Questions tagged [perturbation-theory]

Perturbation theory refers to methods for understanding physical systems by treating them as small modifications to exactly solvable systems.

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Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator

After a perturbation $V(x)$ added to the system, a matrix element $H_{nn}$ calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as: $$\epsilon \hbar \omega_0\begin{...
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Infinitesimal Perturbation in a potential well

If I calculate $<\psi^0|\epsilon|\psi^0>$ and $<\psi^0|-\epsilon|\psi^0>$ separately fro two different intervals, viz. 0 to $\Delta$ and from $a-\Delta$ to $a$ and then add, the ...
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New locution regarding perturbation theory

I am trying to make a sentence more approachable to a general audience by not using technical language. I fear I'm however losing precision in this new language. Original sentence: The use of ...
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Non-linear Perturbations of Minkowski Spacetime

I am reading some of the following paper on the bounded $L^2$ conjecture in general relativity where it mentions non-linear perturbations of the Minkowski metric in the context of quasilinear wave ...
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Where does decay widths in mass mixing matrix come from?

For some time ago I've seen people use complex mass mixing matrix including decay width of the particles. It kind of makes sense, but I could never fully justify it. I would be grateful if you could ...
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What happens to Hubbard chain when perturbation theory blows up? Singular energy for complex interaction

Consider the spinful Hubbard chain: $$ H = - t \sum_{i,\sigma} \left( c^\dagger_{i,\sigma} c^{\vphantom \dagger}_{i+1,\sigma} + h.c. \right) + U \sum_n \left( n_{i,\uparrow} - \frac{1}{2} \right)\left(...
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Perturbations of a Background Spacetime

I am reading some lecture notes on general relativity, where the author talks about perturbation theory applied to GR. In the case of a weak gravitational field, one perturb about the Minkowski ...
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What does the term $\mathcal O(\epsilon^2)$ mean?

In the highest upvoted answer to Where does the $i$ come from in the Schrödinger equation? the author writes the following equation: $$ U^\dagger U=(\mathbb I+\epsilon^* A^\dagger)(\mathbb I+\...
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Minus sign in perturbative expansion via Green's function (Schwartz QFT)?

In Schwartz's QFT textbook Section 3.5, the Lagrangian for the graviton $$\mathcal{L}=-\frac{1}{2}h\Box h+\frac{1}{3}\lambda h^3+Jh$$ with EOM $\Box h-\lambda h^2-J=0$ is perturbatively expanded in $h$...
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Calculating the numerical factor from Feynman diagram

I kind of understood the symmetry factor quite well. However, I just do not understand how one can relate the Feynman diagram to the term (especially the numerical factor in front of it) in the ...
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First order wave function in adiabatic approximation

If the Hamiltonian is slowly varying in time and suppose the initial state is the n-th eigenstate of the initial Hamiltonian H(0), the adiabatic theorem says that the state will still on the n-th ...
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Calculating $\langle \psi_j^0 | \delta\psi_i\rangle$ in perturbation theory [closed]

Within first order (or linear order) quantum perturbation theory, the Schrödinger equation (for a state $i$) can be written: $$\delta H |\psi_i^0 \rangle + H^0 |\delta\psi_i\rangle=\delta\...
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Non-Degenerate Perturbation theory in Sakurai

As stated in the title, I'm studying Non-degenerate Perturbation Theory with the book 'Modern Quantum Mechanics' by J.J. Sakurai. The problem to solve is $$(H_0+\lambda V)|n\rangle = E_n |n\rangle$$ ...
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Diagrammatic Representation of non-Gaussian perturbation expansions

I have no experience in graph theory and am a little confused with how Hugh Osborn represents a perturbation expansion with diagrams on page 15 of these notes. We have a perturbation expansion My ...
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Perturbation expansion with path integrals

This is from Hugh Osborn's 'Advanced Quantum Field Theory' notes, Lent 2013, page 15. I want to evaluate the expression $$ Z = \exp\Big(\frac{1}{2} \frac{\partial}{\partial \underline{x}} . A^{-1} \...
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The stark effect of Hydrogen

I have read the stark effect of Hydrogen (calculating energy levels of the n=2 states of a Hydrogen atom placed in an external uniform electric field along the positive z-direction) from Quantum ...
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Prove that colliding celestial bodies produce perturbations in the fabric of space-time

