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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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Delta function bump in the centre of infinite square well [on hold]

I am learning non degenerate perturbation theory and I came across this question: Suppose we put a delta-function bump in the centre of the infinite square well: $$H'= \alpha \delta \left(x- \...
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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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Question on problem in “Introduction to Quantum Mechanics” by Griffiths [on hold]

In "Introduction to Quantum Mechanics" by Griffiths, on Problem 6.22 it states that the expectation of the spin-orbit term can be reduced to $\hbar^2 m_l m_s$, which is different from the previous ...
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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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67 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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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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1answer
84 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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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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2answers
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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 ...
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1answer
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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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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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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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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
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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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Gauge-invariance of curvature perturbation in uniform density gauge

The perturbed energy conservation equation is given by $$\delta\rho'+3\mathcal{H}(\delta\rho+\delta P)-3(\bar{\rho}+\bar{P})\psi'+(\bar{\rho}+\bar{P})\nabla^2(V+\sigma)=0.$$ If we substitute in the ...
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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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Why can excited determinants be used as corrections in the MP wavefunction?

So many-body perturbation theory is one way of approximately solving the Schrödinger equation. In Moller-Plesset perturbation theory, the Hamiltonian is expressed as a known part, which is a sum of ...
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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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How to solve momentum matrix element (MME)?

I am trying to find the effective mass of manganese sulfide $\left(\text{MnS}\right)$ using the k$\cdot$p method, and now stuck at solving its momentum matrix element. Do I need to solve the MME ...
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1answer
113 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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52 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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Is QFT by nature perturbative (when more than one vacuum exists)?

It is said that because QFT calculations are done around one vacuum - for example, we extract creation/annihilation operators from expansion around particular vacuum state, these calculations break ...
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General rules for calculating symmetry factor associated with a Feynman diagram

Are there a set of general rules to find out the symmetry factor associated with a given Feynman diagram? The field theories I have in mind are (i) real interacting scalar field theory with $\lambda\...
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1answer
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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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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
32 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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33 views

Fermi's golden rule - possible final states

While reading about Fermi's Golden Rule on Wikipedia, I found that claim: "If $H'$ is time-independent, the system goes only into those states in the continuum that have the same energy as the ...
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1answer
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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}) ...
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The physical meaning of radius of convergence [closed]

If y(x) is a power series convergent around $x_0$ with the radius $|x-x_0|<R$, what is the physical meaning? Can I understand that the item $|x-x_0|$ can be one of the dominating factors for the ...