What is Pertubation: Definition and 29 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


{\displaystyle \epsilon }
. The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of


{\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. curiousPep

    Engineering Stability Analysis of Equilibrium Solutions using Small Perturbations

    When I use Lagrange to get the equations of motion, in order to find the equilibrium conditions I set the parameters q as constants thus the derivatives to be zero and then calculate the q's that satisfy the equations of motion obtained. In ordert to check about stability I think I need to add...
  2. M

    Degenerate perturbation theory

    $$ W_{n,n} = \int_0^{2 \pi} \frac{1}{\sqrt{2 \pi}} e^{-inx} V_0 \cos(x) \frac{1}{\sqrt{2 \pi}} e^{inx} dx $$ $$ = 0 $$ $$ W_{n, -n} = \int_0^{2 \pi} \frac{1}{\sqrt{2 \pi}} e^{-inx} V_0 \cos(x) \frac{1}{\sqrt{2 \pi}} e^{-inx} dx $$ $$ = \frac{a n ( \sin(4 \pi n) + i \cos( 4 \pi n) - i...
  3. D

    I Problem: perturbation of Ricci tensor

    I am trying to calculate the Ricci tensor in terms of small perturbation hμν over arbitrary background metric gμν whit the restriction \left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1 Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms...
  4. renec112

    First order pertubation of L_y operator

    Hi, I am trying to solve an exam question i failed. It's abput pertubation of hydrogen. I am given the following information: The matrix representation of L_y is given by: L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1...
  5. K

    I Wavefunctions in first order degenerate pertubation theory

    Suppose we want to solve the Hamiltonian ##H=H_0+\lambda V## pertubatively. Let ##E_1,...,E_n## be the eigenvalues of ##H_0## and ##S_1,...,S_n## the eigenspaces that belong to them. In order to do that, one usually choses an orthonormal Basis ##|\psi_{i,j}>## of each ##S_i## with the property...
  6. Schwarzschild90

    ND Pertubation theory: Second order correction

    Homework Statement Calculate the second-order correction to the ground-state energy of the stationary states of the system. The perturbed Hamiltonian is: H' =- (/gamma /hbar m /omega)/2 (a+ - a-) ^2 2 & 3. Relevant equatio and the attempt at a solution This is not right. I follow the same...
  7. M

    Predicting Peak Displacement in Imbalanced Rotating Drum

    If I am attempting to prevent a suspended motor-rotating imbalanced drum of high mass from colliding with its enclosure, how can I attempt to predict the maximum displacement from a balanced center of rotation if I have only a single sensor that can accurately measure(1000 samples/sec)...
  8. P

    Energy of perturbed harmonic oscillator

    Homework Statement For a quantum harmonic oscillator in an electric field, using ##\hat{V}=q\epsilon\hat{x}##, with the following trial state: $$|\psi\rangle=|0\rangle+b|1\rangle$$ Show that the energy can be written as $$E=\frac{\frac{\hbar...
  9. rkrishnasanka

    Is pertubation a linear operation?

    My question stems from a discussion I had with my colleague today. In Electomagnetic coupling , like in waveguide structures. We apply pertubation theory to find out the coupling between various modes that get coupled in the device. My colleague said that the coupling interaction was...
  10. C

    Linearized Gravity and the smalness of the pertubation

    The approach taken in linearized gravity seems to be to 'perturb' the 'Minkowski metric' such that $$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$ where ##|h_{\mu \nu}| <<1##. As I've understood it, the goal is to get an approximate theory for gravity, i.e. for weak gravitational fields...
  11. T

    Pertubation theory for 3x3 matrix

    Some one knows a study material to diagonalize a matrix mass for 3 neutral scalar using perturbation theory like \begin{equation} M^2=\left(\begin{array}{ccc} 2 \lambda_{\phi} v_{\phi}^2 &\lambda_{\phi \sigma } v_{\phi}v_{\sigma} & \lambda_{\phi\eta} v_{\phi} v_{\eta} \\...
  12. D

    2nd order pertubation theory of harmonic oscillator

    Homework Statement I'm having some trouble calculating the 2nd order energy shift in a problem. I am given the pertubation: \hat{H}'=\alpha \hat{p}, where $\alpha$ is a constant, and \hat{p} is given by: p=i\sqrt{\frac{\hbar m\omega }{2}}\left( {{a}_{+}}-{{a}_{-}} \right), where {a}_{+} and...
  13. J

    Time-Dependent Degenerate Pertubation Theory for 3x3 matrix

    Homework Statement H0 = [2,0,0;0,2,0;0,0,4] H1 = [0,1,0;1,0,1;0,1,0] Find energy eigenvalues to 2nd order. Homework Equations The Attempt at a Solution I know that I need to diagonalize the perturbation in the 2x2 subspace (for my 2 degenerate eignevalues of 2 but I'm not sure...
  14. J