So I was recently asked this question by one of my professors, it really is confusing for me at the moment since I only have a basic grasp of and the ideas Einstein proposed. P.S: I was wondering if ...
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Questions about scattering matrix theory of non-free particles

Hi,I have a problem for scattering matrix theory. Currently, the book I've read is about collision between free particles. What if collision between non-free particles? For example, in lattice, only ...
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Questions about perturbation theory

In time-dependent perturbation theory, when assuming $\hat{V}$ is time-independent, the time development operator is as: $$\hat{U}(t,0)\theta(t)=e^{-i(\hat{H_{0}}+\hat{V})t}=\int \frac{dw}{2\pi}\...
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Ionization rate in atomic hydrogen

In perturbation theory (QM), i found the graph of ionization rate with various parameter like momentum of ejected electron, intensity, frequency, field strength of incident laser field. But i am ...
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Why only loop effects be quantum corrections when the full theory is quantum?

Feynman diagrams with one or more loops in an interacting QFT are diagrammatic representation of corrections to the Green's functions and amplitudes beyond the lowest order in perturbation theory (...
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Perturbative violation of the unitarity: what is it?

Consider the Fermi theory: $$ \mathcal{L} = \frac{G_{F}}{2\sqrt{2}}\bar{n}\gamma_{\mu}(1-\gamma_{5})p \bar{\nu}\gamma^{\mu}(1-\gamma_{5})e $$ The cross section of $2 \to 2$ scattering calculated ...
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Quantum corrections in path integral

I am working the following exercise: Calculate the generating functional $$Z[j]=\int \mathcal{D}\Phi \exp\left(\frac{i}{\hbar}S[\Phi,j]\right),\quad S[\Phi,j]=\int d^4x(\mathcal{L}(\Phi)+j\Phi),$$ $...
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Linearized gravity and perturbation theory

This is a question regarding a calculation in perturbative GR. We have : $g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$ where $h_{\mu\nu}$ is a small perturbation around the flat spacetime metric. In ...
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In relativistic QFT, is it ever possible that the bare mass be finite and equal to the physical mass?

In renormalization, one follows the philosophy that the bare mass is unobservable and could be infinite, and the physical mass comes from the pole of the two-point function. Is it possible that in any ...
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Where can I find relativistic corrections to 2s and 2p levels of Hydrogen Atoms?

I am currently studying for a Quantum Mechanics test, and I want to calculate the 2p and 2s hydrogen atom corrections for the relativistic, spin-orbit and darwin corrections, using perturbation theory....
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Resonant Hamiltonian Mechanics

My question is regarding applying averaging theory to a perturbed Hamiltonian. Now, my Hamiltonian is of the form $$H=H_0 + R(q_i,p_i)$$ Where R is the disturbing potential which is a function of the ...
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Calculating exact energy levels of perturbed Hamiltonian

I wish to find the exact energy levels of the following perturbed hamiltonian. $$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$ I believe that it can be solved by using the ...
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Finding eigenvalues and functions for Hamiltonian (perturbation theory)

I am trying to find the Eigenvalues for the following equation (which comes from the Pauli equation when $p^2/m^2c^2\ll 1$): $$i\hbar\frac{d}{dt}\psi=\left[\frac{\vec{p}^2}{2m}-\frac{e}{2mc}\left(\vec{...
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How should two-photon transitions be modelled? Is second-order perturbation theory required? Or are sequential first-order processes sufficient?

For example, I want to consider the following situation: photon transit from $m$ energy level to $m+2$ after absorption of two phonons with frequency $\Omega$. I want to calculate a transition rate ...
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First-order correction to energy in perturbed harmonic oscillator [closed]

I know, from the perturbation theory, that, if I have the hamiltonian $$ \hat H = \hat H_0 + \lambda \hat W$$ where $\hat H_0$ is the unperturbed hamiltonian of which I know its eigenvectors and ...
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Does the second-order correction to degenerate perturbation theory vanish?

Consider a degenerate two-state system with states denoted by $|1\rangle$ and $|2\rangle$. If we apply a perturbation $H^\prime$, the first order correction to the energy is obtained by choosing two ...
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State coefficient in Kramer-Heisenberg-Dirac formula

I'm following the derivation of the Kramer-Heisenberg-Dirac formula from the book "Modern Optical Spectroscopy" and I'm having trouble understanding the wording and the derivation of the transition ...
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Srednicki Eqs. (6.22) and (9.6). How to get rid of $i\epsilon$ in the interaction term?

