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

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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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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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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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Scalar QED perturbation theory [closed]

Where can I find material about scalar QED? Renormalization, scattering matrix, perturbation theory, etc...
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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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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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188 views

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 ...