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

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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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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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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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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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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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146 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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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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Axion theory and effective loop induced interactions

Suppose we have QCD axion (at this point - above the QCD scale): $$ \tag 1 L = \frac{1}{2}(\partial_{\mu}\theta)^{2} + g_{\theta \gamma}\frac{\theta}{f_{\theta}}F_{EM}\tilde{F}_{EM} + ...
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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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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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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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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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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?
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Nature of energy levels in perturbation theory

Unlike variation method, energies obtained from perturbation theory are in general do not guarantee an upper bound to the ground state energy. Is it possible to say something rigorous about the nature ...
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“Zero overlap” of initial and final states in meson to nucleon + antinucleon scattering of scalar Yukawa

I'm currently studying QFT from David Tong's lecture notes and video lectures. In meson to nucleon + antinucleon decay (section 3.2.1 in this ) in scalar Yukawa theory to order $g$, without using ...
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In perturbation theory, how do I determine the order of an approximation?

The title says it all: I'm confused about the various approximations and their orders. In time-independent perturbation everything is quite explicit and obvious, but, for example, how would it be with ...
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Perturbative series for bosons

I have recently read that ... the perturbation series ... is valid only when the perturbed state is qualitatively similar to (or ‘has the same symmetry as’) the unperturbed state. This means ...
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Solving the quantum an-harmonic oscillator pertubatively?

Background Generally while solving the quantum an-harmonic oscillator: $$ -\frac{d^2 y}{dx^2} + k_1 x^4 y + k_2 x^2 y= E y $$ Most people (I've googled) on the internet always solve this using: ...
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Multiple-scale analysis for 2D Hamiltonian?

I came across a technique called "multiple-scale analysis" https://en.wikipedia.org/wiki/Multiple-scale_analysis where the equation of motion involves a small parameter and it is possible to obtain an ...
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Intuitions on perturbation theory?

I'm a QM newbie and I want to ask some questions on how to accept some peculiar points on this perturbation theory thing naturally. While they can be natural for most of the people, I find it somewhat ...
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Any good textbook on the canonical perturbation theory for Hamiltonian systems?

My teacher of classical mechanics once told us, classical mechanics is more difficult than quantum mechanics in many ways. He used the perturbation theory as an example to illustrate this point. So, ...
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Perturbation theory, eigenvalues and eigenvectors for degenerate case (1st order)

I was trying to understand the perturbation theory, but I was lost in the notation... I have understood that I have to identify the unperturbed kets that are degenerated and find the matrix $V$, ...
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Is this solvable? Time-dependent perturbation theory

The question is A hydrogen atom is placed in a time-dependent homogeneous electric field given by $$ \varepsilon(t) = \varepsilon_0(t^2 + \tau^2)^{-1} $$ where $\varepsilon_0$ and ...
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Lippmann-Schwinger equation and $T$ expansion

Lippmann-Schwinger equation, in operator form, is: $$ T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T, $$ where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
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If we considered chiral perturbation theory with coplex $\phi$-s, wold the next lo leading order renormalization $\gamma$-s change?

The Lagrangian of chiral perturbation theory (with two quark flavors) is written using the following matrix $U$ $$U=e^{i\sigma^i\phi_i/f}$$ where $\sigma^i$ are the Pauli matrices, $\phi_i$ are three ...
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Definition of linear response kernel in terms of wavefunctions (Parr/Yang)

I'm trying to understand the derivation of the linear response kernel in Parr/Yang's "Density-functional theory of atoms and molecules". First some background information: We look at a system of $N$ ...
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Limitation of Rytov approximation for turbulence

I have been working through the textbook "Laser Beam Propagation through Random Media" by Andrews and Phillips and have arrived at an interesting dilemma. For a second-order perturbation in the Born ...
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What justifies the perturbative expansion in chiral perturbation theory?

The Lagrangian of chiral perturbation theory is ordered following a momenta power counting scheme, having terms at leading order (which is two 2 $O(p^2)$) next to leading order ($O(p^4)$) and so on. ...
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Bound states and corresponding elementary fields

Let's have some bound state, like positronium or meson. I need to calculate an amplitude of process which involves bound state in in- or out-state. Is it necessary to introduce corresponding ...