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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Is the Rayleigh–Schrödinger perturbation theory ever useful for a many-body system?

The Rayleigh-Schrodinger perturbation theory is introduced in every textbook on quantum mechanics. It seems that it can yield accurate results for many single-particle systems. Actually, in most ...
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Propagators in Frequency Space and Dyson's Expansion

I am reading the book Topics in Advanced Quantum Mechanics by Holstein. In the book (chapter 1.2) he shows that the time evolution operator $U(t)$ (for a Hamiltonian $H = H_0 + V$, with $V$ time ...
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Normal Ordered Product in Operator Product Expansions

In an example of operator product expansion applied to $\phi^4$ theory of the book QFT an integrated approach, where the Eulerian Lagrangian is $$\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi\right)...
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Integrating by parts the second-order action

before going for the question I should state the problem: $$S = \int d^4x\sqrt{-g} [F(\phi)R + \omega(\phi)X - V(\phi)],$$ where $X = \frac{1}{2}g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi $ is ...
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Do scalar-vector-tensor (SVT) decompositions only apply to maximally symmetric spacetimes only?

I am reading a paper titled "Second-Order Gauge Invariant Cosmological Perturbation Theory" by Kouji Nakamura and I came across the following, (and I rephrase) using the fact that the three-...
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Second-order approximation of density operator

I'm trying to understand how to get the second-order approximation of a density function in the interaction picture given that $$\frac{d \tilde{\rho}}{dt} = -i[\tilde{V}, \tilde{\rho}]$$ $$\tilde{\rho(...
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How to obtain double-well potential's ground energy series expansion in the perturbation theory

In section 1.8 of "Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory"(2015) by Marcos Mariño. The double-well potential is given as $$W(q)=\frac{g}{...
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Problem in the derivation of the Landé $g$-factor

I was watching a lecture on Youtube (https://www.youtube.com/watch?v=NSac7cMQnJw&t=971s) where the professor derives the Landé $g$-factor for the weak-field Zeeman effect. As part of the ...
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How to calculate the second-order pertubation in a Bose gas?

I'm self-learning many body theory and right now I'm trying to solve Problem 1.3 from Quantum Theory of Many-Particle Systems by Fetter and Walecka. Problem: Given a homogeneous system of a spin-zero ...
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How to implement perturbation theory without unperturbed Hamiltonian?

In perturbation theorem, the system Hamiltonian is $H=H_0+H'$, the eigenproblem of $H_0$ is already solved, and $H'$ is a little perturbation compare to $H_0$, then the eigenproblem of $H$ can be ...
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What "really" happens to an evolving state which goes through degeneracy? (and relation to Berry phase)

Say I have a time dependent Hamiltonian $H(t)$ and I let an initial eigenstate $|\psi_n\rangle$ evolve according to the Schrodinger equation, so that at time $t$ the state is $|\psi_n(t)\rangle$ with ...
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Some questions about density perturbations in the early universe

This is really a set of 3 questions in total, if that's fine. They are concerned with the theory of the evolution of small density perturbations in the early universe. Question 1: Consider the ...
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Perturbation Theory for the Hydrogen Atom

The full Hamiltonian of the hydrogen atom (including fine structure) is $$H = H_0 + H_{rel} + H_{s.o},$$ where $H_0 = \frac{p^2}{2m} -\frac {e^2}r$, $H_{rel}$ (resp. $H_{s.o}$) is the relativistic (...
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Why do "good" quantum states remain stationary under perturbation?

I've been reading the degenerate perturbation theory section of Griffiths QM. He introduces the idea that, if we can find an operator $\hat A$ which commutes with $\hat H^0$ and $\hat H'$, then ...
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Perturbation theory amongst supersymmetry transformations of perturbative ground states

Consider a supersymmetric one dimensional sigma-model whose target is a Riemannian manifold $M$. Moreover, assume there is a Morse function $h$ on $M$. In Hori, Kentaro, Cumrun Vafa, Sheldon Katz, ...
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Why is the Interaction Hamiltonian same as classical potential energy? [closed]

In the book Nonlinear Optics by Boyd, the Interaction Hamiltonian when monochromatic wave is incident is given as $$ V=-\mu E(t)= -\mu(Ee^{-iwt}+E^*e^{-iwt}) $$ I know that the classical potential ...
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Why the perturbation Hamiltonian of induced dipole in field still $ d \cdot E$? [closed]

We used to calculate the potential energy of permanent dipole in the uniform field is like $-p \cdot E$ ... and when we put a perturbation Hamiltonian of an external field on the neutral atom, that ...
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The effective action in the linear sigma model

I am reading the section 11.4 of Peskin and Schroeder's book (page 373), and there is a step I could not follow. To calculate the effective action of linear sigma model, the determinant of $\frac{\...
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Perturbation theory, Feynman diagrams and Dyson equation for non-instantaneous two-particles interaction

In non-relativistic regime, the Coulomb interaction $W$ is instantaneous. In second quantization we can write it as (I am neglecting spin) $$W=\sum_{r_1,r_2} W(r_1,r_2)\, \psi^{\dagger}(r_1)\psi^{\...
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Does classical simple harmonic motion violate thermodynamics?

