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

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Can an azimuthally symmetric perturbation lift the 2l+1 degeneracy of angular momentum eigenstates?

Assume the initial Hamiltonian of a spinless, non relativistic particle is $$H_0(r,\theta,\phi)=\frac{{\bf p}^2}{2m}+V_0(r)$$ Such that the eigenstates are angular momentum eigenstates $|n,l,m>$, ...
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On the Stability of Circular Orbits

Bertrand's Theorem characterizes the force laws that govern stable circular orbits. It states that the only force laws permissible are the Hooke's Potential and Inverse Square Law. The proof of the ...
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Why Does Renormalized Perturbation Theory Work?

I've read about renormalization of $\phi^4$ theory, ie. $\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{\lambda}{4!}\phi^4\,,$ particularly from Ryder's book. But I am ...
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Electronic or excitonic band structure?

Usually, in the papers the electronic band structure for monolayers $WS_2$ is something like in the figure below: As you can see the direct bandgap is around ~2.0 eV. When we excite electrons at the ...
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Details of Newtonian Prediction for Mercury's Precession

Could anyone point me to a book or outline the methods used to actually calculate the 532 arcseconds per century that Newtonian theory apparently predicts for Mercury's precession. I am completely ...
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Degenerate perturbations: why is it not necessary that $ [H_0,H']=0$?

Suppose we are doing a degenerate Rayleigh-Schodinger perturbation problem. Let's say the Hamiltonian $H_0$ is perturbed by a small perturbation $H'$, and we want corrections to the energy eigenstates/...
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Higher order corrections to hydrogen and their consquence

I was wondering if there have to be higher order corrections for the quantum mechanical description of the Hydrogen atom. I'm aware that there are relativistic corrections and also QED corrections. ...
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Wavefunctions “adapted” to the perturbation ? Relation to Faraday effect

I came accross the following statement in a book: If one wants to switch on a magnetic field, one must first choose the appropriate complex unperturbed wave functions (that are "adapted" to the ...
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First order time-independent perturbation theory: How to practically calculate the perturbed wave-function

This is one of the problems that draws the line between academically learning something, and having to use it. While I learned the formulas relevant to this, I just want to make sure I'm using them ...
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How to properly use Perturbation Theory in classical systems?

Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom: $$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$ We can solve it ...
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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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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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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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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 \begin{equation}\Delta\...
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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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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 0+...
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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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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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138 views

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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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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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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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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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 combination ...
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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} F_{\mu\...
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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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168 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} Tr\...
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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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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> - <k\,|<k_2\,|\,u\,|\,k>|\,k_2&...
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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 $\tilde{F}_{EM}$ ...