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


{\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. ergospherical

    A Synchronous to Newtonian gauge

    In the context of cosmology, you can perturb around the FRW background, conventionally:$$g = a^2(\tau)[(1+2A)d\tau^2 - 2B_a dx^a d\tau -(\delta_{ab} + h_{ab}) dx^a dx^b]$$with ##a,b## being spatial indices only (1,2,3). You can do gauge transformations ##\tilde{x} = x + \xi## of the coordinates...
  2. H

    Linearization Euler's equation

    I'm trying to linearize (first order) the Euler's equation for a small perturbation ##\delta## Starting with ##mna (\frac{\partial}{\partial t} + \frac{\vec{v}}{a} \cdot \nabla ) \vec{u} = - \nabla P - mn \nabla \phi## (1) ##\vec{u} = aH\vec{x(t)} + \vec{v(x,t)}## Where a is the scale factor...
  3. Rayan

    Time-dependent perturbation theory

    So I have the solution here and trying to understand what happened at the beginning of the second row! How did we get the exponential $$e^{i(\omega_m - \omega_0 ) t' }$$ ?
  4. G

    I Help with some confusions about variational calculus

    I had some several questions about variational calculus, but seems like I can't get an answer on math stackexchange. Takes huge time. Hopefully, this topic discussion can help me resolve some of the worries I have. Assume ##y(x)## is a true path and we do perturbation as ##y(x) + \epsilon...
  5. ermia

    A charge inside a ring, small oscillation

    This is the picture of the problem. I attach my solution. I first used a trick with gauss's law to calculate the radial electric field at first order of r. ( where r is small ) ( we can assume ##small r=\delta r##) I used a cylinder at the center of the ring then i calculated the ##\hat{z}##...
  6. C

    I Linearising Christoffel symbols

    Carroll linearising by perturbation ##g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}## has: (Notes 6.4, Book 7.4) ##\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\lambda}\left( {\partial_{ \mu}}g_{\nu\lambda}+{\partial_{ \nu}}g_{\lambda\mu}-{\partial_{...
  7. cwill53

    Question on Time-Independent Perturbation Theory

    I'm currently reading this passage to review perturbation theory. Just before Equation (A.4), this passage tells me to take the inner product of the proposed eigenstate ##|\psi _j\rangle## with itself. Writing this out, I got: $$1=\left \langle \psi _j| \psi _j\right \rangle=\left ( |\psi^0...
  8. V

    A Cosmological Density Perturbation vs Homogeneity: Questions Answered

    When arriving at the standard model of cosmology, i.e. the exapnding universe, we assume based on experirmental data that the cosmos is homogenous on large enough scales. But when we go back in time, when the galaxies are beginning to form, we note that because of the growth of density...
  9. B

    A Time dependent perturbation theory applied to energy levels

    Hello! I am reading this paper and in deriving equations 6/7 and 11/12 they claim to use second oder time dependent perturbation theory (TDPT) in order to get the correction to the energy levels. Can someone point me towards some reading about that? In the QM textbooks I used, for TDPT they just...
  10. codebpr

    A The kinetic term of the Hamiltonian is not positive definite

    I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation: $$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$ where they...
  11. C

    I Calculating Relative Change in Travel Time Due to Spacetime Perturbation

    Suppose you have the following situation: We have a spacetime that is asymptotically flat. At some position A which is in the region that is approximately flat, an observer sends out a photon (for simplicity, as I presume that any calculations involved here become easier if we consider a...
  12. breadlover98

    Runge-Lenz vector with perturbation potential

    For the case that there is only a potential ##\sim 1/r##, I have already proven that the time derivative of the Lenz vector is zero. However, I'm not sure how I would "integrate" this perturbation potential/force into the definition of the Lenz vector (as it is directly defined in terms of the...
  13. K

    I More doubts in perturbation theory

    Townsend, quantum mechanics " In our earlier derivation we assumed that each unperturbed eigenstate ##\left|\varphi_{n}^{(0)}\right\rangle## turns smoothly into the exact eigenstate ##\left|\psi_{n}\right\rangle## as we turn on the perturbing Hamiltonian. However, if there are ##N## states ##...
  14. K