If SHM allows for motion to occur forever, we can consider it perpetual motion, does this imply that the second law of thermodynamics is violated? Or does the presence of an external force act on the ...
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Adding an Impurity as a Perturbation to a Simulation

I am working on some plots corresponding to superconductors (of many kinds). If we insert a vortex, we can view this as an impurity and observe the Quasi-Particle-Interference (QPI). How I go about ...
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Second order degenerate perturbation theory for a 4-level system

I am trying to find the second-order energy correction for a perturbation to a 4-level system $$H = H_0 + \lambda V.$$ I only care about corrections to the lowest energy, which is non-degenerate, ...
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Second-Order Perturbation in electron gas

I was trying to figure out the solution to exercise 1.4 on Quantum theory of many particles systems by Fetter & Walecka and I read through this question and its answer. But a point made in both ...
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Gell-Mann Low formula vs time independent perturbation

Consider a nonperturbed Hamiltonain $H_0$ and an eigenstate $|\Psi\rangle$ satisfying $$H_0|\Psi\rangle=E_0|\Psi\rangle.$$ Now consider the perturbed Hamiltonian $H=H_0+\lambda H_1$ and let $H_\...
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Time-dependent canonical transform step in Hamiltonian perturbation theory (Percival problem 8.20)

In Percival and Richards's great book, "Introduction to Dynamics", problem 8.20 asks the following question. Any insight on how to solve this would be appreciated: Consider a system with ...
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Relevance of perturbation theory at high coupling

Is there any possibility in any field theory model that at coupling >= 1.0, perturbation theory can be used to calculate, say field propagators beyond tree level? What is the room left for ...
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Orthogonality and degeneracy relation when considering time independent Perturbation

Initially we consider the case of no degeneracy and we test the orthogonality of an eigenstate of the Hamiltonian, given in the following way: $|n\rangle_i$ where $n$ is the n-th eigenstate, while $i$ ...
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Choosing diffeomorphisms for the pullback metric in the Weak Field approximation

In the weak field approximation of the EFEs $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we take $g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu}$. The $\eta_{\mu\nu}$ term is just the flat space Minkowski metric and $...
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How to diagonalize the following perturbation?

Consider an infinite cubic well: $V(x,y,z)=0$ for $0<x<L, 0<y<L, 0<z<L$ and $V=\infty$ otherwise The eigenstates of this system are given by: $$\Psi(x,y,z)=\bigg(\frac{2}{L}\bigg)^{\...
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Why does the first-order non-degenerate energy perturbation fail if there is degeneracy in the unperturbed energy eigenstates?

Griffiths (Introduction to Quantum Mechanics, 3rd edition, §7.1.2) presents the following derivation for first-order energy perturbations in nondegenerate time-independent perturbation theory. Suppose ...
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Express the power spectrum by displacement field in Lagrangian perturbation theory

Recently I'm reading this paper, Resumming Cosmological Perturbations via the Lagrangian Picture, to learn the application of Lagrangian perturbation theory in the modelling of large-scale structures. ...
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How to diagrammatically show this Green's function from a tridiagonal Hamiltonian?

Consider the following electronic Hamiltonian $$ \begin{bmatrix} \ddots & \ddots & & & \\ \ddots & H_{n-1} & V^- & & \\ & V^+ & H_n & V^- & \\ &...
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Is there a counterexample to this pattern in QM perturbation theory?

Consider some $1d$ system $H_0$ with energy levels $E_n^{(0)}$ that has been perturbed by $\lambda V$, i.e. $H = H_0 + \lambda V$ for some $\lambda >0$. Consider the ratio of the first order ...
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Perturbed hydrogen atom transition probabilities

Given the perturbed hydrogen atom having Hamiltonian $$ \hat{H} = \frac{\vec{p}^2}{2m}-\frac{e^2}{|\vec{r}|}+\lambda \vec{S}\cdot \vec{r} $$ (1) say which operators commute with the Hamiltonian among $...
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Is the $(1s)(2s)$, $S=1$ excited Helium state metastable?

