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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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Perturbed Ricci scalar in Modified Gravity

When getting the perturbed Ricci scalar in a Modified Gravity theory of the form $\mathcal{L}_{gr}=F\left(\phi,R\right)R$, $\phi$ being a scalar field, it is easy to arrive at an expression of it in ...
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Relation Asymptotic Series and perturbative effects

Perturbative expansions of a function $f(x)$ around say $x=0$ cannot determine contributions from a function such as $e^{-1/x}$ since its Taylor series vanishes to all orders. This kind of ...
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Cosmology - Demonstration for equation of the evolution of the density contrast

In a context of cosmology, I need help about a differential equation that I can't get to demonstrate: The growth of density fluctuations obeys a second order differential equation. At early enough ...
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Heuristic physics journals? [on hold]

I am in the last year of a Phd about perturbations in General Relativity --probably also the year of physics in my life-- and I am searching for a refereed journal in order to know if some intuitions ...
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Mechanism behind spin orbit interaction

The spin orbit coupling can be explained through two different frames one is the electron frame and the other, the lab frame. In the electron frame magnetic field produced by proton current current ...
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What happened to the factor of $\pi$ in this question?

$\\ $ I was going through the answer to this problem, when I noticed that a factor of $\pi$ in the denominator disappeared and a factor of 4 appeared in the numerator when the author started ...
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Why do we need the coupling small when doing perturbative QFT calculation?

I don't really understand why, when we calculate say the 2-point Greens function in a scalar QFT with interaction $\lambda \phi^4$, we need the coupling constant $\lambda$ to be small? Everywhere I ...
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How to express the wavefunction of a harmonic oscillator in a perturbing electric field?

So I am looking at the problem of the (charged) harmonic oscillator in a weak electric field - the problem that defines e.g. the polarizibility of the oscillator. Let the fieldless Hamiltonian be: $...
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Gaussian integral extended to multi-dimensions

In Quantum Field Theory in a Nutshell by A. Zee, the following integral $$Z(J)=\int_{-\infty}^{+\infty} d q e^{-\frac{1}{2} m^{2} q^{2}-\frac{\lambda}{4!} q^{4}+J q}$$ is solved perturbatively by ...
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Precise conditions for Apply Degenerate vs. Non-Degenerate Perturbation Theory?

Is there a precise way to phrase when you are allowed to apply non-degenerate perturbation theory versus degenerate perturbation theory? When you have a Hamiltonian of the form $$ H = H_0 + \lambda V(...
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Pertubation of Riemann tensor in a general curved space-time

It is a direct and simple question. I am fully developing the perturbation of Einstein Field Equations, and I need to calculate the perturbation of the Riemann tensor. However the background metric is ...
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Linear perturbations of the energy conservation in FLRW spacetime

Recently i have some troubles regarding linear stability analysis in GR, especially matter conservation equation. First order perturbations of the Hubble parameter and energy density are: $$H=H_b(1+\...
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Teukolsky (Bardeen-Press) equation ingoing coordinates

EDIT: I can't seem to delete this question, so I've posted the solution below (I must have made an algebra error-someone checking all this would still be appreciated!). I've left the question as is. ...
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Renormalization group and summation of diagrams

Currently I'm studying renormalization group, and I'm having trouble understanding the following statement which I see almost everywhere in books on QFT: renormalization group sums a series of ...
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Derivation of Perturbation Terms in Thermodynamic Perturbation Theory

In the "A critical evaluation of perturbation theories by Monte Carlo simulation of the first four perturbation terms in a Helmholtz energy expansion for the Lennard-Jones fluid" paper by T. van ...
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Time-dependent pertubation theory assigning the order of expansion to squares of the solution

What is the square of a solution from time dependent pertubation theory? Assume we have found the corrections up to second order such that $$ |\psi(t)\rangle \approx |\psi^0(t)\rangle + |\psi^1(t)\...
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Linearity of Schrödinger equation and perturbation theory

So, I was studying quantum mechanics and reached the point where perturbation theory is discussed. It is my first time in this topic, and something called my attention: it was said that we need ...
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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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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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1answer
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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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1answer
63 views

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|...