    I Doubt in understanding degenerate perturbation theory

    McIntyre, quantum mechanics,pg360 Suppose states ##\left|2^{(0)}\right\rangle## and ##\left|3^{(0)}\right\rangle## are degenerate eigenstates of unperturbed Hamiltonian ##H## Author writes: "The first-order perturbation equation we want to solve is ##...
  15. C

    A Perturbation of Maxwell stress from voltage distribution

    I have a voltage distribution ##V(x,y) = V_{dc}(x,y)+ V_{ac}(x,y) \cos(\omega t)##, I have derived the Matrix e. But I do not know how to extract it from the voltage, meaning I do not know how to find ##E_{x0} , E_{y0}, \delta E_{x}, \delta E_{y}## in terms of ##V_{dc}(x,y), V_{ac}(x,y)##...
  16. Samama Fahim

    I Time Dependent Sinusoidal Perturbation Energy Conservation

    The transition probability -- the probability that a particle which started out in the state ##\psi_a## will be found, at time ##t##, in the state ##\psi_b## -- is $$P_{a \to b} = \frac{|V_{ab}|}{\hbar^2} \frac{sin^2[(\omega_0 - \omega)t/2]}{(\omega_0 - \omega^2}.$$ (Griffiths, Introduction...
  17. ThiagoSantos

    Diagonalizing q1ˆ3q2ˆ3 with Degenerate Perturbation Theory

    I tried to use the degenerated perturbation theory but I'm having problems when it comes to diagonalizing the perturbation q1ˆ3q2ˆ3 which I think I need to find the first order correction.
  18. M

    Perturbation Methods: Guarantee a boundary layer?

    Hi, I was working on the following problem: Question: A small parameter multiplying the highest derivative does not guarantee that the solution will have a boundary layer for small values of ##\epsilon##. This may be due to the form of the differential equation, or the particular boundary...
  19. M

    Perturbation Methods: Asymptotic Matching Question

    Hi, I was working on a problem and I can't seem to make much progress with it. From a high level my steps are: 1. Use the naive method to find the outer solution which can satisfy one of the two boundary conditions 2. Introduce a boundary layer (via stretched coordinates) to find inner solution...
  20. Kaguro

    A Degenerate Perturbation Theory: Correction to the eigenstates

    Given the unperturbed Hamiltonian ##H^0## and a small perturbating potential V. We have solved the original problem and have gotten a set of eigenvectors and eigenvalues of ##H^0##, and, say, two are degenerate: $$ H^0 \ket a = E^0 \ket a$$ $$ H^0 \ket b = E^0 \ket b$$ Let's make them...
  21. M

    I How are good quantum numbers related to perturbation theory?

    Hello folks, I am currently studying from Griffiths' Introduction to Quantum Mechanics and I've got a doubt about good quantum numbers that the text has been unable to solve. As I understand it, good quantum numbers are the eigenvalues of the eigenvectors of an operator O that remain...
  22. ooleonardoo

    Perturbation Theory: Calculating 1st-Order Correction

    Of course, this question consisted of two parts. In the first part, we needed to calculate the first-order correction. It was easy. In all the books on quantum mechanics I saw, only first-order examples have been solved. So I really do not know how to solve it. Please explain the solution method...
  23. dRic2

    Time dependent perturbation theory (Berry phase)

    If I plug the solution into the Schrodinger equation I get $$(i \hbar \partial_t - H)\ket{\psi} = 0$$ Since I know that the zeroth-order expansion is lambda is already a solution I think this is equal to $$(i \hbar \partial_t - H)e^{i\phi} e^{-i\gamma}\ket{\delta n} = 0$$ If now I carry on with...
  24. S

