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

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“Zero overlap” of initial and final states in meson to nucleon + antinucleon scattering of scalar Yukawa

I'm currently studying QFT from David Tong's lecture notes and video lectures. In meson to nucleon + antinucleon decay (section 3.2.1 in this ) in scalar Yukawa theory to order $g$, without using ...
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In perturbation theory, how do I determine the order of an approximation?

The title says it all: I'm confused about the various approximations and their orders. In time-independent perturbation everything is quite explicit and obvious, but, for example, how would it be with ...
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Perturbative series for bosons

I have recently read that ... the perturbation series ... is valid only when the perturbed state is qualitatively similar to (or ‘has the same symmetry as’) the unperturbed state. This means ...
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Solving the quantum an-harmonic oscillator pertubatively?

Background Generally while solving the quantum an-harmonic oscillator: $$ -\frac{d^2 y}{dx^2} + k_1 x^4 y + k_2 x^2 y= E y $$ Most people (I've googled) on the internet always solve this using: ...
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Multiple-scale analysis for 2D Hamiltonian?

I came across a technique called "multiple-scale analysis" https://en.wikipedia.org/wiki/Multiple-scale_analysis where the equation of motion involves a small parameter and it is possible to obtain an ...
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Intuitions on perturbation theory?

I'm a QM newbie and I want to ask some questions on how to accept some peculiar points on this perturbation theory thing naturally. While they can be natural for most of the people, I find it somewhat ...
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244 views

Any good textbook on the canonical perturbation theory for Hamiltonian systems?

My teacher of classical mechanics once told us, classical mechanics is more difficult than quantum mechanics in many ways. He used the perturbation theory as an example to illustrate this point. So, ...
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112 views

Perturbation theory, eigenvalues and eigenvectors for degenerate case (1st order)

I was trying to understand the perturbation theory, but I was lost in the notation... I have understood that I have to identify the unperturbed kets that are degenerated and find the matrix $V$, ...
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57 views

Is this solvable? Time-dependent perturbation theory

The question is A hydrogen atom is placed in a time-dependent homogeneous electric field given by $$ \varepsilon(t) = \varepsilon_0(t^2 + \tau^2)^{-1} $$ where $\varepsilon_0$ and ...
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83 views

Lippmann-Schwinger equation and $T$ expansion

Lippmann-Schwinger equation, in operator form, is: $$ T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T, $$ where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
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If we considered chiral perturbation theory with coplex $\phi$-s, wold the next lo leading order renormalization $\gamma$-s change?

The Lagrangian of chiral perturbation theory (with two quark flavors) is written using the following matrix $U$ $$U=e^{i\sigma^i\phi_i/f}$$ where $\sigma^i$ are the Pauli matrices, $\phi_i$ are three ...
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Definition of linear response kernel in terms of wavefunctions (Parr/Yang)

I'm trying to understand the derivation of the linear response kernel in Parr/Yang's "Density-functional theory of atoms and molecules". First some background information: We look at a system of $N$ ...
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18 views

Limitation of Rytov approximation for turbulence

I have been working through the textbook "Laser Beam Propagation through Random Media" by Andrews and Phillips and have arrived at an interesting dilemma. For a second-order perturbation in the Born ...
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74 views

What justifies the perturbative expansion in chiral perturbation theory?

The Lagrangian of chiral perturbation theory is ordered following a momenta power counting scheme, having terms at leading order (which is two 2 $O(p^2)$) next to leading order ($O(p^4)$) and so on. ...
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35 views

Bound states and corresponding elementary fields

Let's have some bound state, like positronium or meson. I need to calculate an amplitude of process which involves bound state in in- or out-state. Is it necessary to introduce corresponding ...
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34 views

Proton spin independent fine structure “Hamiltonian” $W_f$

To find the perturbation correction (fine structure) in the case of a degenerate energy $E_n^0$, we can diagonalize the operator $W_f^n$, the restriction of $W_f$ to the eigen-space associated to ...
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Is the perturbation Hamiltonian an observable?

