Questions tagged [perturbation-theory]

Perturbation theory refers to methods for understanding physical systems by treating them as small modifications to exactly solvable systems.

Filter by
Sorted by
Tagged with
1
vote
0answers
8 views

Schrieffer-Wolff Transformation for conventional superconductors

I was trying to follow the discussion in Radi A. Jishi's book (Feynman Diagrams in Condensed Matter Physics), Chapter 12 on superconductors. They basically have a Hamiltonian that comprises of a ...
0
votes
1answer
76 views

Rigorous proof of Bertrand's Theorem for orbits under central force

I have read through several proofs of Bertrand's Theorem, including the one on Wikipedia. A typical proof can be found here (Santa Cruz Institute for Particle Physics). Almost all proofs using ...
0
votes
0answers
11 views

Diffeomorphism for Equation of motion in Brans Dicke theory

I am trying to see the gauge invariance, (diffeomorphism) for equations of motion in Brans-Dicke theory. Frist the equation of motion is given as follows, [I just copied from Wikipedia] \begin{...
1
vote
1answer
364 views

What is the difference between Lamb shift and the Darwin term?

I am currently learning for an exam and I wonder what the difference between the Lamb shift and the Darwin term is. What I know is that the Darwin term affects only the s-states of the hydrogen atom ...
0
votes
0answers
21 views

What does it mean to study transition probability?

I'm starting to study time-dependent perturbation theory. The book applies time-dependent perturbation on the hamiltonian and it says that for this reason quantum transition are allowed. Then it ...
0
votes
0answers
34 views

Borel Resummation on a finite asymptotic series

I am reading http://users.physik.fu-berlin.de/~kleinert/kleiner_reb8/psfiles/16.pdf I am a little confused on the use of the Borel Transform in perturbative series. I know that Green's Functions in ...
0
votes
0answers
11 views

couplings between mesons and baryons (singlet and octet)

What is the basic meaning and good reference for the deep and simple meaning(in simple language) of couplings between mesons and baryons, e.g., g1 and g8 couplings (singlet and octet) from where these ...
2
votes
1answer
160 views

What are “the background equations” in cosmology?

We're currently working on perturbations within cosmology. There is something I have not heard before which has cropped up, that is: a reference to the term "the background equations". Are these just ...
-2
votes
1answer
760 views

What is Wick's theorem and what this is use for? [closed]

I am reading Wick's theorem but although I look for it to clearly understand in some textbooks and youtube videos but still it is unclear to me. I cannot get my head over what is normal ordering ...
2
votes
1answer
55 views

Perturbation Theory - Exact Solutions and Good Quantum States

I'm having a problem with the following question: Problem: Consider the unperturbed, degenerate Hamiltonian $H_0=\bigg(\begin{matrix} E &0\\ 0& E\end{matrix}\bigg)$. Add the perturbation $...
4
votes
2answers
305 views

Struggling to understand degenerate perturbation theory

As far as I gather, before a perturbation is applied, the eigenspace associated with the degenerate energy is multidimensional but after applying the perturbation this space 'splits' into different ...
1
vote
2answers
79 views

How to ascertain that the Rayleigh-Ritz variational method gives the exact value of the ground state energy?

So the Rayleigh-Ritz variational method can be used to calculate the ground state energy of a quantum system. If $\phi(x)$ is a suitable (square integrable) and normalised function of the coordinates ...
0
votes
0answers
18 views

First Order vs. Second Perturbative Transition Probabilities with Unruh deWitt Detectors

I have been working with perturbative expressions for the transition probability/rate of an Unruh deWitt detector. Many papers such as this one - https://arxiv.org/pdf/gr-qc/0606067.pdf - seem ...
1
vote
1answer
177 views

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 ...
1
vote
2answers
226 views

Derivative with respect to perturbation in QM

The book "Introduction to Computational Chemistry" by Frank Jensen claims the following (Eq. 10.35): Let $H(\lambda) = H_0 + \lambda V$ be a Hamiltonian parametrized by a perturbation strength of $\...
0
votes
1answer
26 views

What's the most general approach to Zeeman effect?

