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

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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 ...
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33 views

Energy gap in phonons and violation of perturbation theory

In a 1 dimensional chain of similar ions which are connected to each other with similar springs there is just one corresponding frequency for each wave vector. But solving the problem of one ...
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44 views

Time independent perturbation theoery

Why do we talk of transitions only in time dependent perturbation theory when the eigen states are corrected even in time independent perturbation theory? If we can,for sake of argument,say : eigen ...
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53 views

Bending of Light in General Relativity using Perturbation

It is standard textbook calculation (e.g. Schutz's First Course in General Relativity page 294) that we can find a total angular change in light deflection due to gravity to be ...
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Time-evolution operator of a perturbed system

How do we evaluate the time-evolution operator of a perturbed system with time-independent perturbation ? For example: In a two state system acted up on by a time-independent perturbation, let's say ...
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The Hilbert space that contains the first order correction to the state vector in Time-independent Perturbation Theory

When deriving the expression for the first order correction to the state vector of the new hamiltonian( H = H0 + H' ) we assume that $|\psi$n1> = $\sum_{m \neq n}$ C$_m$(n) $|\psi ^0 _m>$ ...
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Can Feynman diagrams be used to represent any perturbation theory?

In Quantum Field Theory and Particle Physics we use Feynman diagrams. But e.g. in Schwartz's textbook and here it is shown that it applies to more general cases like general perturbation theory for ...
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89 views

Fermi's golden rule and infinite probablity?

I am slightly confused about the application of Fermi's golden rule. Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: ...
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Perturbation theory for a particle in a weak potential

I have a basic question about quantum mechanics, maybe it has a basic answer. Take a free particle in a quartic potential, $L=\frac{1}{2}\dot{x}^2-\lambda x^4$ This is massless $\phi^4$ theory in ...
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45 views

Perturbation Theory - 1D potential well

Consider an electron in a one-dimensional potential well of width Lz, with infinitely high barriers on either side, and in which the potential energy inside the potential well is parabolic, of the ...
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20 views

Nonequilibrium Green's functions weakly interacting two-component Bose gas

I am planing to describe time evolution of two-component BEC. I was thinking about non-equilibrium Green's functions, but I don't if the method can be applied to the problem describe below. I know ...
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Normal Modes for Standing Waves in 1-D Acoustic Ducts with Arbitrary (but real) Impedance Jumps

Let's say we have a 1-D duct, such as this: Where $Z_i \equiv \frac{P}{US}$ is the acoustic impedance, L is the length of the duct in question, and S is the area of the cross-section. In general, ...
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51 views

Particle creation through a time dependent Hamiltonian

We know that a time dependent Hamiltonian can create particles. We know this for instance from field theory in curved spacetime, where for instance in an expanding or contracting universe creation and ...
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53 views

Perturbation by electrical field in infinite potential well: difference in first energy corrections because of difference in the limits of the well

In time independent perturbation theory we can calculate the first and second energy corrections resulted by a potential V in the Hamiltonian $ H=H_o + λV , $ , λ<<, by the expressions: $$ε_1 = ...
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55 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 ...
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26 views

Book Recommendations on Perturbation Theory

I am interested in studying Quantum Electrodynamics and figure I should begin by learning Perturbation theory and Asymptotic expansions. If anyone could recommend some books, or starting points for ...
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Griffiths Intro to QM Section 9.1.2: What type of approximation is he using here and what is the justification for it?

I really do not understand Griffiths logic in this section and was wondering if someone could help. This is basically a 1st order coupled system of ordinary differential equations but I haven't seen ...
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Degenerate perturbation theory

I don't understand the part about turning off the perturbation. What is meant by or what is he referring to when he says "upper" and "lower" states. Why must the "upper" state reduce to a ...
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Why is stat mech involved in mean field theory?

Mean field theory involves approximating an interacting system by a non-interacting one, by replacing some operators with their expectation value. However, to my surprise, free energy is involved in ...
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Why are the perturbation term in Zeeman effect not diagonalized?

In the case of weak field Zeeman effect (anomalous Zeeman effect) in hydrogen atom, the unperturbed Hamiltonian reads as $$ H_0 = \frac{\hat{p}^2}{2m} + \frac{C_1}{r} + f(r)\mathbf{L}\cdot\mathbf{S} ...
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What are non-perturbative effects and how do we handle them?

Schwartz's QFT book contains the following passage. To be precise, total derivatives do not contribute to matrix elements in perturbation theory. The term $$\epsilon^{\mu\nu\alpha\beta} ...
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106 views

Higher orders in perturbation theory

I would like to compute an energy level up to many orders in perturbation theory. My difficulty right now is not in the calculation itself but in understanding the algebraic structure of the higher ...
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Is the Fermi golden rule really accurate for calculating the life time of an atomic level?

In my impression, Fermi golden rule is routinely used in calculating the life time of an excited atomic level. But it is based on the first order perturbation theory, so it is not expected to be ...
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Why are many physicists trying to develop non-perturbative quantum theories? [closed]

I would like to briefly know where (and why) does perturbation theory fail and why are physicists so desperate looking for non-perturbative theories.
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1st order perturbation of energy for quantum harmonic oscillator [closed]

I am trying to do part B of Griffiths QM 2nd edition problem 6.2 It asks to find the 1st order correction to the energy for a perturbation of a quantum harmonic oscillator where the new spring ...
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Relation between the reduced Green's function and the full Green's function

Let us assume that we have some Hamiltonian and we know its spectrum $$H_0 \psi_n = E_n \psi_n .$$ We define the Green's function in as $$ G(x,y,E) =\sum_m \frac{\psi_m^*(x)\psi_m(y)}{E-E_m}, $$ and ...
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Why drop the kinetic energy operator in Griffiths QM 2nd edition Example 6.1?