I am reviewing properties of atoms, and I am trying to understand the concept of forbidden transitions better. My understanding is mostly at the level of Griffiths quantum mechanics. My motivation is ...
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Deriving matrix for quadratic Stark effect

We need to evaluate this matrix, however I'm unsure as to how this is done. I could use the selection rules but we have not studied those yet. They are unintuitive. The matrix somehow simplifies to - ...
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Are the 'wrong' states eigenstates of perturbed Hamiltonian?

Townsend quantum mechanics In our earlier derivation we assumed that each unperturbed eigenstate $\left|\varphi_{n}^{(0)}\right\rangle$ turns smoothly into the exact eigenstate $\left|\psi_{n}\right\...
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Doubt regarding Degenerate perturbation theory

McIntyre, quantum mechanics,pg360 Suppose states $\left|2^{(0)}\right\rangle$ and $\left|3^{(0)}\right\rangle$ are degenerate eigenstates of unperturbed Hamiltonian $H$. The first-order perturbation ...
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Reference Request- Black Hole Perturbation Theory (Teukolsky formulation)

I am involved in a project where I need to calculate a metric resulting from a perturbation. In particular, I am doing perturbation on a Kerr Geometry. I am new to black hole perturbation theory. I ...
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Peskin and Schroeder Exercise 10.2 - Yukawa Theory Renormalization

I am having some trouble with exercise 10.2 in Peskin and Schroeder, on the renormalization of Yukawa theory. Part a) of the exercise says show that the theory contains a superficially divergent 4$\...
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Why are the eigenkets of perturbed hamiltonian the eigenkets of the perturbation matrix?

I've just started studying perturbation theory, and of course have now encountered the case where degeneracy arises. I understand why we have to diagonalize the perturbation matrix, and the concept of ...
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Transitions in constant perturbations using time independent perturbation theory

The perturbed Hamiltonian is given as. $$H=\begin{cases} H^{(0)}&\text{for }t\leq 0 \\ H^{(0)}+V(x)&\text{for }t>0\end{cases}.$$ Here $V(x)$ does not depend explicitly on time but it can ...
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Dimensions of perturbative parameter in $\varphi^3$ theory?

In QFT, $\lambda\varphi^4$ is one of the most studied interactions for the scalar field. The parameter $\lambda$ is adimensional, which makes the perturbative treatment straightforward. In the case of ...
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Unlinearity of second order Stark effect in comparison to diagonalized Hamiltonian

I am extremely confused right now; please forgive the long title. From H.W. Kroto's Molecular Rotation Spectra (ISBN: 9780486672595) pp. 166 I read that in the second order perturbation theory, the ...
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Generating functional in $\phi^4$ theory calculation upto 1st order

This question is based on section $1.2$ of Gauge Theory of Elementary Particle Physics by Ta-Pei Cheng and Ling-Fong Li. In $\phi^4$ or $-\frac{\lambda}{4!}\phi^4$ theory let $W[J]$ be the vacuum-to-...
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Why is the functional derivative of the Lagrangian wrt. field evaluated at the classical field equal to negative of source current at lowest order?

I am having difficulty showing this equation in Peskin & Schroeder's Introduction to Quantum Field Theory (Section 11.4 p.340): We wish to compute $\Gamma$ as a function of $\phi_{\text{cl}}$. ...
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Harmonic oscillator in pertubation theory-cosmology

I have a doubt at the following point, in the book "the primordial density pertubation-David H. Lith, Andrew R. Liddle" on page 383, it does that: $$\delta \ddot{\phi}_{k} + 3H\delta \dot{\...
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When is the motion of the perturbative term resonant with the Hamiltonian?

I am given the following Hamiltonian, $H$, which is a perturbed version of $H_0$, $$ H(\theta,I) = H_0(I) -\epsilon \cos(\theta-\Omega t)$$ where $H_0 = \frac{I^2}{2}$, $\epsilon << 1$ and $(I,\...
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Confusion about Kubo Formula Derived in Interaction Picture

I was reading up on "Many-body quantum theory in condensed matter physics" by Henrik Bruus and Karsten Flensberg. I'm however having some trouble understanding the meaning of the Kubo ...
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Cases for weak field approximation for EFEs being applied to non-weak fields

Consider the Einstein Field Equations $$G_{\mu\nu}=\kappa T_{\mu\nu}.$$ These equations are a set of 10 independent highly coupled non linear hyperbolic-elliptic second order PDEs. This implies that ...
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