What is Perturbation theory: Definition and 263 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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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}...
  6. 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 =...
  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

    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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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.
  27. 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...
  28. 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.
  29. 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...
  30. Hepth

    Chiral Perturbation Theory : pi0 pi0 Z vertex?

    Not sure if anyone has any experience with chiral perturbation theory, but I'm trying to see what all of the vertices are for interactions with a single Z boson. I've looked at the lagrangian up to order p^4 so far, and it seems that the Z only interacts with charged pions/kaons. I'm using...
  31. J

    Perturbation Theory: Calculating for the correction on the ground state energy

    Homework Statement Homework Equations E_{1}=<ψ_{1}|V(r)|ψ_{1}> The Attempt at a Solution That is equal to the integral ∫ψVψd^3r So I'll just perform the integral, correct ? But r is not constant here right? So, I' ll keep it inside the integral? How should I continue? Please...
  32. J

    Perturbation Theory (Non-Degenerate)

    If I have V(x)=\frac{1}{2}m\omega^{2}x^{2} (1+ \frac{x^{2}}{L^{2}}) How do I start to solve for the hamiltonian Ho, the ground state wave function ?? Calculate for the energy of the quantum ground state using first order perturbation theory?
  33. S

    Time Dependent Perturbation Theory - Klein Gordon Equation

    Hey, I'm struggling to understand a number of things to do with this derivation of the scattering amplitude using time dependent perturbation theory for spinless particles. We assume we have some perturbation 'V' such that : \left ( \frac{\partial^2 }{\partial t^2}-\triangledown ^2 +...
  34. J

    Energy Levels According To Second-Order Perturbation Theory

    Homework Statement If E1≠E2≠E3, what are the new energy levels according to the second-order perturbation theory? Homework Equations H' = α(0 1 0) (1 0 1) (0 1 0) ψ1= (1) (0) (0) ψ2= (0)...
  35. Hepth

    Chiral Perturbation Theory : Some quick questions

    I just want to make sure that I am doing some things correctly. I'll be using http://www.physics.umd.edu/courses/Phys741/xji/chapter5.pdf from about 5.64 on. The kinetic term : \frac{f^2}{4} Tr[D_{\mu} \Sigma D^{\mu} \Sigma^{\dagger}] Now if I want to expand this out, as \Sigma =e^{i...
  36. D

    Perturbation theory infinite well

    in the infinite well with small potential shown in the attachment. I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x. E=h^2/8mL^2 +∫ ψkψ dx ψ=√(2/L) sin⁡(∏/L x) when integrating ∫...
  37. D

    Perturbation theory infinite well

    in the infinite well with small potential shown in the attachment. I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x. E=h^2/8mL^2 +∫ ψkψ dx ψ=√(2/L) sin⁡(∏/L x) when integrating ∫...
  38. P

    MHB Solving for Roots of a Cubic Equation Using Perturbation Theory

    Question: obtain 2-term expansions for the roots of x^3+x^2-w=0 , 0<w<<1. I assumed an expansion of the form x=a+bw+... and from this obtained a=-1, b=1 as one solution. How do I work out the form of the other 2 expansions? Thanks.
  39. E

    Deuteron Ground State / Perturbation Theory

    Homework Statement The deuteron ground state is made up of l = 0 and l = 2 states; a)Show this mixture cannot be an eigenstate of a central potential Hamiltonian b)Using first-order time independent perturbation theory, argue the potential must contain a term proportional to some combination of...
  40. A

    Time-Dependent Perturbation Theory and Transition Probabilities

    I'm rather stuck on this problem. I seem to be having issues with the simplest things on this when trying to get started. Homework Statement There is a particle with spin-1/2 and the Hamiltonian H_0 = \omega_0 S_z. The system is perturbed by: H_1 = \omega_1 S_x e^{\frac{-t}{\tau}}...
  41. B

    First order perturbation theory problem

    Ok so I have a classic particle in a box problem. If a particle in a box, the states of which are given by: ψ = (√2/L) * sin(nπx/L) where n=1,2,3... is perturbed by a potential v(x) = γx , how do I calculate the energy shift of the ground state in first order perturbation I'm guessing that...
  42. M

    Optical theorem and renormalised perturbation theory (c.f. Peskin 10.2)

    Hi all, On p.327 in my second edition of Peskin and Schroeder, I have an expression for the one loop correction to the 4-point amplitude of phi^4 theory: i\mathcal{M}=-i\lambda - \frac{i \lambda^2}{32 \pi^2}\text{[Complicated integral]} Mathematica can do the integral for me, and all that...
  43. B

    Specific question on Goldstein section on Time-independent perturbation theory

    I apologize that this is rather specific, but hopefully enough people have used Goldstein. I have a basic grasp of action-angle variables, and I'm going through the time-independent perturbation theory section in Goldstein (12.4). In this section we seek a transformation from the unperturbed...
  44. L

    Perturbation theory (qualitative question)

    Homework Statement How does the energy change (negative, positive or no change) in the HOMO-LUMO transition of a conjugated polyene where there are 5 double bonds when a nitrogen is substituted in the center of the chain? The substitution lowers the potential energy in the center of the box...
  45. alemsalem

    Weak field zeeman effect, why don't we still use degenerate perturbation theory?

    the spin orbit coupling removes the degeneracy but not completely, should we still use the degenerate perturbation theory. is it because of relativistic corrections? Thanks!
  46. M

    Question about Quantum + Thermodynamic Perturbation theory

    The following comes from Landau's Statistical Physics, chapter 32. Using a Hamiltonian \hat{H} = \hat{H}_0 + \hat{V} we get the following expression for the energy levels of a perturbed system, up to second order: E_n = E_0^{(0)} + V_{nn} + \sideset{}{'}{\sum}_m \frac{\lvert...
  47. A

    Fine structure constant and perturbation theory

    Hi all, I have a question about perturbation theory and the fine structure constant. Consider an electron moving through the vacuum - this wil induce vacuum polarization, and (if I understand correctly) perturbation theory can be used to analyze the situation. My question is essentially: if...
  48. P

    Potential well with inner step, perturbation theory

    hey, say you have a infinite potential well of length L, in the middle of the well a potential step of potential V and length x. Inside the well is a particle of mass m. why are the first order energy corrections large for even eigenstates compared to odd ones? also, say well...
  49. P

    Perturbation Theory: Time-Independent, Non-Degenerate Results

    time-independent, non-degenerate. I am referring to the following text, which I am reading: http://www.pa.msu.edu/~mmoore/TIPT.pdf On page 4, it represents the results of the 2nd order terms. In Eqs. (32), (33) and (34) I don't understand the second equality, i.e. basing on which formula he has...
  50. N

    Degenerate Perturbation Theory

    Hi I am reading about Degenerate Perburbation Theory, and I have come across a question. We all know that the good quantum numbers in DPT are basically the eigenstates of the conserved quantity under the perburbation. As Griffiths he says in his book: "... look around for some hermitian...
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