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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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177 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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255 views

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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81 views

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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29 views

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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54 views

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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80 views

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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182 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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170 views

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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148 views

$U(1)$ 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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162 views

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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415 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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53 views

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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151 views

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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408 views

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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97 views

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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226 views

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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108 views

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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216 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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456 views

Can we get full non-perturbative information of interacting system by computing perturbation to all order?

As we know perturbative expansion of interacting QFT or QM is not a convergent series but an asymptotic series which generally is divergent. So we can't get arbitrary precision of an interacting ...
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114 views

Is there any proof that any result from perturbation theory is necessary an asymptotic series?

I know that almost all the series coming from perturbation theory are divergent, such as those from eigenvalue problems or the S-matrix in quantum field theory. The lore is that the series are ...
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Relationship between power spectra, density perturbations and temperature?

I am trying to understand how density perturbations relate to temperature fluctuations in the CMB. I understand the physical effects involved, but my question is how is the density power spectrum ...
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122 views

Prerequisites for chiral Perturbation theory

I am Ms final year student. Could you pls guide me for the prerequisites of the chiral Perturbation theory? I have studied Quantum Field theory up to large $N$ gauge theory and renormalizations. Now I ...
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96 views

Why time-independent non-degenerate perturbation theory problems are not solved with the secular equation?

The usual way of solving a QM problem with a small perturbation operator $V$ is done in the following way: Of course I assume that the solutions (eigen-functions $\psi^0$ and eigenvalues $E^0$) of the ...
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76 views

Time-dependent perturbation at time $t$

If I use this solution: $|\varphi(t)\rangle=\sum_n b_n(t)e^{-iE_nt/\hbar}|\varphi_k\rangle$ in the time dependent Schrödinger equation, and expand $b$ in $\lambda$, $$b_n(t)=b_n^{(0)}(t)+\lambda b_n^{...
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98 views

Born Oppenheimer approximation and perturbation theory

In the book Molecular Physics by Demtroder there is an explanation of the Born Oppenheimer approximation and the adiabatic approximation in terms of a perturbative series. The Hamiltonian is $H_0 + T_\...
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125 views

Heisenberg's Perturbation Theory

In Heisenberg's The Physical Principles of the Quantum Theory he has a section of Perturbation Theory, where he develops Perturbation theory on the Matrix Theory he's developed in the earlier sections ...
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135 views

Find the eigenkets of degenerate perturbation theory

I guess Im missing something BIG, because it is not explained in any book. When I study the correction of a perturbed degenerated Hamiltonian I must find first the energy correction. This is well ...
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87 views

Perturbation theory with eigenvalue-dependent perturbation

Suppose I have an operator equation of the sort $$ [H_0 + H_1(\lambda)]\mathbf{v} = \lambda\mathbf{v}. $$ Here, $H_0$ defines unperturbed eigensolutions $\{\lambda_0, \mathbf{v}_0\}$, and $H_1(\...
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241 views

Perturbed wavefunctions

I've been studying a number of TISE perturbation problems, where the Hamiltonian is $H = H_{0} + \epsilon H^{\prime}$, the wave function for bound state $n$ is $|n\rangle = \sum_{m=0}^{\infty} \...
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27 views

Linearising with a density fluctuation

I have some trouble understanding orders. Starting with the continuity equation $\partial_t\rho=-\nabla_r .(\rho \vec{u})$ and applying a peturbation to the density $\rho(\vec{r},t)=\bar{\rho}(\vec{r})...
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330 views

Second order perturbation of a degenerate system with no first order correction

Consider the following Hamiltonian, in arbitrary units: $$ H = \begin{bmatrix} 0 & 0 & g\\ 0 & 0 & g\\ g & g & 1 \end{bmatrix}$$ where $g<<1$. It is relatively ...
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471 views

One-loop Correction to Effective Action

This might be a stupid question. In Bailin and Love's "Cosmology in gauge field theory and string theory", the authors are describing how to calculate the effective potential at a finite temperature ...
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196 views

Goldstein on Bertrand's theorem

Bertrand's theorem (Wikipedia) Regarding central force motion, Bertrand's theorem states that only inverse-square and linear force law produce stable closed, if bounded, orbits. Wikipedia's proof of ...
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215 views

Assumptions in basic perturbation theory

I am teaching myself the basics of perturbation theory, mainly from Sakurai's 'Modern Quantum Mechanics', but also looking up lecture notes online. I am puzzled by one thing from the start of the ...
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1answer
57 views

How would a layer of hot air affect the normal frequencies in a pipe?

