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

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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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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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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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How to find 2nd order pertubation to wave function? [closed]

Today, I'm looking for how to find the 2nd perturbation to the wave function in Rayleigh Schrödinger Perturbation Theory (RSPT). SETUP Starting from the 2nd order perturbation in Dirac's notation: ...
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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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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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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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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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Stationary Perturbation Theory

We write first order correction in the wavefunction as a linear sum of eigenstates of unperturbed hamiltonian. Why is this possible?
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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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Final States in time dependet pertubation theory

I am trying to understand how to get over a common contradiction in time dependent perturbation theory. In time dependent perturbation theory we assume, that the perturbation only changes the ...
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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 ...
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Definition of “nonlinear” in the context of perturbation of gravity

What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric? I have seen so called "linear perturbations" which look like $$ds^2 = -(1+2\Phi)dt^2 ...
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Time evolution of two orthogonal states in Time Dependent Perturbation Theory

Given the two orthogonal states for $H_0$ , $|n(t)>_I, |m(t)>_I$, in the interaction picture, we want to find the probability of transforming from one to the other after time t, aka: $ \ (1) \ ...
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Is the energy expectation value comparable to the equation from power series ansatz?

The Hamiltonian is given by $$ H = H_0 + \lambda H_1 $$ where $H_0$ is the unperturbed Hamiltonian, which solves the Schrödinger Equation $$ H_0 \left |n^{(0)} \right \rangle = E_n^{(0)} \left ...
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General two state system with general pertubation

I am trying to solve this two-level system with time independent perturbation problem Consider an atomic system with only two stationary states $|1\rangle$ and $|2\rangle$ , of respective energies ...
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Question about derivations in Sakurai's Quantum mechanics, section 5.8 [duplicate]

If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He ...
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Sakurai QM section 5.8

If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He ...
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Transition Probabilities for the Perturbed Harmonic Oscillator

I consider the following Hamiltonian $$H=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\Theta(t)Fx,$$ where $F$ is an external constant force. So the Hamiltonian describes an unperturbed harmonic oscillator ...
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On GR with perturbation

Could anyone explain to me what I have misunderstood/missed when trying to understand this paper on GR perturbation? The paper is http://arxiv.org/pdf/0704.0299v1.pdf In equation 25 for $R_{00}$, ...
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A question about first order perturbation in the Stark effect.

Consider ground state of a hydrogen atom which influenced with external uniform weak electric field. What does mean the following statement? $$\left< 100|\quad Z\quad |100 \right> =\int { |{ ...
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Integrability of the many body problem

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
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Renormalization of perturbation theory in non-relativistic quantum mechanics

A simple example: In calculating the nonlinear polarization of an atom, in perturbation theory, we typically get something like: $$p^{(n)} = \sum_{m=0}^n \left< \psi^{(m)} \right| \mu \left| ...
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Gauge invariance of Fermi's golden rule

I am having some issues with gauge invariance of Fermi's golden rule. Say we have a system Hamiltonian for a particle in an electric field and some additional potential $V$ with ...
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What does the first order energy correction formula in non-degenerate perturbation theory means?

I'm studying for a test in quantum mechanics and I'm currently trying to learn about perturbation theory and I've realized that I don't quite understand what I'm doing when I'm doing my calculations. ...
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Time Dependent Perturbation Theory Probabilities

(This is taken from Griffiths Quantum Mechanics): So suppose I have two states $\psi_{a}$ and $\psi_{b}$, and the particle starts out in the state $\psi_{a}$: $$ c_{a}(0)=1\qquad c_{b}(0)=0. $$ To ...
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Cubic perturbation to coupled quantum harmonic oscillators

I recently came across this two-dimensional problem of a particle in a potential of the form $$V = \displaystyle{\frac{1}{2}m \omega^2} \big(y^2 + x^2y \big) - \alpha y,$$ where $x$ and $y$ are known ...
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Wavefunction renormalisation in first order perturbation theory

I just read the following in the context of scattering amplitudes in QFT: Note that the wavefunction renormalisation factor $Z$ itself is of the form $1 + \mathcal{O}(\lambda)$ in perturbation ...
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Applications of the study of Hamiltonians with constant magnetic fields

I am interested in understanding possible applications for the study of quantum systems with constant magnetic fields. For definiteness, consider the Landau Hamiltonian $$H_{0} = ...
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Why does the time-independent perturbation theory become no longer useful when its order gets larger?

