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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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Selection rule in three dimensional harmonic oscillator in representation $(n,l,m)$

I'm now doing exercises about selection rule for transition probability of first order perturbation. But in the one where a periodic perturbation Axcos(ωt) add to a three dimensional harmonic ...
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Question coming from Cosmological Perturbation

We consider the following scalar perturbation on the FRW metric: $$ ds^2 = -(1 + 2\phi)dt^2 +2a\partial_i B dx^i dt + a^2 \left( (1 - 2\psi)\delta_{ij} + 2\partial_{ij}E\right) dx^i dx^j $$ where $\...
Shivam Mishra's user avatar
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Clarification of Weinberg's cosmology book eqns 5.1.44-5.1.47 for scalar perturbation

Has anyone clarified the equations in Weinberg's cosmology book for scalar perturbation for nonzero $F$ and $B$, eqns 5.1.44-5.1.47. I am not sure why there are terms with $\nabla^2 \dot{B}$ and $\...
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Local linearization of general Fokker-Planck equation with spurious drift?

The most general Fokker-Planck equation for a probability density $f$ over $2d$-dimensional phase space is $\partial_t f = Lf$ for the differential operator $$L f = -\partial_i(u_i f) + \frac{1}{2} \...
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Problem with linearized gravity on flat background in spherical coordinates

I am solving the linearized Einstein equations with a flat background in spherical coordinates, i.e $ds^2 = -dt^2 + dr^2 + r^2 d\Omega^2$ and writing $h_{\mu\nu}$ in terms of spherical harmonics. ...
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What is the relation between the Faraday effect and the Zeeman effect?

The Faraday Effect basically says that certain materials under a magnetic field have different indexes of refraction for right and circular polarized light. Linear light which is a superposition of ...
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A question on perturbed flux function

Given a magnetic field $\mathbf{B}=B_{0}\left(\dfrac{xz}{z_{s}^{2}},\dfrac{yz}{z_{s}^{2}}, 1-2\left( \dfrac{x^{2}}{r_{s}^{2}}+\dfrac{y^{2}}{r_{s}^{2}}\right) -\dfrac{z^{2}}{z_{s}^{2}}\right)$ with the ...
gambling addict's user avatar
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What does it mean to "resum" the large logarithms?

I am struggling to understand the concept of resummation of large logarithms in QFT; from what I learnt so far the problem relies on the fact that if a full theory defined in the UV contains much ...
Filippo's user avatar
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Unperturbed eigenvector combination for degenerate case in perturbation theory

My question has arised from the previously asked question. In short: I have Hamiltonian with a perturbation such that $\hat{H} = \hat{H_0} + \lambda \hat{V}$. I know eigenvectors for the unperturbed ...
Марина Лисниченко's user avatar
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Weisskopf and self-energy

I am working my way through the 1934 paper by Weisskopf on the self-energy of the electron and is much helped by the English translation found here. I do have some difficulties with section 2 of this ...
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Effects of Localized Medium Changes on Field Propagation

I've studied various theories related to fields. These theories often include equations describing how the activity of a source is transmitted to other locations. The properties of the medium ...
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How to determine if gravity is roughly linear?

The Einstein field equations are famously nonlinear, which is one of the properties that makes them difficult to solve. I know (or at least I believe) that a linear system's behavior is roughly ...
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Why do we use perturbation theory instead of statistical error?

In perturbation theory, the solution to a given quantum physical problem cannot be solved to an exact precission. But the approximate function which yields similar solutions can be improved by adding ...
groaking's user avatar
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In degenerate perturbation theory why can we assume that matrix elements above and below the degenerate subspace disappear?

The picture shows some original Hamiltonian H which has some degeneracies. Suppose I have some perturbation V to the system and I want to find the new energies and eigenstates of the system. Then from ...
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Relation between spontaneous emission and natural linewidth

I have a question regarding the relation between the spontaneous emission and the corresponding natural linewidth of a two-level system. The way that I have seen people deriving the lineshape from the ...
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Solving for unitary operation using perturbation theory

Let the time-dependent Hamiltonian be \begin{equation} H(t) = H_0(t) + \lambda H_1(t), \end{equation} where $\lambda$ is a small parameter. In the interaction picture (i.e. rotating frame w.r.t ...
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Perturbation theory of Anderson impurity model

I’ve been learning about DMFT(Dynamical mean field theories) these days and have encountered rather simple questions in many-body perturbation theory. It is about IPT Impurity solver (applying ...
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Why is Perturbative expansion of gravity in terms of $GE^2$?