Imagine that you have a pipe of length $L$ with one open end and one closed end. If the sound speed inside the pipe is $v_s$, then the fundamental frequency is: $$f_1=\frac{v_s}{4L}$$ and the ...
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74 views

Why acoustic glitches in stars translate into extra oscillatory components in the normal frequencies?

Acoustic glitches are locations inside the star where the sound speed changes abruptly compared to the wavelength of the acoustic waves that propagate through. Examples are the ionization zones and ...
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How do you find the 2nd order perturbed energy shift from the quantised dipole hamiltonian?

Without using the Rotating wave approximation how do you find the trapping potential $U(\mathbf r)$ experienced by an atom at position $\mathbf r$ for arbitrary laser frequencies? This can be done ...
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143 views

Dealing with degeneracy in Paschen-Back Effect

Suppose there's a strong external magnetic field applied on a Hydrogen-like atom. The Hamiltonian due to spin-orbit coupling will have much less effect compared to the other Hamiltonians. I would ...
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1answer
75 views

Canonical Perturbation theory of Keplerian orbits

Preamble The motion of a test particle around a point mass $\mu$ is governed by the Hamiltonian $$ (*)\qquad\qquad H(r,p_r,p_\phi) = \frac{p_r^2}{2} + \frac{p_\phi^2}{2r^2} - \frac{\mu}{r} $$ which ...
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248 views

Derivative term in Lagrangian

In QFT we sometimes have to treat Lagrangian which contain interaction derivative terms in the form $$ \left(\partial_\mu \partial ^{\mu}\right)^a \phi^b$$ or also something like $$ \partial_{\mu} \...
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287 views

Eigenkets of degenerate perturbation theory

Suppose the original Hamiltonian is $H$ and we perturb it by a small potential $V$. The basis kets of the original hamiltonian $H$ contains some degeneracy. Since there's some degeneracy, we take ...
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1answer
164 views

Solving linear response in frequency domain

My question stems from a derivation given in chapter 12 of R. McWeeny's Methods of Molecular Quantum Mechanics for solving linear response equations via variational perturbation theory. (Linear ...
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231 views

Some questions about the Brillouin-Wigner form of perturbation theory

I'm attempting to gain an understanding of the Brillouin-Wigner formulation of perturbation theory in quantum mechanics (I intend to write a brief summary of it for a class project encouraging us to ...
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Why can we not continue physics calculations on false vacuum?

I understand that starting with some vacuum state, it may transition to another vacuum if it exists. Howver we can technically represent a vacuum in terms of another vacuum. So why can we not do this, ...
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51 views

A question on perturbative terms involved in the hyperfine structure of hydrogen

In studying the hyper-fine structure of the hydrogen atom at the 2n level in my notes the following is stated ; $$\langle W_{mv} \rangle _{2s}=\langle n=2, l=0|- \frac{\hat{P^4}}{8m_e^3c^2}| n=2, ...
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91 views

Quantum mechanics perturbation and the orthogonality of energy states [closed]

Consider the following question and its solution: My question is concerning the solution of $a_{nm}$. Surely if the energy eigenstates are orthogonal then $a_{nm}$ must be equal to zero. WHy is this ...
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1answer
427 views

Lifting degeneracy in degenerate perturbation

What is the idea behind finding a set of commuting observables to lift the degeneracy in perturbation theory? I just started a course in quantum mechanics and I do not understand how it works. My ...
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1answer
71 views

Result of bra-kets with multiple spins

I'm working on an exercise where I'm calculating the transition probability of a system consisting of two spin-1/2 particles. This system has a Hamiltonian $$ \hat{H} = H_z (\hat{S}_{1x}+\hat{S}_{2x}) ...