    Time-Dependent Degenerate Pertubation Theory for Spin System

    Homework Statement Consider the so-called Spin Hamiltonian H=AS2Z+B(S2X-S2Y) for a system of spin 1. Show that the Hamiltonian in the SZ basis is the 3x3 matrix: \hbar2*[(A,0,B; 0,0,0; B,0,A)].Find the eigenvalues using degenerate pertubation theory. Homework Equations Spin Pauli MatricesThe...
  15. K

    Time dependent pertubation theory

    Okey, so I´m taking a course in QM and I feel that I got a grip of most of it. But then we arrive at this formulea i\hbar\frac{\partial}{\partial t} c_n(t) = \sum_m \hat{V}_{nm} e^{i\omega_{nm} t}c_m(t), where \omega_{nm} \equiv \frac{(E_n - E_m)}{\hbar}. In other words time dependent...
  16. K

    Time Independant Pertubation Theory - QM

    Homework Statement An electron is confined to a 1 dimensional infinite well 0 \leq x \leq L Use lowest order pertubation theory to determine the shift in the second level due to a pertubation V(x) = -V_0 \frac{x}{L} where Vo is small (0.1eV). Homework Equations [1] E_n \approx...
  17. L

    1st order pertubation on 2 level system

    Hi all. I m stucked on the question followed, any helps will be greatly appreciated. A perturbation has the form H'=z act on a two level system which they have different parity. So the first order correction to the energy level 1 and 2 are give by: E11=<ψ1| H' |ψ1> Same for level 2...
  18. L

    1st order Pertubation energy and wavefunction

    Hi all, I must misunderstood somewhere, couldn't figure out the following, any helps will be greatly appreciated. The first order correction of the pertubated energy is: \leftψn0\langle H'\rightψn0\rangle Where: ψn0 Is the solution of the unpertubated Hamiltonian. My question is can ψn0 be...
  19. G

    Perturbation Theory: Calculating 2nd Approx of E in Hydrogen 2s State

    hi i want to calculate the second approximation of the energy by a potential V between two hydrogen atoms in a 2s state, but I do not know how to apply pertubation theory correctly? Landau Lifgarbagez says: E_n^2= \sum_m ' \frac{|V_m_n|^2}{E_n^0-E_m^0} (where the prime means that the...
  20. MathematicalPhysicist

    What Are the Best Textbooks for In-Depth Study of Rigorous Perturbation Theory?

    Anyone has any recommendation for a textbook/s that doesn't shun away from proofs of theorems? I read Murdock's text, but he says himself that he doesn't cover it all. And Bender's methods is more on exercising the methods than understanding them. Any? Thanks.
  21. L

    Pertubation Theory - 3D harmonic oscillator

    Homework Statement A particle with mass m moves in the potential: V(x,y,z) = \frac{1}{2} k(x^{2}+y^{2}+z^{2}+ \lambda x y z) considering that lambda is low. a) Calculate the ground state energy accordingly to Pertubations Theory of the second order. b) Calculate the energies of...
  22. V

    Perturbation Theory: Deciphering Missing Lines of Explanation

    Again, I am having difficulty deciphering my class notes - in this case there are missing lines of explanation. If we consider a system of particles that approach and interact, the Heisenberg representation of the interacting field is: \phi(\vec{x} , t) = U^{-1} (t) \phi_{a} (\vec{x} , t)...
  23. Z

    Degenerate pertubation theory when the first order fails

    The basic algorithm of degenerate perturbation theory is quite simple: 1.Write the perturbed Hamiltonian as a matrix in the degenerate subspace. 2.Diagonalize it. 3.The eigenstates are the 'correct' states to which the system will go as the perturbation ->0. But what to do if the first...
  24. K

    Pertubation and density matrix

    Hi there, I am reading a text by Robert W. Boyd "Nonlinear optics", in page 228, he used pertubation theory on two-level system and let the steady-state solution of the dynamics equation of density matrix as w = w_0 + w_1 e^{-i\delta t} + w_{-1}e^{i\delta t} where w=\rho_{bb} - \rho_{aa}...
  25. K

    How to Simplify Equations Using Perturbation Method?

    Suppose I have a equation of the form F(x, k) = G(x, k) which is unsolvable analytically. We apply the method of pertubtaion (k is small quantity) and let the first order solution approximated as x = x^{(0)} + kx^{(1)} we need to put this back to the equation and find out x^{(1)}...
  26. T

    Sagging Bottom Quantum Box and Pertubation

    Homework Statement Consider a particle in a one-dimensional “box” with sagging bottom v(x) = -V_0sin(\pi x/L) for 0 \leq x \leq L infinity outside of thius (x > L, x < 0) a) Sketch the potential as a function of x. b) For small V_0 this potential can be considered as a...
  27. H

    Asymptotic methods and pertubation method

    Hi all! I am currently reading a book on asymptotic and pertubation methods for finding approximate solutions. How asymptotic methods and pertubaion methods are related? I mean, can I skip asymptotic methods and directly jump to study pertubation methods? (I know it may be better to have a...