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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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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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135 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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29 views

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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63 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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36 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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103 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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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
63 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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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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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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98 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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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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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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What maximizes angular momentum matrix elements?

Given a quantum particle in $\Bbb R^3$ with eigenstates $$\begin{align} \hat{h}\psi=& \;e \psi \\ \hat{h}\psi'=&\;e'\psi'\\ \dots\end{align}$$ I would be interested on the implications of $$...
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What is a good book for an introduction to perturbation theory in quantum mechanics? [duplicate]

So I have been learning Quantum Mechanics from the great book "Introduction to Quantum Mechanics by Griffiths", I just stumbled on the chapter of "Perturbation Theory" (Chapter 9). I find the ...
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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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1answer
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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 ...
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Solving a 2x2 Perturbed Hamiltonian Exactly

Problem Consider Hamiltonian $H = H_0 + \lambda H'$ with $$ H_0 = \Bigg(\begin{matrix}E_+ & 0 \\ 0 & E_-\end{matrix}\Bigg) $$ $$ H' = \vec{n}\cdot\vec{\sigma} $$ for 3D Cartesian vector $\...
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Using relativistic QFT to compute energy levels

I've taken a year of QFT so far, and although there seems to be a lot of attention paid to scattering amplitudes and decay rates and perhaps bound states, I view computing energy spectra as certainly ...
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two point function between momentum eigenstates $\langle p\rvert T\{\phi(0)\phi(x)\}\rvert p\rangle$

I am reading a skript about the Operator product expansion. There appears the following expansion: $\langle p\rvert T\{\phi(0)\phi(x)\}\rvert p\rangle \sim 1+a \lambda^2ln(x^2p^2)+..., ~~~ a \in \...
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Can we show that the ground state of the He atom is a spin singlet rather than triplet?

The ground state of Helium atom is a state in which the space part of the wavefunction is symmetric and the spin part is antisymmetric under the exchange of the electrons. Therefore, the ground state ...
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1answer
78 views

Path integral formulation of the density matrix ρ

In Feynman's Statistical Mechanics - A Set of Lectures, upon the introduction of the path integral, a series of approximations are made in order to calculate integrals. I am unsure how exactly to get ...
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1answer
91 views

Gravitational attraction between quantum particles [closed]

Let's say we have a quantum particle with mass $m$ in a 1-Dimensional box. The potential outside the box is infinite. Say that $n=2$, so that $|\psi|^2$ will have two maxima. How would the ...
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How do we calculate the first order energy correction with total spin?

We're trying to calculate matrix elements for a perturbation theory problem. One element looks like (I've left off the $B_z$ field and some $ \hbar $ factors): $W_{ab} = <1 1 | (1 - (S_2)^2 + (S_1)...
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130 views

Feynman propagator for Dirac fields and $i\epsilon$ prescription for analytic continuation

The analytic continuation from Euclidean space to Minkowski spacetime is perturbatively well (uniquely) defined to all orders for the Feynman propagator for Dirac fields with the so called "$i\epsilon$...
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When is an anomaly one-loop exact?

There are many examples of quantum anomalies that are one-loop exact, and many examples of anomalies that have contributions to all orders in perturbation theory. I haven't been able to identify a ...
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Griffiths Intro to QM Section 9.1.3: How did he get this answer and am I misunderstanding something?

In Section 9.1.3 Griffiths develops time-dependent perturbation theory, but I don't understand how some extra terms are popping into his equations. I searched around for some answers and found this ...
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Gravitational Lenses in External Shear Fields

I am reading Massimo Meneghetti's notes on gravitational lenses, available here: http://www.ita.uni-heidelberg.de/~massimo/sub/Lectures/gl_all.pdf On page 38 he begins discussing embedding a lens in ...
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Asymptotic behavior of canonical perturbation theory for the classic anharmonic oscillator

What do we know about the asymptotic behavior of the perturbative expansion for the classical anharmonic oscillator? The Hamiltonian is $$ H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4 $$ ...
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Time-independent perturbation theory: why i'th order perturbations are orthogonal to base state?