In fine structure calculation we use the perturbation theory. The basic Hamiltonian $H_0$ is perturbed as: $H = H_0 + W$ First, the basic problem assume that $H_0$ is an observable. That allows to ...
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Quantum perturbation theory recommendations

What are some concise resources, in particular, online resources, for perturbation theory in quantum mechanics? I want something like a crash course to perturbation theory in quantum mechanics that is ...
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142 views

Effective hamiltonian for the second-order degenerate perturbation theory

I'm currently trying to figure out the way we arrive to the Hamiltonian of a topological insulator. In an article by Xiao-Liang Qi (arXiv: http://arxiv.org/abs/1005.1682) in a process of arriving to ...
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164 views

Perturbation theory in second quantization

I am dealing with electron/phonon interaction in QM. In particular, given the Hamiltonian of a solid, $$H=H_\text{el}+H_\text{ion}+H_\text{el-ion}$$ we have that the el-phonon Hamiltonian is treated ...
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49 views

How is translational symmetry related to Fourier decomposition?

The book (The Cosmic Microwave Background By Ruth Durrer) about cosmological perturbations says that because of translational symmetry of the background at a constant time, we can decompose our ...
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135 views

How was this one probability amplitude derived by Mattuck?

I'm reading A Guide to Feynman Diagrams in the Many-Body Problem by Richard D. Mattuck (2nd edition). You can look at the relevant pages here. On page 45, he presents a formula for $D_t c_p(t)$. ...
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Corrections in Perturbation theory

Is there a way to construct a bound on the perturbative corrections to a problem in perturbation theory? For example, if I have the standard 1st order correction to the eigensolutions of a problem ...
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Rigorous Proof of General Relativity's Non-renormalizability?

The answer to this question and the comments on it implies that general relativity has not been rigorously shown to be non-renormalizable for all loop diagrams -- only shown for two loops. However, ...
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Perturbation in Linear Response Theory (classical formalism)

A N-particles system is described by the following Hamiltonian $$H=H_0+H'(t)$$ where $H_0$ is the unperturbated Hamiltonian and $H'(t)$ is the perturbation $$H'(f)=-A\cdot\mathcal{F}(t)$$ written as ...
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How do the renormalization enter the actual amplitude calculation in QFT?

I have studied QFT from Peskin and Schroeder and from a few other books and lectures and I think I understand the procedure of renormalizing various parameters in the Lagrangian like mass, coupling ...
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97 views

Kubo formula for general observables

In the wiki page about Kubo formula, the expectation of some observable under weak time-dependent perturbation is derived. However, from my point of view, some crucial steps are missing. I did the ...
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Capturing (perturbatively) non-equilibrium field theory effects using “elementary” methods

I am considering a system of two interacting scalar fields: $\psi$, and $\phi$. The Lagrangian is given by: \begin{equation} ...
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Expectation value in spin-orbit coupling

So I was just trying a question where it asked to find the Energy shift due to a spin-orbit coupling Hamiltonian to first order using perturbation theory. The Hamiltonian is $$H_{LS} = ...
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35 views

What is a diffusionless fluid?

I'm taking a course in astrophysical fluid dynamics and have come across a problem involving "small diffusionless disturbances" of a fluid. Based on the nature of the course I expect the examiner to ...
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88 views

What really are perturbation expansions?

I'm unsure if this question belongs here or at Math.SE, but since I've got to it by reading some articles about Physics I'm going to post it here anyway. In this particular article (Theoretical ...
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How should I interpret degenerate $\pm m_j$ states under the Stark effect?

