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. 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 \ \ \ ; \ \ \...
  2. 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...
  3. 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?
  4. 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...
  5. Diracobama2181

    Time Dependent Perturbation Problem

    I am assuming this is the interaction picture, so I start with $$|\psi>=c_1(t)|1>+c_2(t)|2>$$. Plugging this into the Schrodinger equation, I get the equations $$i\hbar c_1(t)=<1|H'|2>c_2(t)$$ and $$i\hbar c_2(t)=<1|H'|2>c_1(t)$$. I am assuming H' (the perturbation) is $$H'= − f(t)[...
  6. Diracobama2181

    A Time Dependent Perturbation of Harmonic Oscillator

    An electric field E(t) (such that E(t) → 0 fast enough as t → −∞) is incident on a charged (q) harmonic oscillator (ω) in the x direction, which gives rise to an added ”potential energy” V (x, t) = −qxE(t). This whole problem is one-dimensional. (a) Using first-order time dependent perturbation...
  7. Baibhab Bose

    Infinitesimal Perturbation in a potential well

    If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term. SO how should I approach this? And how the Δ is relevant in this calculation?
  8. Robin04

    Perturbation theory for solving a second-order ODE

    I have to solve the equation above. I haven't heard about an exact method so I tried to apply perturbation theory. I don't know much about it so I would like to ask for some help. First I put an ##\epsilon## in the coefficient of the non-linear ##\xi^2(t)## term: ##\ddot{\xi}(t)=-b\xi...
  9. saadhusayn

    Perturbation expansion with path integrals

    I expanded the exponential with the derivative to get: ## Z = \Bigg(1 + \frac{1}{2} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} + \frac{1}{4} \frac{\partial}{\partial x_{i}} A^{-1}_{ij} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{k}} A^{-1}_{kl}...
  10. M

    I Time independent perturbation theory in atom excitation

    Hello! In Griffiths chapter on Time independent perturbation theory, he has a problem (9.20) in which he asks us to calculate the first order contribution to the electron Hamiltonian in an atom if one takes into account the magnetic dipole/electric quadrupole excitations, beside the electric...
  11. Futurestar33

    Shelf in a box, treating the shelf as a weak perturbation

    In this problem I am supposed to treat the shelf as a weak perturbation. However it doesn't give us what the perturbed state H' is. At the step V(x) = Vo, but that is all that is given and isn't needed to determine H'. This isn't in a weak magnetic field so I wouldn't you use H'=qEx and then...
  12. A

    I Perturbation to Flat Space Metric: Geodesic Equation

    From the geodesic equation d2xμ/dΓ2+Γμ00(dt/dΓ)2=0,for non-relativistic case ,where Γ is the proper time and vi<<c implying dxi/dΓ<<dt/dΓ. Now if we assume that the metric tensor doesn't evolve with time (e,g gij≠f(t) ) then Γμ00=-1/2gμs∂g00/∂xs. If we here assume that the metric components of...
  13. V

    Gauge invariance in GR perturbation theory

    I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the metric. We use an arbitrary small vector field to have the effect of a chart transition map or...
  14. iVenky

    What happens when injecting a current at different instants in an oscillator?

    Hi, I am reading the Hajimiri-Lee phase noise model, and got a question on that. If you have an LC tank circuit that is free-running and I inject a current i(t) (dirac current) at instants either t1 or t2 (shown in the figure), depending on when you inject the phase of the output changes (as...
  15. N

    A Perturbation solution and the Dirac equation

    I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$ The solution up to first order is $$ \psi(x) =...
  16. BeyondBelief96

    2nd Order Non-Degenerate TI Perturbation Theory Corrections

    Homework Statement Show that the 2nd order nondegenerate perturbation theory corrections are given by: ##E_n^2 = \sum_{k \neq n}^{\infty} \frac{|\left < \phi_n | \hat{H} | \phi_k \right> |^2}{E_n^0 - E_k^0}##[/B] and ## C_{nm}^2 = \frac{C_{nm}^1 E_n^1 - \sum_{k \neq n}^{\infty} C_{nk}^1...
  17. W

    B Time independent perturbation theory

    This isn't explained anywhere so it must be super basic and I'll probably kick myself for not getting it, but on the wiki page for time independent perturbation theory, section 3.1: https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics) It talks about first order corrections and...
  18. C

