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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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1D Infinite well with perturbation using matrix methods [on hold]

I've been really struggling to solve this worksheet and applying Dirac notation. I missed the last few classes and really don't understand how to proceed, any help would be greatly appreciated.
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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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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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69 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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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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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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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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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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Where to read about retarded response functions in the context of quantum mechanics?

I was going through this paper by Walter Kohn https://journals.aps.org/prb/pdf/10.1103/PhysRevB.13.2270 where they derive the Van Der Walls interaction between a particle and a solid surface. On ...
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Lagrangian perturbation theory

I have a system of coupled non-linear differential equations which stem from a Lagrangian and the Euler-Lagrange Equations. I want to solve them with perturbation theory. I know that in Quantum ...
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Electron density of atom in laser field

I would like to consider the atomic or molecular electron density dynamics in some laser field. But the explicit numeric solution of the time-dependent Schrödinger equation seems complicated to ...
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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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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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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 & -...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...