I have a question regarding the Zeeman effect and perturbation theory in the hydrogen atom. We have hamiltonian of the hydrogen atom is given by $H_0$, that of spin-orbit coupling given by $H_{\text{...
0
votes
0answers
56 views

Problem in odd-even decomposition of a generic metric

The metric of the unit two-sphere is given by $ \Omega_{\mu \nu} = \begin{equation} \begin{pmatrix} 1 & 0 \\ 0 & \sin^2 \theta \end{pmatrix}. \end{...
0
votes
1answer
1k views

Inversion of a metric

I am currently reading a paper by Bredberg $et.al$ arXiv:1101.2451 titled "From Navier-Stokes to Einstein". In this paper, the authors have considered a metric of the form \begin{eqnarray}ds^2_{p+2} = ...
2
votes
1answer
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 ...
0
votes
1answer
76 views

Perturbed Ricci scalar in Modified Gravity

When getting the perturbed Ricci scalar in a Modified Gravity theory of the form $\mathcal{L}_{gr}=F\left(\phi,R\right)R$, $\phi$ being a scalar field, it is easy to arrive at an expression of it in ...
1
vote
2answers
453 views

Perturbation theory with infinite potential

I'm trying to solve an excercise that involves first order perturbation theory and an infinite potential. To ease the problem, I tried to consider an easier one dimensional model. Consider an infinite ...
0
votes
1answer
20 views

Relation Asymptotic Series and perturbative effects

Perturbative expansions of a function $f(x)$ around say $x=0$ cannot determine contributions from a function such as $e^{-1/x}$ since its Taylor series vanishes to all orders. This kind of ...
4
votes
3answers
795 views

Relativistic correction to Hydrogen atom - Perturbation theory

Given the relativistic correction $$ H_1' = - \frac{p^4}{8m^3 c^2} $$ to the Hamiltonian (i.e. a perturbation), what does it mean when $[H_1', \mathbf{L}] = 0$? The book I'm reading says this implies ...
0
votes
0answers
31 views

Cosmology - Demonstration for equation of the evolution of the density contrast

In a context of cosmology, I need help about a differential equation that I can't get to demonstrate: The growth of density fluctuations obeys a second order differential equation. At early enough ...
0
votes
0answers
24 views

Mechanism behind spin orbit interaction

The spin orbit coupling can be explained through two different frames one is the electron frame and the other, the lab frame. In the electron frame magnetic field produced by proton current current ...
0
votes
1answer
45 views

What happened to the factor of $\pi$ in this question?

$\\ $ I was going through the answer to this problem, when I noticed that a factor of $\pi$ in the denominator disappeared and a factor of 4 appeared in the numerator when the author started ...
2
votes
1answer
134 views

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 ...
4
votes
1answer
276 views

Why do we need the coupling small when doing perturbative QFT calculation?

I don't really understand why, when we calculate say the 2-point Greens function in a scalar QFT with interaction $\lambda \phi^4$, we need the coupling constant $\lambda$ to be small? Everywhere I ...
1
vote
0answers
30 views

How to express the wavefunction of a harmonic oscillator in a perturbing electric field?

So I am looking at the problem of the (charged) harmonic oscillator in a weak electric field - the problem that defines e.g. the polarizibility of the oscillator. Let the fieldless Hamiltonian be: $...
3
votes
1answer
312 views

Renormalization group and summation of diagrams

Currently I'm studying renormalization group, and I'm having trouble understanding the following statement which I see almost everywhere in books on QFT: renormalization group sums a series of ...
3
votes
1answer
203 views

Difference between positron and electron scattering in Coulomb field

In first order of perturbation theory the S-matrix amplitude for electron scattering in the Coulomb field will be (up to normalization factors) $$ S_{fi} = \frac{iZ q^2}{\sqrt{2E_{f}2E_{i}}}\bar {u}(...
2
votes
1answer
299 views

Can this kind of TDSE be solved by series?

Note: some notation was changed according to the comments, after the first answer was posted. I have a particular kind of time dependent Schrodinger equation: $$i \hbar \frac{\partial}{\partial t} \...
0
votes
1answer
424 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 ...
10
votes
3answers
620 views

Use my example to explain why loop diagram will not occur in classical equation of motion?

