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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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How to pick a boundary layer coordinate or stretching transformation

I am following Introduction to Perturbation Methods by Holmes and am unsure how I to pick the power in my boundary layer coordinate if my governing equation is the Laplace equation given by \begin{...
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Reference for Feynman diagram technique(position space) in Thermal Field Theory

I am trying to study perturbative expansion of Sachdev-Ye-Kitaev model, where I know that the dominant terms are the Melonic diagrams. I am interested in seeing how perturbative corrections affect the ...
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What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. ...
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Free energy: comparison between exact solution and perturbation theory

I'm studying a system in condensed matter for which there is an exact solution for the free energy $F(\lambda)$, where $\lambda$ is the parameter of the interaction. I can expand $F(\lambda)$ at some ...
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Perturbation theory with a continuous degenerate spectrum

Let's assume that the unperturbated system $H_0$ is a free particle . It has the following energy spectrum $$ E = \frac{p^2}{2m} $$ and the set $\{ \vert k \rangle \} $ forms a complete basis for $H_0$...
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Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields $$ \langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\...
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What if $\omega =0$, which is the frequency of the perturbation term?

In analytic mechanics, when we found a equilibrium position of the system, to determine the stability of that configuration, we apply $q \to q_0 + \epsilon \eta$ with $|\eta| \ll 1$ s.t $q_0$ is the ...
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Is there such a thing as complex conjugate of the first order correction of a wave function?

If I am given the first-order correction to a wave function as $$\left|\psi_n^{(1)}\right\rangle = \sum_{m \neq n} \frac{ \left\langle\psi_m^0\right|H'\left|\psi_n^0\right\rangle }{E_n^0 - E_m^0}\...
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The form of the wave function in time-dependent perturbation theory

I'm trying to understand why the wave function is expressed as follows in a time dependent perturbation. I understand that since the $c(t)$ are unspecified functions, it is mathematically reasonable ...
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Perturbation theory development for the ground state of the QM particle in the box with a centered dirac-delta spike

In the course of a discussion in the chat there emerged an interesting problem, namely a particle in an infinite well with an additional Dirac-delta function spike of scalable hight: $$ H = -\frac{\...
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A question about Dyson's argument of convergence radius of QED

Related question Doubt in Dyson's argument about the divergent nature of the perturbative expansion in QED My question is, for the part $e^2>0$, I may imagine virtual pair productions. Since ...
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Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]

I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form $$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$ where the interest is in the ...
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How do one find the first correction order of $\lambda$ for scattering cross section?

I have a lagrangian $\mathcal{L}=\mathcal{L_0+\mathcal{L_1}}$, where $\mathcal{L_1}$ is a perturbation given by:$$\mathcal{L_1}=-(1/3!) \lambda \phi^3-(1/4!) \lambda \phi^4 $$ and$$\mathcal{L_0}=-\...
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Rigorous justification for non-relativistic QM perturbation theory assumptions?

In perturbation theory for non-relativistic quantum mechanics, you begin with a Hamiltonian of the form $$H=H_0+\lambda H'$$ and assume that the perturbed eigenstates and eigenvalues can be written as ...
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Perturbation to the flat space metric

from the geodesic equation for non-relativistic case where $$v_i\ll c$$ $$\frac{dx^i}{dt}\ll1,{\rm for }\ c =1$$ $$\frac{dx^i}{d\tau}\ll\frac{dt}{d\tau}$$using this the geodesic equation for proper ...
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Dirac solution with coulomb-field (perturbation theory)

The dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$ The solution up to first order is $...
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Why is perturbation theory used in quantum mechanics?

I don't seem to understand what perturbation theory really is, and what it is needed for. Can someone please provide an explanation for what it is, and why it is needed in QM?
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Gauge invariance in GR perturbation theory

I have been following this video lecture on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the ...
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Optical theorem in $\phi^4$: which poles contribute to discontinuity in Feynman amplitude?

Section 7.3 ("The Optical Theorem") in Peskin and Schroeder's QFT text contains a leading order verification of the optical theorem in $\phi^4$ theory by calculating the (discontinuity across the ...
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264 views

Why don't very high order Feynman diagrams contribute significantly?

In a particle physics lecture I had today it was stated that the magnetic moment, $g$, is not quite equal to 2, and the difference is accounted for by QED. Later it was stated that we can see this ...
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Solution of the coupled non-linear oscillators by using perturbation theory [closed]

The integration shown here, $$∫_{-\infty}^{+∞}x^r\mathrm{Exp}[−x^2]\mathrm{H_n}^2[x]\mathrm{d}x,$$ appears when we try to calculate the spectrum of the perturbed non-linear oscillators by using ...
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1answer
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Hydrodynamic interaction between two spheres in $Re\ll 1$ flow

I am studying the interaction between two spherical particles of radius $a$ in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces ...
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1answer
48 views

First Order Approximation of the Navier-Stokes Equation: Order of Magnitude of the Gradients of First-Order Fields

I am currently working on a project in acoustics and I am studying first and second-order approximations to the Navier-Stokes equation. I have been reading the book 'Theoretical Microfluidics' by ...
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Rotating wave approximation : Why do we need both weak coupling and one of the frequency higher with respect to the other one

I have read Rigorous justification for rotating wave approximation to have an idea of a rigorous proof of RWA approximation. The main idea I have from this is that you can see it if you write a ...
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The fine structure constant and the strength of interaction between two particles

In my notes, the following is mentioned: We consider the scattering of a beam particle with energy $E$, momentum $p$ and charge $ze$ off a charge distribution $\rho (x)$ of total charge $Ze$. We ...
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1answer
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Question about the perturbation solution involved Rashba spin-orbit coupling?