    MHB Regular perturbation nonlinear problem

    Hi all, I have this (nondimensionalised) system of ODEs that I am trying to analyse: \[ \begin{align} \frac{dr}{dt}= &\ - \left(\alpha+\frac{\epsilon}{2}\right)r + \left(1-\frac{\epsilon}{2}\right)\alpha p - \alpha^2\beta r p + \frac{\epsilon}{2} \\ \frac{dp}{dt}= &\...
  19. 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...
  20. N

    A TD perturbation - state initially in continuous part

    Hi everyone, I am doing a time dependent perturbation theory, in a case when the electron is prepared in a state of the continuous part of the energy spectrum. Existence of the discrete part and the degeneracy of the continuous part is irrelevant at the moment and will not be considered...
  21. E

    First order perturbation energy correction to H-like atom

    Homework Statement Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential $$ \phi(r) = \begin{cases} \frac{Ze}{R}(\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}) &\quad r<R\\ \frac{Ze}{r} &\quad r>R \\...
  22. JuanC97

    I Restrictions of 1st Order Perturbation Theory

    Hello guys, I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory. I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski...
  23. S

    I Degenerate Perturbation Theory

    Hello! I am reading Griffiths and I reached the Degenerate Time Independent Perturbation Theory. When calculating the first correction to the energy, he talks about "good" states, which are the orthogonal degenerate states to which the system returns, once the perturbation is gone. I understand...
  24. T

    Perturbation of a Magnetic Field

    Homework Statement Could someone please see if my working are correct for this question, I have never actually done a question of this nature before, and after reading up about the derivation on the perturbation I thought I give ago and apply, my final answer dose not seem correct, as I believe...
  25. Warda Anis

    How Does a Massive Photon Affect Hydrogen Atom Energy Levels?

    Homework Statement The photon is normally assumed to have zero rest mass. If the photon did have a tiny mass, this would alter the potential energy the electron feels in the hydrogen atom (due to the Coulomb interaction with the proton). The potential then becomes yukawa potential...
  26. I

    Hydrogen Ionization Rates Using Time Dependent Perturbation

    Homework Statement [/B] Calculate the rate of ionization of a hydrogen atom in the 2p state in a monochromatic external electric field, averaged over the component of angular momentum in the direction of the field. Ignore the spin of the particles. In this case we can write...
  27. Necmi

    Mathematica Solve Perturbation problem with mathematica

    u'(1-epsilon(u')^2)=-y uo(1)=0 u1(1)=0 I need solve this problem with mathematica.
  28. V

    Show metric perturbation transformation

    Homework Statement Consider following transformation: Transformation: $$X^{\mu}\rightarrow \tilde{X^{\mu}}= X^{\mu}+\xi^{\mu}(\eta, \vec{x})$$ where ##\xi^0=T, \xi^i=L_i## Show transformation of metric perturbation ##B_i\rightarrow \tilde{B_i}=B_i+\partial_iT-\partial_{\eta}L_i## Homework...
  29. diegzumillo

    A LSZ, perturbation and renormalization

    My current understanding of renormalization is that the LSZ formula requires normalized fields. So when you normalize them you get some extra parameters from the regularization procedure you encounter along the way. It's an upgrade on my previous understanding of it as some artificial way of...
  30. S

    How to Determine Parameters for Hydrogen Transition Probability?

    Hello! I have the following problem I'm trying to solve: Homework Statement An Hydrogen atom in the state |100> is found between the plates of a capacitor, where the electric field (weak and uniform) is: E(t) = \epsilon e^{-\alpha t / \tau}. Calculate the parameters of the potential...
  31. Tursinbay

    A String perturbation equations from Polyakov action

    General physical perturbations of string is derived by A.Larsen and V.Frolov (arXiv:hep-th/9303001v1 1March 1993). An arbitrary string configuration is in 4-dimensional gravitational background. Starting point is Polyakov action $$ S = \int d \tau d\sigma \sqrt {-h} h^{AB} G_{AB}$$. Here is...
  32. Fips

    Atomic Textbook about Perturbation Theory

    Hi I was hoping someone could advise me on a textbook/platform where I can learn more about the perturbation theory applied to helium and the perturbation theory time depedant. Thanks
  33. S

    A Calculating the power spectra of scalar perturbation

    I'd like to numerically calculate the power spectra of the scalar perturbation at the Hubble crossing in warm inflation, my problem is that I don't know how to do it. As I know, the Hubble crossing happens at the onset of warm inflation where the different modes become larger than the Hubble...
  34. F

    I Higher order terms in perturbation theory (QFT)