    I The “philosophical cornerstone” of the Moller-Plesset perturbation theory

    In quantum chemistry, the MP rows (MP2, MP3, MP4, etc) can converge both quickly and slowly, and for some cases (e.g. CeI4 molecule) they even diverge instead of converging. My question is quite philosophic: what is the “mathematical cornerstone”, or “philosophical cornerstone” of the...
  25. George Keeling

    I Linearized Gravity & Metric Perturbation when Indices Raised

    I have just met linearized gravity where we decompose the metric into a flat Minkowski plus a small perturbation$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\ \ \left|h_{\mu\nu}\ll1\right|$$from which we 'immediately' obtain $$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$$I don't obtain that. In my rule book...
  26. Viona

    Time-dependent Perturbation Theory

    I was reading in the Book: Introduction to Quantum Mechanics by David J. Griffiths. In chapter Time-Dependent Perturbation Theory, Section 9.12. I could not understand that why he put the first order correction ca(1)(t)=1 while it equals a constant.
  27. T

    A QFT with vanishing vacuum expectation value and perturbation theory

    In This wikipedia article is said: "If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a...
  28. M

    I Energy shifts in time dependent perturbation theory

    Hello! I saw in many papers people talking about the effects of a time dependent perturbation (usually an oscillating E or B field) on the energy levels of an atom or molecule (for now let's assume this is a 2 level system). Taking about energy makes sense when the hamiltonian is time...
  29. A

    I Perturbation theory with two parameters?

    Hello, I am looking for a reference which describe perturbation theory with two parameters instead of one. So far, I did not find anything on the topic. It might have a specific name and I am using the wrong keywords. Any help is appreciated. To be clear, I mean I have ##H =...
  30. DaniV

    RG flow of quadrupole coupling in 6+1 dimension electrostatic problem

    I tried to do a Euler Lagrange equation to our Lagrangian: $$\frac{S_\text{eff}}{T}=\int d^6x\left[(\nabla \phi)^2+(\nabla \sigma)^2+\lambda\sigma (\nabla \phi)^2\right]+\frac{S_{p.p}}{T}$$ and then I would like to solve the equation using perturbation theory when ##Q## or somehow...
  31. mcas

    Perturbation theory and the secualr equation for double degeneration

    I've been assigned to do a problem from Landau which you can read below: I have no problem with finding the energy. Then I write down the equations: \begin{equation*} \begin{cases} (V_{11}-E^{(1)})|c_1|e^{i\alpha_1} + V_{21}e^{i\alpha_2}|c_2| = 0\\ V_{12}e^{i\alpha_1}|c_1| +...
  32. 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...
  33. Mayan Fung

    Perturbation from a quantum harmonic oscillator potential

    For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to...
  34. P

    A Gauge Invariance of Transverse Traceless Perturbation in Linearized Gravity

    In linearized gravity we define the spatial traceless part of our perturbation ##h^{TT}_{ij}##. For some reason this part of the perturbation should be gauge invariant under the transformation $$h^{TT}_{ij} \rightarrow h^{TT}_{ij} - \partial_{i}\xi_{j} - \partial_{j}\xi_{i}$$ Which means that...
  35. M

    I Confused about perturbation theory

    Hello! Let's say we have 2 states of fixed parity ##| + \rangle## and ##| - \rangle## with energies ##E_+## and ##E_-## and we have a P-odd perturbing hamiltonian (on top of the original hamiltonian, ##H_0## whose eigenfunctions are the 2 above), ##V_P##. According to 1st order perturbation...
  36. A

    I Perturbation Theory and Zeeman Splitting

    Suppose we have a hamiltonian $$H_0$$ and we know the eigenvectors/values: $$H_0 |E_i \rangle = E_i|E_i \rangle $$ We then add to it another perturbing Hamiltonian: $$H’$$ which commutes with $$H_0.$$ According to nondegenerate first order perturbation theory: $$\langle H \rangle \approx...
  37. K

    I Finding matrices of perturbation using creation/annihilation operators

    "Given a 3D Harmonic Oscillator under the effects of a field W, determine the matrix for W in the base given by the first excited level" So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result: W = 2az2 - ax2 - ay2 + 2az+ 2 -ax+...
  38. B

    Finding energy eigenvalues with perturbation

    I know the basis I should use is |m_1,m_2> and that each m can be 1,0,-1 but how do I get the eigenvalues from this?
  39. Z

    Perturbation Theory - expressing the perturbation

    Hi, I just need someone to check if I am on the right track here Below is a mutual Coulomb potential energy between the electron and proton in a hydrogen atom which is the perturbed system: ##V(x) = \begin{cases} - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} \text{for } 0<r \leq 0 \\ -...
  40. Phylosopher

    I Why perturbation theory uses power series?