From General Relativity by Weinberg p.797 edited by Hawking & Israel: This is to be used to generate a perturbation series in powers of $GE^2$ or $G/r^2$ (where $E$ and $r$ are an energy and a ...
Arevilov 3's user avatar
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Fermi's Golden Rule: Interpreting the Dirac Delta in Transition Probabilities [duplicate]

I am trying to understand an aspect of Fermi's golden rule in the case of a constant perturbation, $V$. The formula for the transition probability from an initial state $i$ to a final state $f$ is ...
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Are "good" states in perturbation theory eigenstates of both the unperturbed and perturbed Hamiltonian?

In my quantum course, my professor asked us the true/false question: "Are 'good' states in degenerate perturbation theory eigenstates of the perturbed Hamiltonian, $H_0 + H'$?" I was ...
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What are the similarities and differences between the Magnus expansion and the Schrieffer-Wolff transformation?

The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates ...
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$Z_1=Z_2$ and its relation to vertex renormalization in QED

I have been working on the full renormalization of scalar QED with self-interactions, following the steps of Schwartz’s treatment on spinor QED (Chap 19). I have 3 main questions regarding this: Need ...
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Understanding the dynamics of a perturbed quantum harmonic oscillator system

I'm trying to understand how quantum systems behave when they are perturbed, and I'm using the quantum harmonic oscillator as a model. I start by implementing a symmetric gaussian shaped bump in the ...
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How to study regularity of a Green's function when solving field equations perturbatively?

Preliminaries Consider a nonlinear differential operator $\mathcal{O}$ acting on a field $\phi$, with source $\rho$ $$\mathcal{O}(\phi)=\rho$$ Let's say the charge density is small, so we can define $\...
P. C. Spaniel's user avatar
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Calculating Eigenkets of Perturbed Matrix for Second-Order Correction

Q: Find the eigenvalues of the 3x3 symmetric matrix $H$ using perturbation theory where all of the elements on the diagonal of $H$ are an order greater than the elements not on the diagonal. We can ...
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Existence of eigenstates in the context of continuous energies in the Lippmann-Schwinger equation

In the book QFT by Schwartz, in section 4.1 "Lippmann-Schwinger equation", he says that: If we write Hamiltonian as $H=H_0+V$ and the energies are continuous, and we have eigenstate of $H_0$...
Gao Minghao's user avatar
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Perturbation theory for the strong interaction in the operator formalism

In the operator formalism, we start with the Hamiltonian whose density is given by \begin{eqnarray} {\cal H}=\pi_A^{\mu}\partial_0A^a_{\mu}+\cdots-{\cal L}_{\rm QCD}. \end{eqnarray} The canonical ...
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How to calculate dissociation rate of molecular hydrogen ion?

Suppose I have a positive molecular hydrogen ion in a bath of electrons with a temperature $T$. How do I calculate the dissociation rate of the molecule due to electron impact? Should I just use Fermi'...
S.T. Zweig's user avatar
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What goes wrong with strongly coupled theories?

Let $\lambda$ be the coupling constant of a quantum field theory. It is said that Perturbation theory is only valid when the theory is weakly coupled ($\lambda \ll 1$). In most cases, the series of ...
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Second-Order Perturbation Correction for Helium atom

I am trying to calculate the second order perturbation correction for helium atom: $$E^{(2)}_n = \sum^\infty_{i, i \neq n} \frac{|\langle \Psi_n | V | \Psi_i \rangle|^2}{E_n - E_i}$$ with the ...
Mariam Ibrahim's user avatar
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1 answer
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In a scalar QFT, how are Feynman diagrams drawn if the interaction Lagrangian has multiple terms?

In my introductory course for QFT we have covered many different interaction Lagrangians using scalar fields, for example $\phi^3$ theory. However, so far we've only covered Lagrangians with a single ...
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Does the functional form of a perturbative Hamiltonian indicate how many nonzero terms are in energy corrections?

Suppose we add a perturbative Hamiltonian to a quantum system. In principle, we can compute high order corrections to the energy levels with perturbation theory. However, I've come across some systems ...
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Is Sakurai's derivation of the Lippmann-Schwinger equation correct?

I am using Sakurai's Modern Quantum Mechanics 3rd ed. The following is from the beginning of chapter 6. The defining equation for the $T$-matrix is $$\langle \vec{k}' \lvert U_I(t, t_0) \lvert \vec{k} ...
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Does Sakurai's definition of $S$-matrix assume a particular type of scattering?

