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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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Schrieffer-Wolff Transformation for conventional superconductors

I was trying to follow the discussion in Radi A. Jishi's book (Feynman Diagrams in Condensed Matter Physics), Chapter 12 on superconductors. They basically have a Hamiltonian that comprises of a ...
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Rigorous proof of Bertrand's Theorem for orbits under central force

I have read through several proofs of Bertrand's Theorem, including the one on Wikipedia. A typical proof can be found here (Santa Cruz Institute for Particle Physics). Almost all proofs using ...
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Diffeomorphism for Equation of motion in Brans Dicke theory

I am trying to see the gauge invariance, (diffeomorphism) for equations of motion in Brans-Dicke theory. Frist the equation of motion is given as follows, [I just copied from Wikipedia] \begin{...
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What does it mean to study transition probability?

I'm starting to study time-dependent perturbation theory. The book applies time-dependent perturbation on the hamiltonian and it says that for this reason quantum transition are allowed. Then it ...
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Infinitesimal displacement and taylor series [closed]

Why does the infinitesimal shear strain approximated by Taylor series? Why the partial derivative multiplied by (dx)? can somebody elaborate more on this?
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34 views

Borel Resummation on a finite asymptotic series

I am reading http://users.physik.fu-berlin.de/~kleinert/kleiner_reb8/psfiles/16.pdf I am a little confused on the use of the Borel Transform in perturbative series. I know that Green's Functions in ...
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couplings between mesons and baryons (singlet and octet)

What is the basic meaning and good reference for the deep and simple meaning(in simple language) of couplings between mesons and baryons, e.g., g1 and g8 couplings (singlet and octet) from where these ...
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55 views

Perturbation Theory - Exact Solutions and Good Quantum States

I'm having a problem with the following question: Problem: Consider the unperturbed, degenerate Hamiltonian $H_0=\bigg(\begin{matrix} E &0\\ 0& E\end{matrix}\bigg)$. Add the perturbation $...
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2answers
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Struggling to understand degenerate perturbation theory

As far as I gather, before a perturbation is applied, the eigenspace associated with the degenerate energy is multidimensional but after applying the perturbation this space 'splits' into different ...
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First Order vs. Second Perturbative Transition Probabilities with Unruh deWitt Detectors

I have been working with perturbative expressions for the transition probability/rate of an Unruh deWitt detector. Many papers such as this one - https://arxiv.org/pdf/gr-qc/0606067.pdf - seem ...
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26 views

What's the most general approach to Zeeman effect?

I have a question regarding the Zeeman effect and perturbation theory in the hydrogen atom. We have hamiltonian of the hydrogen atom is given by $H_0$, that of spin-orbit coupling given by $H_{\text{...
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Problem in odd-even decomposition of a generic metric

The metric of the unit two-sphere is given by $ \Omega_{\mu \nu} = \begin{equation} \begin{pmatrix} 1 & 0 \\ 0 & \sin^2 \theta \end{pmatrix}. \end{...
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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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31 views

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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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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1answer
45 views

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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1answer
274 views

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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33 views

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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89 views

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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1answer
42 views

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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1answer
34 views

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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1answer
309 views

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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1answer
35 views

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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1answer
60 views

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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29 views

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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54 views

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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68 views

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
44 views

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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162 views

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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1answer
49 views

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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39 views

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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1answer
61 views

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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1answer
78 views

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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27 views

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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1answer
40 views

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 ...
2
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1answer
65 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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1answer
48 views

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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2answers
95 views

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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1answer
37 views

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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2answers
94 views

Quantum corrections in path integral

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