In Griffiths Quantum Mechanics Example 6.1 on page 252, the problem is just a simple square well where the potential floor is raised from zero to $V_0$. Thus he states that the Hamiltonian for this ...
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Expansion of wave function and energy in terms of small parameter

In time-independent perturbation theory, the Hamiltonian is perturbed with a perturbation of the form $\lambda V$, and the eigen-energies and wave-functions of the unperturbed Hamiltonian are ...
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61 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 ...
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26 views

Quadratic versus linear Stark shift

I'm trying to understand why the Stark shift changes from quadratic to linear as the applied electric field increases. I think there is some kind of connection to whether the induced dipole moment is ...
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91 views

Doubt in Dyson's argument about the divergent nature of the perturbative expansion in QED

I am trying to understand Dyson's argument about the divergent nature of the perturbative expansion in QED. Quoting his own words [...] let $$F(e^2)=a_0+a_1e^2+a_2e^4+\ldots$$ be a physical ...
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Why do we still use perturbation theory, when we have advanced numerical methods and fast computers?

If my question sounds ignorant or even insulting, I apologise. I may be completely wrong, since I'm not a theoretical physicist. So, I understand why perturbation theory was originally used in ...
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Time independent perturbation theory for a 1D simple harmonic oscillator system

I have been looking through my notes and it says in a footnote that the approximation of energy levels using perturbation theory is more accurate when the energy shift of the energy levels due to the ...
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154 views

Finite temperature correlation functions in QFT

Suppose that we want to calculate this imaginary time-ordered correlation function for an interacting system (in Heisenberg picture) : $$\langle \mathscr{T} A(\tau_A)B(\tau_B) \rangle =\frac{1}{Z} ...
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37 views

Radiative corrections and stability

What is meant by the terms radiatively stable and radiatively unstable? I know that when calculating physical observables in quantum field theory, such as the mass of the electron, to obtain a more ...
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Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
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Derivation of Hartree-Fock equations using 2nd quantization [closed]

I derived the following effective Hamiltonian: $$ H_{eff} = \sum_k{ \left( \, \epsilon_k + \sum_{k_2}{\left(<k \, |<k_2 \, |\,u\,| \, k_2>|\, k> - ...
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How does one get the first few terms of the S-matrix expansion?

According to a set of notes I'm reading $$\langle p_f | S | p_i \rangle = \delta(p_f-p_i) + 2 \pi \delta(E_f-E_i) \bigg[\langle p_f | V | p_i \rangle + \cdots\bigg] \tag{1.29}$$ I don't understand ...
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Rayleigh-Schrodinger perturbation of double-well

Might be a silly question but anyway. I know how to use the Rayleigh-Schrodinger method when the total Hamiltonian as $H=H_0 +H'$ where the first term is known and the second term is proportional to a ...
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Symmetry of interaction lagrangian and symmetry of full lagrangian

Suppose we have lagrangian $$ \tag 1 L = \frac{\theta}{f_{\gamma}}F_{EM}\tilde{F}_{EM} +\frac{1}{2}(\partial_{\mu}\theta)^2 - \frac{1}{2}m_{\theta}^2\theta^2 + L_{SM}, $$ where $\tilde{F}_{EM}$ ...
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Perturbation Theory Question

How can you work out the average perturbation, from a normal hamiltonian, of all states that rely on the quantum numbers of s = __ and l = __, with the perturbation being proportional to the product ...
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128 views

Number of distinct Feynman Diagrams for different orders of $\phi^4$ theory for 2 point function

There is 1 distinct Feynman diagram for zeroth order and 2 distinct diagrams for first order in $\phi^4$ theory for two point function. I want to know is there a way to predict the number of distinct ...
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Operators for a Perturbed Hamiltonian: Heisenberg Picture ($\hat{x}$, $\hat{p}$)

Problem I am trying to calculate the Equations of Motion in the Heisenberg picture for $\hat{x}$ and $\hat{p}$ in a perturbed Hamiltonian, $$ \tag{1} \hat{H} = \hat{H}_0 + \hat{H}' $$ Assume ...
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60 views

Perturbation theory of $\lambda q^4$ perturbed harmonic oscillator

For a perturbed Hamiltonian $$ H = H^{(0)} + H' $$ the perturbation theory approach $$ \Psi = \Psi^{(0)} + \lambda \Psi^{(1)} + ... \\ E = E^{(0)} + \lambda E^{(1)} + ... $$ leads to the equations ...
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Renormalization group resummation

I'm having trouble in understeanding a mathematical feature of RG, namely how it provides a way to resum the perturbation series and how that's defined mathematically. From a conceptual point of view ...
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What is the advantage of the canonical transformation when getting the effective hamiltonian?

Assume we have hamiltonian $$ H = H_0 + \lambda V$$ where $ H_0 $ is unperturbed hamiltonian which we know the eigenstastes, and $ V$ is a perturbation. In the effective hamiltonian approach using ...
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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) ...
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What is the exact meaning that QED perturbative series is only asymptotic and eventually diverges at very high orders?

When I read paper PRB89, 235431 about the effective field theory of graphene, there is a statement that QED perturbative series is only asymptotic and eventually diverges at very high orders (e. g. ...
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153 views

Mathematical proof that $\exp(-1/|g|)$ is always related with formation of bound states through scales?

I know that this function ($g$ means coupling) is non-analytical in $g=0$, so this function is only appreciable under non-perturbative calculations, so is a non-perturbative phenomena. This function ...
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110 views

Hydrogen anion system [closed]

For the H- system,how we calculate the energies correct to the first order for the ground state and first excited state. Is it possible for a single proton to hold two electrons and still be stable?