I'm studying qft from Srednicki's book. If one writes down the full $i\epsilon$ terms, passing from Eq. (6.21) (non-perturbative definition) to Eq. (6.22) (perturbative definition) yields $$\left<0|...
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Deriving the continuity equation for a perturbation from the continuity equation for the full density

I am attempting to derive the continuity equation for a density perturbation $\delta$, given the continuity equation for the full density $\rho(\mathbb{x}, t)$. This is in the context of cosmological ...
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Help with calculating the Ricci tensor for the PPN formalism

I'm trying to follow the calculation done by Will in his book Theory and experiment in gravitational physics, and I was hoping for some help in calculating the Ricci tensor components in Section 5.2 (...
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Learning resources for the PPN formalism

I'm trying to learn more about the PPN formalism, and I was hoping to find some papers that show how the various parameters are calculated. Basically, I want to go through the usual calculations of ...
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Feynman rules for a general Lagrangian

How do I find the Feynman rules for a general Lagrangian density? For example the Lagrangian $$L = \partial_\mu \psi \partial^\mu \psi +a \psi\partial_\mu \psi \partial^\mu \psi+b \psi^2 \partial_\mu ...
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Inversion of a metric

I am currently reading a paper by Bredberg $et.al$ arXiv:1101.2451 titled "From Navier-Stokes to Einstein". In this paper, the authors have considered a metric of the form \begin{eqnarray}ds^2_{p+2} = ...
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Supersymmetry Perturbation Theory

Source:Mirror Symmetry p.198 I have the Hamiltonian $$H = \lambda\bigg( \frac{1}{2} \tilde{p} + \frac{1}{2}h''(x_i)^2(\tilde{x}-\tilde{x_i})^2 + \frac{1}{2}h''(x_i)[\overline{\psi}, \psi] \bigg) + \...
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Why is the “fine structure” correction called that way?

I'm working on the fine structure correction to the Hydrogen atom. I have more of a conceptal, maybe historical question, why is this correction called this way? and why is the fine structure constant ...
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Estimating error in perturbation theory

Is there a simple way to estimate the error in the eigenvalues when approximating a hamiltonian by its $n^{th}$ order perturbation expansion?
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How to ascertain that the Rayleigh-Ritz variational method gives the exact value of the ground state energy?

So the Rayleigh-Ritz variational method can be used to calculate the ground state energy of a quantum system. If $\phi(x)$ is a suitable (square integrable) and normalised function of the coordinates ...
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Where does this Differential Equation comes from?

Im studying Stark Effect and im trying to prove that the second order correction to the ground state of hydrogen like atoms goes like \begin{equation} \delta E^{(2)}_{100}= -\frac{1}{4}a_o^3 E^2(4+5Z^...
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For degenerate perturbation theory, how do we interpret the eigenvectors and eigenvalues of $\hat V$?

For the eigenvectors that are unmixed by the matrix $\hat V$, the eigenvalues are the energy corrections of this eigenbasis. However, the eigenbasis tends to always be (as far as I'm aware) a linear ...
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QFT perturbation theory

I would like to clarify the following statement: Perturbation theory (PT) in QFT is derived with several assumptions such as: adiabatic interaction, spectrum is bounded downward... This statement ...
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Different purposes for using the Large-$N$ Expansion

I've started studying the Large-$N$ expansion and there seems to be several different reasons for using it. In the context of the SYK model, the limit is useful because it reorganizes the Feynman ...
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Horizon entry, Meszaros suppression and start of perturbation growth

I thought that the onset of perturbation growth was determined by horizon entry of the perturbation (because there won't be a gravitational collapse of an over dense region not causally connected to ...
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Expression of proper time with light perturbation on Minkowski metric

In my lecture on General relativity, it is said that, by taking the following metric : $$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x),~ {\rm with}\ h_{\mu\nu}\ll1$$ one has the definition below of ...
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What is Wick's theorem and what this is use for? [closed]

I am reading Wick's theorem but although I look for it to clearly understand in some textbooks and youtube videos but still it is unclear to me. I cannot get my head over what is normal ordering ...