    I'm fairly new to QFT and I'm currently trying to understand perturbation theory on this context. As I understand it, when one does a perturbative expansion of the S-matrix and subsequently calculates the transition amplitude between two asymptotic states, each order in the perturbative...
  35. D

    Quick question about hydrogen atom perturbation

    Homework Statement I have already solved the problem, but I don't really understand why the orbital angular momentum in the z-direction has to be taken to 0 ? Homework EquationsThe Attempt at a Solution Suppose the component of orbital angular momentum in the z-direction is...
  36. D

    Time-dependent perturbation theory

    Homework Statement The problem consists of 2 parts,the first one(I have done it) is on the following website: https://www.physicsforums.com/threads/transition-probability-from-two-states.804343/ Q1: I calculated the desired result p(t) = sin^2(Ut/h). However,I don't understand why <1,t | 2 >...
  37. Leechie

    What is the Perturbation in the Modified Coulomb Model of a Hydrogen Atom?

    Homework Statement Suppose there is a deviation from Coulomb's law at very small distances, with the mutual Coulomb potential energy between an electron and a proton being given by: $$V_{mod}(r)= \begin{cases} - \frac {e^2} {4 \pi \varepsilon_0} \frac {b} {r^2} & \text {for } 0 \lt r \leq b \\...
  38. Leechie

    Using perturbation to calculate first order correction

    Homework Statement I'm trying to evaluate the following integral to calculate a first-order correction: $$\int_0^\infty R_{nl}(r)^* \delta \hat {\mathbf H} R_{nl}(r) r^2 dr$$ The problem states that ##b## is small compared to the Bohr radius ##a_o## Homework Equations I've been given...
  39. S

    I First order perturbation derivation

    In lectures, I learned that in first order perturbation, \hat{H}_0 term cancels with E_0 term because \hat{H}_0 is Hermitian. What property does Hermitian operators hold that cancels with the unperturbed energy?
  40. acdurbin953

    Time-Dependent Perturbation of a 1D Infinite Square Well

    Homework Statement At t < 0 we have an unperturbed infinite square well. At 0 < t < T, a small perturbation is added to the potential: V(x) + V'(x), where V'(x) is the perturbation. At t > T, the perturbation is removed. Suppose the system is initially in the tenth excited state if the...
  41. V

    Linear perturbation to harmonic oscillator

    Homework Statement Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##. Homework Equations First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...
  42. L

    I Why do we need perturbation theory

    Why do we need perturbation theory in quantum mechanics?
  43. P

    Time-Dependent Perturbation Theory

    Note this isn't actually a homework problem, I am working through my textbook making sure I understand the derivation of certain equations and have become stuck on one part of a derivation. 1. Homework Statement I am working through my text (Quantum Mechanics 2nd Edition by B.H Bransden & C.J...
  44. G

    I Time dependent perturbation(determination of order)

    I have a question about time dependent perturbation. In time dependent perturbation, unlike time independent perturbation, there is no lamda which is used for comparing order. So, I`m confused how can I determine order. Is there any explanation which use lambda or some other method for...
  45. L

    Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

    Homework Statement How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x) The attempt at a solution H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x) I know how to find the matrix of the normal...
  46. S

    Perturbation Methods + Airfoils

    First of all, thanks for reading. I'd like to ask if some of you here can suggest a reference text concerning perturbation methods on thin airfoil? I'm using the text of Katz and Plotkin as well as Van Dyke but for now, I find it challenging to follow their discussion. I hope that finding a...
  47. F

    I Gauge transformation in cosmological perturbation

    Based on this lecture notes http://www.helsinki.fi/~hkurkisu/CosPer.pdf For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system...
  48. bananabandana

    I Degenerate Perturbation Theory

    I'm struggling to understand degenerate perturbation theory. It's clear that in this case the 'normal' approximation method fails completely seeing as you get a divide by zero. I follow the example for a two state system given in e.g D.J Griffiths "Introduction to Quantum Mechanics" However...
  49. B

    First Order Perturbation Theory - QM

    Homework Statement The ground state energy of the 1D harmonic oscillator with angular frequency ##\omega## is ##E_0 = \frac{\hbar \omega}{2}##. The angular frequency is perturbed by a small amount ##\delta \omega##. Use first order perturbation theory to estimate the ground state energy of the...
  50. A

    A Canonical perturbation for infinite chain

    I've been Dealing with a problem of perturbation of the movement of an infinite chain of harmonic oscillator and I tried to apply the von Zeippel-Poincare formalism of canonical perturbation theory just to see what I get. This was too naive since I quickly stumbled into the problem of defining...
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