many parts. For example, (5.1.24) is...
  49. M

    Diagonal Matrix & Perturbation Theory in Quantum Mechanics

    What does it mean for a matrix to be diagonal, especially in Quantum Mechanics, where we get to Perturbation theory (Degeneracy). I don't get it. Please if you can explain in 'simple' language.
  50. B

    First-Order Perturbation Theory Derivation in Griffiths

    Homework Statement On page 251 of Griffiths's quantum book, when deriving a result in first-order perturbation theory, the author makes the claim that <\psi^0|H^0\psi^1> = <H^0\psi^0|\psi^1> where H^0 is the unperturbed Hamiltonian and \psi^0 and \psi^1 are the unperturbed wavefunction and its...
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