I have been learning about time independent perturbation theory (non-degenerate for the moment), and am not satisfied about a particular point: the justification for setting $\langle n^i|n^0\rangle = ...
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65 views

Questions on Stark Effect on Hydrogen

Suppose that a hydrogen atom is subject to a weak uniform electric field $\vec{E}=\epsilon \hat{z}$. Let's neglect the effect of electron spin. The perturbation added to the original hamiltonian $H_0$ ...
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Perturbation theory: justifying expansion in terms of eigenstates of the basis Hamiltonian

I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this ...
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$U(1)$ Scalar Field Theory: Why no $| \phi |$ term?

When we write down the lagrangian of a general $U(1)$ scalar field theory we generally write $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi^* - \frac{m^2}{2}\phi \phi^* - V(|\phi|^2)$$...
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Feynman diagrams included in Hartree-Fock approximation

Given a hamiltonian, I compute the Hartree-Fock self-energy. Let's say I now compute the second order self-energy with diagrams. Some of them are just like the Hartree or Fock diagrams of first order, ...
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1answer
130 views

Diagrams involved in 1-loop electron self-energy in QED

I'm following the derivation of electron self-energy at 1-loop in QED on Peskin-Schroeder, page 216. To second order in the coupling the considered diagram (7.15) is The 2-point correlator at second ...
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1answer
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Feynman, Hibbs Perturbations and Energy

I am currently self-studying from Feynman & Hibbs’ Quantum Mecahnics and Path Integrals, but having an issue understanding a step in their development of first-order perturbations. They define $$...
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Break down of time independent perturbation formula of quantum mechanics in quantum field theory

The following paragraph is from Schwartz Sec 4.2.1 Using OFPT we would calculate the energy shift using $$\Delta E_n = \langle\psi_n\rvert H_{int} \rvert \psi_n\rangle +\sum_{m,m \ne n} \frac{...
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Drive frequency for second order quantum transitions

Summary $ \newcommand{\ket}[1]{\left \lvert #1 \right \rangle} \newcommand{\bra}[1]{\left \langle #1 \right \rvert} \newcommand{\braket}[2]{\left \langle #1 | #2 \right \rangle} \newcommand{\bbraket}[...
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Procedure for Effective Hamiltonian using Perturbation Theory? (Bilayer Graphene model)

Sorry if this is a dumb question as I'm just starting out, but in this paper https://arxiv.org/pdf/1803.08057.pdf on Twisted Bilayer Graphene, the authors claim to use "standard perturbation theory" ...
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Perturbation theory in Griffiths

In Griffith's (page 222), the perturbed Hamiltonian has been written as $H + \lambda H'$ Where $\lambda$ is apparently a small number that they will later crank up, and $H'$ is the extra portion ...
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Perturbative Techniques In Finding Electric Field of Symmetric Distributions

Lets say we have a uniform sphere of charges at the origin (at retarded time = 0) with some velocity and we are interested in the field at a point along the x-axis (normal to the surface of the sphere)...
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What do the dashed lines represent in this figure from the discussion of the Zeeman effect in Griffiths?

Consider the figure below (figure 6.12 from Griffiths, Introduction to Quantum Mechanics, p 249 in the 1995 edition), which shows the Zeeman effect on the $n=2$ eigenvalues of hydrogen. The figure ...
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How to construct a minimal model based $\vec{k} \cdot \vec{p}$ method and symmetry arguments?

Currently, I am repeating the results of this famous paper written by Di Xiao. In this paper, the authors construct a minimal band model based symmetry arguments and $\vec{k}\cdot\vec{p}$ method. The ...
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1answer
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Perturbation Theory of Liquids: Weeks Chandler Anderson Model

To put simply, what is the big deal about the WCA model of describing solutes in liquid theory? I understand that the WCA model splits the potential into a repulsive force component, and an attractive ...
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115 views

How can a Dirac delta function that does not occur under an integral be used to describe a transition rate?

In his excellent notes (found here), Mark Tuckerman shows that the transition rate of absorption between quantum states i and f, coupled by operator B, can be expressed as the fourier transform of the ...
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Perturbative diagonalization of matrix

I want to perturbatively diagonalize the matrix $ \begin{pmatrix} \mu & 0 & -M_z A & M_z B \\ 0 & M_2 & M_z C & -M_z D \\ -M_z A & M_z C & 0 & -\mu \\ M_z B & -...