    I am revising perturbation theory from Griffiths introduction to quantum mechanics. Griffiths uses power series to represent the perturbation in the system due to small change in the Hamiltonian. But I see no justification for it! Other than the fact that it works. I searched on the internet a...
  41. G

    I Perturbation lines for the 1s hydrogen electron

    Does anyone know theory about how the perturbation lines are for 1s hydrogen electron? By perturbation I mean the perturbation that is caused by moving an electron so that the E-field lines it emits becomes dragged. by perturbation I mean for example dragging a charge as described below Above...
  42. electrogeek

    I 2nd Order Perturbation Theory Energy Correction

    Hi everyone, I'm struggling with the proof for the second order energy correction for perturbation theory when substituting in the first order wavefunction. I have attached an image of my current proof for it below, but I'm not sure whether this is the correct approach for it (the H's in the...
  43. G

    I Confused by Notation? Perturbation Theory Explained

    Looking at. <psi|AB|theta>, under what conditions would this be equal to <psi|A|theta> * <psi|B|theta> I’m just getting into perturbation theory and am running into confusing notation. Thanks john
  44. patric44

    Perturbation treatment of hydrogen molecular ion

    hi guys i am a the third year undergrad student and in this 2nd semester in my collage we should start taking quantum mechanics along with molecular physics , our molecular physics professor choose a book that we are going to take which is " molecular physics by wolfgang Demtroder " when i...
  45. Arman777

    I Jeans Instability in an Expanding Universe

    I am trying to derive the equation for a case, where we have a dust(zero-pressure) in an expanding universe. There are 4 equations but I think exercising on one of them would be helpful for me. I am trying to derive the equation for a case, where we have a dust(zero-pressure) in an expanding...
  46. JD_PM

    Deriving the wave equation using small perturbations

    Note that the wave equation we want to derive was introduced by Alfven in his 1942 paper (please see bottom link to check it out), but he did not include details on how to derive it. That's what we want to do next. Alright, writing the above equations we assumed that: $$\mu = 1 \ \ \ ; \ \ \...
  47. W

    I Raising/Lowering Metric Indices: Explained

    If I have a metric of the form ##g_{\mu \nu} = f_{\mu \nu} + h_{\mu \nu}## where ##f_{\mu \nu}## is the background metric and ##h_{\mu \nu}## the perturbation, how do I raise and lower indices of tensors? For instance, I was told that ##G_{ \ \nu}^{\mu} = f^{\mu \nu '} G_{\nu ' \nu }##. But...
  48. Diracobama2181

    Selection Rules (Time Dependent Perturbation Theory)

    I suppose my question is, since X commutes for H, does this mean that the selection rules are $$<n',l',m'|X|n,l,m>=0$$ unless $$l'=l\pm 1$$ and $$m'=m\pm 1$$, as specified in Shankar?
  49. Diracobama2181

    Applying Selection Rules to Determine Non-Zero Ground State Perturbations

    Since E_i=0 for the ground state, and $$E_f=\frac{(\hbar)^2l(l+1)}{2I}$$, $$w_{fi}=\frac{E_f-E_i}{\hbar}=\frac{(\hbar)l(l+1)}{2I}$$. So, $$d_f(\infty)=\frac{i}{\hbar}\int_{-\infty}^{\infty}<f|E_od_z|0>e^{\frac{i\hbar l(l+1)t}{2I}+\frac{t}{\tau}}dt$$ My question is in regards to...