Currently, I am reading the original paper about spin field effect transistor proposed by Supriyo Datta and Biswajit Das. In the last part of this paper, to obtain a larger overall current modulation ...
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1answer
53 views

Consistency of time-dependent and time-independent perturbation theory

I am confused as to how time-independent and time-dependent perturbation theories in QM give consistent results at the instant the perturbation is switched on. Suppose I have a two-level system which ...
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1answer
32 views

Perturbation of Diracs equation (first order)

I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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37 views

Constant Density of States and Perturbation theory

Given a constant Density of States (DOS) corresponding to a one-electron hamiltonian, $\text{DOS}(\omega)=\dfrac{1}{2D} \chi_{[-D,D ]}(\omega),$ where $\chi$ is the indicator function, I want to know ...
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164 views

Non-dimensionalization and perturbative expansion

I need to expand an equation, of the form $$\dot{r} = \gamma(a,\mu) F_1 + g(\mu,\ell,h,R) F_2$$ in powers of $\epsilon = a/\ell$. So I thoughts I would non-dimensionalize it first. I know that $$\...
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Contribution from $u$-channel and $t$-channel processes in OPE analysis for deep inelastic scattering

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.633 the moment sum rules for the deep inelastic form factors are discussed $$\int_0^1 dx x^{n-1}f_f^+(x,...
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Spring with oscillating support (Goldstein chapter 11, problem 2)

The problem: A point mass m hangs at one end of a vertically hung hooke-like spring of force constant k. The other end of the spring is oscillated up and down according to $z=a\cos(w_1t)$. By ...
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186 views

Why can we add counterterms?

I'm having a hard time understanding why renormalized perturbation theory works. Why is it permissible to add counterterms to the Lagrangian? Terms which are often divergent themselves and carry ...
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Time-dependent perturbation theory in a degenerate system

In the derivation of probability transition of time-dependent perturbation theory (see for example these notes, from Ben Simons from Cambridge University), I have only encountered treatments of non-...
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1answer
64 views

Second Order Correction to the Perturbative Approximation of the Transition Amplitudes in RQM

I am studying Relativistic Quantum Mechanics from my professor's notes. When calculating the second order perturbative correction to the transition coefficient $T_{fi}$* in a scattering process by a ...
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Proving the collapse of a many body system (Fetter and Walecka problem 1.2)

I was trying to solve the problem 1.2 from Quantum theory of many-body systems by A. Fetter and J. D. Walecka. I succeeded in the first part, obtaining the suggested formulation for the expectation ...
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1answer
52 views

The stark effect on ground state [closed]

When considering the Stark Effect, we consider the effect of an external uniform weak electric field which is directed along the positive $z$-axis, $\vec E = E \hat z$, on the ground state of a ...
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1answer
169 views

Why first-order Born Approximation doesn't satisfy optical theorem?

First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential: $$ f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
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2answers
123 views

Derive Effective Hamiltonian Directly Using Perturbation Theory?

I am struggling with the concept of deriving an effective Hamiltonian using perturbation theory. Say we have $$ \hat{H} = \hat{H}_0 + \hat{V} $$ and suppose we know the energies $E_n^{(0)}$ and ...
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1answer
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How to choose a boundary layer coordinate or stretching transformation in matched asymptotic expansion

In matched asymptotic expansions how should one properly chose a boundary layer coordinate or stretching transformation. At the moment I am following example 2.3 from Introduction to Perturbation ...
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1answer
85 views

Need for normalization in non-degenerate perturbation theory

I'm currently taking a class in QM and we came across the topic of non-degenerate perturbation theory. Let us for further discussions assume that $H_0$ is the unperturbed Hamiltonian with solutions of ...
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1answer
67 views

Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?

Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
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1answer
76 views

Does the vanishing of the one-loop beta-function imply no running to all orders?

This question sounds ridiculous, but bear with me. I am having a hard time reconciling the following two facts: Classical global symmetries can become anomalous upon quantization, and the anomalous ...
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Problems on perturbation theory with spin

Problems on perturbation theory with spin (solve all perturbations up to only its first-order approximation): Taking into account the relativistic correction of electron kinetic energy and spin-orbit ...
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1answer
106 views

Why is the Fermi Golden rule called so?

I was studying time dependent perturbation theory and this rule came under the case of constant (weak) perturbations. I understood the rule and the derivation but what I cannot understand is that is ...
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Diffraction of electron from single slit using time dependent perturbation theory

While reading about time dependent perturbation theory, I began wondering if it was possible to obtain the diffraction pattern by treating the walls as an infinite potential barrier and the slit as a ...
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3answers
62 views

What does it mean for a perturbation expansion to break down?

For an anharmonic oscillator with Hamiltonian $H = {\hat p_x^2\over 2m}+{1\over 2}m\omega^2\hat x^2+b\hat x^4$ I found the first order shift in energy is: $$E_n^{(1)}={3 \hbar^2 b\over 4m^2\omega^2}\...
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2answers
57 views

In perturbation theory, why can perturbed eigenfunctions be expanded into the basis set of the unperturbed eigenfunctions?

So I am studying non-degenerate time independent perturbation theory and I came across the derivation of the first order correction to the wavefunction. So the notes given to me affirm that the each "...
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Why don't the corrections to energy disappear in perturbation theory?

The corrections to the wavefunctions and energies depend on $<\psi_m^0\,| \,H'|\psi_n^0>$ to some order. I would've thought that $<\psi_m^0\,| \,H'|\psi_n^0> \, =\, <H' \psi_m^0\,|\...
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133 views

Linearizing the Einstein-Hilbert action; shortcuts?

I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the ...