In Griffith's Introduction to Quantum Mechanics p. 256, after figuring out $$E_n^2=\sum_{m\neq n} \frac{|\langle\psi_m^0|H'|\psi_n^0\rangle|^2}{E_n^0-E_m^0}$$ he says We could go on to calculate ...
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Hartree-Fock correction to $e$-$e$ interaction

The corrections to the energy per electron in a jellium model (uniform distribution of positive ion charge approximation to the regulated long range order ionic array) is given by (in units of Ry) ...
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What's the importance of background field gauge?

Recently I've read that background field gauge is very convenient for gauge theories, because it fixes the connection between normalization constants of gauge field and gauge coupling constant one. I ...
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WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
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Projectile motion, solved with perturbation theory [closed]

A ball is dropped from rest, the initial height is $h_1$. There is air resistance given by $-mkv$. In the downward motion, the equation of motion can by written as $$\dot v = -g-kv$$ with the y ...
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Fine Structure Degenerate Perturbation Theory Hydrogen

Note: We are dealing with perturbation on the states $|nlm_lm_s>$ where n is the principle quantum number, l is the angular momentum quantum number, and $m_l$ and $m_s$ are the eigenvalues of $L_z$ ...
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On nonlinearity parameter in Nonlinear Schrodinger Equation (NLS)

While I am studying the wave propagation in fluids, the amplitude modulation seems to be governed by the NLS equation. In some of the journal papers the nonlinearity parameter, $N$ seems to be of high ...
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LSZ reduction vs adiabatic hypothesis in perburbative calculation of interacting fields

As far as I know, there are two ways of constructing the computational rules in perturbative field theory. The first one (in Mandl and Shaw's QFT book) is to pretend in and out states as free ...
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“Good” States In Degenerate Perturbation Theory

During the section on Degenerate Perturbation Theory, Griffiths (Introduction to Quantum Mechanics 2ed) starts with a general linear combination of two orthogonal eigenfunctions of $H_0$. He walks ...
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Textbooks on algorithms for the perturbative calculation of High energy physics

For the perturbative calculation of High energy physics, I have known some packages such as FeynArts, FeynCalc, MadGraph, CompHEP, GiNaC, and so on. But I am wondering whether there exists a textbook ...
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How to carry out the perturbation expansion of an anharmonic oscillator to high orders?

I think this is a standard problem in quantum mechanics. Consider the anharmonic oscillator $E \psi = \left(- \frac{1}{2} \frac{\partial^2}{\partial^2 x } + \frac{1}{2}x^2 + \epsilon x^4 \right) ...
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Estimating volume of moduli space of genus-g Riemann surface with n marked points

I wanted to know how can I estimate the volume of the moduli space of a Riemann surface of genus $g$ and having $n$ marked points. I am reading some old string theory papers which discuss divergences ...
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Coulomb Exhaustion

In thinking about a perturbation model for Coulomb's law, one could imagine photons spewing off of a charge in all directions. The chance of interaction with a near by charge being proportional to ...
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Splitting of degenerate energy levels with a perturbed particle in a box

Suppose you have a particle in a square box $[0,L]\times[0,L]$. As the box is a square, the (2,1) and (1,2) eigenfunctions will have the same energy. If you were to apply an oscillating electric field ...
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Electron photon interaction potential in old fashioned perturbation theory (OFPT)

In this PDF on old fashioned perturbation theory (OFPT) we find from equation (14) the potential describing the interaction between the electron and photon: $$ V = \frac{1}{2}e \int \mathrm{d}^3x\, ...
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Expansion in Quantum Fluctuations of the Path Integral

In this post: Dimensionless Constants in Physics there is a discussion about dimensionful vs. dimensionless constants in physics. In the context of this discussion, I'm wondering about the ...
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A question about the asymptotic series in perturbative expansion in QFT

Related post I heard about the argument that the perturbative expansion in QFT must be asymptotic, such as http://ncatlab.org/nlab/show/perturbation+theory#DivergenceConvergence Roughly this can ...