I am using Sakurai's Modern Quantum Mechanics 3ed. In chapter 6, Sakurai defines the $T$-matrix via the equation $$\langle \vec{k}' \lvert U_I(t, t_0) \lvert \vec{k} \rangle = \delta_{k'k} - \frac{i}{\...
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3 votes
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What do you get when you Taylor expand a Magnus expansion?

The Magnus expansion and Dyson series are very similar to each other, in that they both give a way to approximate a time-evolution operator as a series expansion $$U(t) = \mathcal{T}\left(\exp\left[-i\...
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Perturbation of central field potential

i`d like to consider system with Coulomb potential: $U = -\frac{\alpha}{r}$ and constant magnetic field.It is easy to write Lagrangian function: $$ L = \frac{m}{2}(\dot{\rho}^2 + \rho^2\dot{\phi}^2) + ...
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On the choice of observables in linear response theory

For linear response theory I need two observables. The idea is to see how the change in one observable changes the other under weak perturbations in equilibrium state. Suppose I want to see the change ...
Rafi Ullah's user avatar
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1 answer
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Power-series expansion in coupling/Planck constant

By using Feynman rules of the interacting theory, one obtains the scattering amplitude $$\mathcal{M} = \mathcal{M}_0 + \mathcal{M}_1 + \cdots = \sum^{\infty}_{i = 0}\mathcal{M}_i\tag{1}$$ Where $\...
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3 answers
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How do vacuum bubbles "dress" terms in the $S$-matrix numerator?

I am self-studying QFT using the book "A modern introduction to quantum field theory" by Maggiore. On page 124-125 he's doing the calculation in the interaction picture for a process with ...
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Perturbative analysis of pendulum via virial theorem

I am following Sivardiere, Jean. "Using the virial theorem." American Journal of Physics 54.12 (1986): 1100-1103. The pendulum with length $L$ has equation of motion $\ddot{x}=-\omega_0^2 \...
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Degeneracy in time-independent perturbation theory

I have a problem understanding the reasoning of my professor in Quantum Mechanics. The topic is time-independent perturbation theory. Let $H = H_0 + \lambda H_1$ be a perturbed Hamilton-Operator and ...
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Possible decays of a scalar particle, Feynman diagrams and calculation of branching ratio - Explenation of solution required

Consider the Lagrangian describing two Dirac fields of masses $m_1$ and $m_2$ and a scalar neutral field of mass $M$, which in addition to the free terms contains the following interaction term \begin{...
Camthalian's user avatar
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How can I count diagrams (in this scalar QED example) at a particular order without drawing all the Feynman diagrams?

Here's a 3-loop diagram for light-by-light scattering in scalar QED (from Schwartz textbook question 9.2): The question 9.2 asks approximately how many other diagrams contribute at the same order in ...
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Solutions for nonrelativistic-matter perturbations

I'm studying the nonrelativistic-matter perturbations if the expansion of the Universe is driven by a combination of components. I'm currently Following this document (The growth of density ...
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Derivation of ODE for $c_n(t)$ in Time-dependent Perturbation Theory [closed]

I'm going through the notes on Time-dependent perturbation theory from MIT OCW and want to make sure I understand how we're going from the first equality to the second in the picture attached below: ...
Keshav Balwant Deoskar's user avatar
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Is there an error bound for perturbation theory?

It is usually said that perturbation theory is valid when the perturbation is much smaller than the spacing between energy levels. However, I was thinking whether there exists an rigorous error bound. ...
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Scale dependent density growth in sub-horizon scales

In standard cosmology, we use \begin{equation} \ddot{\delta} + 2 H \dot{\delta} - 4 \pi G_N \rho \delta=0 \tag{1} \end{equation} to study the structure growth in sub-horizon scales. However, at the ...
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Transformation under coordinate transformation of scalar perurbation of FLRW metric

For the past few days I've been studying perturbation in cosmology. More specifically I am now busy with chapter 6 in Dodelson's Modern cosmology. In this book the perturbed FLRW metric which only ...
luki luk's user avatar
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Ratio of scale factors during matter-radiation equality

For the past few days I've been studying cosmology using Dodelson's book. I am now busy with chapter 8 and was making some exercises. However I got stuck with exercise 8.6 (2nd edition) which goes as ...
luki luk's user avatar
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Expression for $k_\mathrm{eq}$, the wavenumber entering the horizon at matter-radiation equality

For the past few days I have been busy with studying cosmology more specifically chapter 8 in the book by dodelson on modern cosmology. I have the following question concerning equation (8.70) which ...
luki luk's user avatar

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