We always say that tree levels are classical but loop diagrams are quantum. Let's talk about a concrete example: $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is ...
3
votes
1answer
633 views

Calculation of electron vertex correction in Peskin and Schroeder

I am trying two days now to simplify the numerator of the electron vertex correction in the one-loop contribution. My problem is to prove that $$\bar{u}(p')\left[-\frac{1}{2}\gamma^\mu l^2+(-y \gamma ^...
0
votes
1answer
33 views

Gaussian integral extended to multi-dimensions

In Quantum Field Theory in a Nutshell by A. Zee, the following integral $$Z(J)=\int_{-\infty}^{+\infty} d q e^{-\frac{1}{2} m^{2} q^{2}-\frac{\lambda}{4!} q^{4}+J q}$$ is solved perturbatively by ...
0
votes
0answers
28 views

Precise conditions for Apply Degenerate vs. Non-Degenerate Perturbation Theory?

Is there a precise way to phrase when you are allowed to apply non-degenerate perturbation theory versus degenerate perturbation theory? When you have a Hamiltonian of the form $$ H = H_0 + \lambda V(...
0
votes
2answers
89 views

Pertubation of Riemann tensor in a general curved space-time

It is a direct and simple question. I am fully developing the perturbation of Einstein Field Equations, and I need to calculate the perturbation of the Riemann tensor. However the background metric is ...
1
vote
1answer
34 views

Teukolsky (Bardeen-Press) equation ingoing coordinates

EDIT: I can't seem to delete this question, so I've posted the solution below (I must have made an algebra error-someone checking all this would still be appreciated!). I've left the question as is. ...
1
vote
1answer
42 views

Linear perturbations of the energy conservation in FLRW spacetime

Recently i have some troubles regarding linear stability analysis in GR, especially matter conservation equation. First order perturbations of the Hubble parameter and energy density are: $$H=H_b(1+\...
0
votes
0answers
29 views

Derivation of Perturbation Terms in Thermodynamic Perturbation Theory

In the "A critical evaluation of perturbation theories by Monte Carlo simulation of the first four perturbation terms in a Helmholtz energy expansion for the Lennard-Jones fluid" paper by T. van ...
0
votes
1answer
204 views

First-Order Perturbation of Energy Eigenfunction

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector ...
1
vote
1answer
35 views

Time-dependent pertubation theory assigning the order of expansion to squares of the solution

What is the square of a solution from time dependent pertubation theory? Assume we have found the corrections up to second order such that $$ |\psi(t)\rangle \approx |\psi^0(t)\rangle + |\psi^1(t)\...
3
votes
2answers
267 views

Pauli- Villars regularization in the Electron Vertex Function: Evaluation

I'm studying one loop contribution for electron vertex function form Peskin and Schroeder's book " An introduction to quantum field theory " Section: 6.3. I have some troubles with Pauli- Villars ...
0
votes
1answer
85 views

Linearity of Schrödinger equation and perturbation theory

So, I was studying quantum mechanics and reached the point where perturbation theory is discussed. It is my first time in this topic, and something called my attention: it was said that we need ...
1
vote
1answer
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 $H_0$ ...
2
votes
1answer
489 views

Perturbation Method in Mechanics: Average velocity of a small mass on a vibrating inclined plane [closed]

I've stumbled across this delightful and difficult collection of problems, by Jaan Kalda. The following problem has stumped me. (It's probem 16 on the sheet, which I have provided as a link) http://...
1
vote
2answers
467 views

Second order perturbation theory

The result for the non-degenerate second order perturbation theory is $$ E_n'= \sum_{m\neq n}\frac{\left|\langle m | H' | n\rangle\right|^2}{E_n-E_m} $$ But does this mean that is doesn't matter if ...
0
votes
1answer
61 views

Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator

After a perturbation $V(x)$ added to the system, a matrix element $H_{nn}$ calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as: $$\epsilon \hbar \omega_0\begin{...
0
votes
0answers
29 views

New locution regarding perturbation theory

I am trying to make a sentence more approachable to a general audience by not using technical language. I fear I'm however losing precision in this new language. Original sentence: The use of ...