I'm thinking about the Stark effect in Alkalis where fine structure is important (Cs, Rb, etc). The Stark effect doesn't lift the degeneracy of the $\pm m_j$ states. So should I interpret a state ...
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Post-Minkowskian expansion of some quantities in Post-Newtonian theory

I'm studying Post-Newtonian theory on the book "Gravity" by Poisson and Will and I found a few formulas that I can't obtain by myself. I'm pretty sure it must be quite simple, but can't find the right ...
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Fermi Golden Rule

First order time dependent perturbation theory tells us that under the influence of a perturbation $Ve^{i\omega t}$, a system that started in the state $|n\rangle$ at time $t=0$ has probability ...
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Variational principle perturbation

I am trying to learn 2D SHO and free particle variational principles. However, I came across a perturbation like so: $Ax^2y^2$. The particle is simple harmonic in $x$ and a free particle in $y$? I'm ...
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Second-order correction in Quantum-Confined Stark effect

In the wikipedia article, there is a second-order correction in the Quantum-Confined Stark Effect. I could not understand how it was solved. I did not understand the meaning of 2(0) and 1(0) and how I ...
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Perturbation theory in quantum harmonic oscillator [closed]

This question concerns the quantum harmonic oscillator: (a)Express the operator $\hat B = \hat x \hat p + \hat p \hat x + \hbar$ in terms of $\hat a_{\pm}$ and $\hbar$ (b)Write the matrix ...
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125 views

Perturbations in linear response theory

I've been working on applications of linear response theory to condensed matter systems, and I've got quite far into the literature on the subject. However, there is an identity which seems to be ...
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173 views

Perturbation theory : quadratic external field

I'm trying to derive the explicit form of S-matrix of an interaction Hamiltonian $$H' = \frac{1}{2} \lambda \left[ \int d^3 x \rho({\vec x}) \phi({\vec x}, t)\right]^2\tag{1}$$ Even though the ...
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782 views

How to tell the order of a Feynman diagram?

How can we know the order of a Feynman diagram just from the pictorial representation? Is it the number of vertices divided by 2? For example, I know that electnro-positron annihilaiton is first ...
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55 views

Linear Perturbation theory in General Relativity, what to do with products of derivatives?

I'm trying to do a problem in which I am given the Einstein tensor for the following metric: $$ ds^2 = -e^{2\Phi}(d{x^0})^2 + e^{2\Psi}\delta_{ij}dx^idx^j, $$ And then asked to find the Einstein ...
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Applicability of perturbation theory

Consider some system in some initial state $|k^{(0)}\rangle$. The probability that such a state makes a transition to some other state $|m^{(0)}\rangle$ can be computed to various orders in time ...
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Gauge invariant quantities [closed]

In the context of cosmological perturbation one write the most general perturbed metric as $$ ...
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113 views

Performing one-loop “triangle” integral

Let's have integral $$ \tag 1 I_{\alpha \beta \gamma \delta} = \int d^{4}qD^{V}_{\alpha \beta}(k + q)D^{V}_{\gamma \delta}(q + k - p)D^{f}(q), $$ where $D^{V}_{\alpha \beta}$ corresponds to massive ...
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123 views

Energy levels in close-proximity of each other in time-independent degenerate perturbation theory

I've applied second order time-independent degenerate perturbation theory corrections to the energy with the method presented in Modern Quantum Mechanics by J.J. Sakurai. I shortly summaries this ...
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592 views

Non-degenerate or degenerate perturbation theory for a non-degenerate level of a system with other levels degenerate?

To decide whether I have to use non-degenerate or degenerate perturbation theory, I have to look only on whether the energy level I am calculating corrections to is degenerate, the degree of ...
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Decay of some particle involved quarks vs mesons as outgoing states

Let's have decay width of some mother particle into the state which involves hadrons. For simplicity, let's assume that creation of hadrons (on diagram) is possible only through electroweak vertices ...
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Quadratic order perturbation terms in the expansion of Ricci tensor [closed]

I want to expand Einstein-Hilbert action for the metric $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} $$ up to quadratic order in $h_{\mu \nu}$. For this purpose I need to calculate the Ricci tensor ...
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What is the sum over the transition rates?

I was looking at the solution to an exercise, and I came over this expression: $$P_{i\to f} = \sum \limits_{f} {2 \pi \over \hbar }\; |\langle f |\hat V | i \rangle |^2 \delta(E_{fi}-E),$$ where ...
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279 views

Does the fact that we cannot exactly solve the Standard Model undermine the validity of QFT?

I have seen discusstions of this types before: there is a question about photons or virtual particles or vaccuum, etc. And there is usually a good and clear explanation from the point of view of ...