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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Perturbation theory for an asymmetric top
I am working through Prof. Littlejohn's notes on perturbation theory, and I'm trying exercise number 2, which asks to analyze an asymmetric top (exact problem statement in the notes). I'm confused ...
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"Good" eigenstates for hydrogen atom in first excited state in an electric field
Consider a hydrogen atom in the first excited state that is being kept in an external electric field of strength $\lambda E_{ext}\mathbf {\hat{k}}$ . This electric field causes the splitting of the ...
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How does transition rate behave under $T \rightarrow \infty$ limit
I am supposed to learn Fermi's Golden Rule, and the book I am using for that is Modern Particle Physics by Mark Thomson. On page 52, he goes :
The transition rate $d\Gamma_{fi} = \frac{1}{T}|{T_{fi}^...
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Feynman rules for scalar QED [closed]
Can i have a derivation for the Feynman rules of scalar QED?
I don't understand the ones i find online such as the one in the link https://canvas.harvard.edu/files/936391/download?download_frd=1&...
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Nearly Free Electron Model (Perturbation Theory)
I am having difficulty understanding the degenerate perturbation theory treatment of the nearly free electron model. So for a free electron, the energy dispersion is relation is $E^{0}=\frac{h^{2}k^{2}...
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Why does degenerate perturbation theory work? [duplicate]
As is well-known, the first-order correction to the $n$th unperturbed eigenstates is given by
$$|\psi^n_1 \rangle = \sum_m \frac{\langle m| H_1 |n \rangle}{\varepsilon_n - \varepsilon_m},$$
where I ...
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Vector perturbations in an expanding universe
Why vector perturbations in an expanding universe decay while the scalar and tensor perturbations don't?
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Do we have an analytic calculation to derive $\frac{F^2}{4}\,\text{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\}$ from the QCD Lagrangian?
I have studied the quark condensate and chiral perturbation theory. However, I am not sure where the "kinetic term" of the pion
$$\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^...
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Solve the perturbed Boltzmann Einstein equations in cosmology using RK4 method
I'm trying to learn to numerically solve the perturbed Boltzmann-Einstein equations in cosmology using the RK4 method.
These are the equations:
\begin{align}
\dot{\Theta}_{r,0}+k\Theta_{r,1}&=-\...
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Perturbative calculation of hierarchy problem
I've been trying to understand the origin of the hierarchy problem for the Higgs mass but I've tied myself into some pretty nasty knots and I'm hoping someone can shed some light on this.
So as I see ...
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Connection between a saddle point approximation and plain perturbation theory
I am currently studying functional integration in the context of classical and quantum equilibrium thermodynamics. However one thing puzzles me:
In the book "Phase Transitions and Renormalization ...
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Time-Independent Perturbation: Correction Components are Orthogonal to the Original State (Townsend)
I am reading Townsend's A Modern Approach to Quantum Mechanics, Second Edition. While developing the first order correction to the eigenstate (in time independent, non-degenerate Perturbation theory), ...
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Darwin Term (Fine Structure) and the Taylor expansion of the electric potential energy
I am trying to derive the Taylor expansion for the potential $U(\vec r + \delta \vec r)$. The general expression for the Taylor expansion is: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n....
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2D Q.H.O and degenerate time independent Perturbation theory
In the case when we consider the 2D quantum harmonic oscillator, we also consider an additional disturbance operator of the form $W=Kxy$.
If we consider the eigenvalue $E_1$, where $n=1$, these ...
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An instance of the equality of the transition rates $W_{i\to f}$ and $W_{f\to i}$ from Fermi's Golden rule
Fermi's Golden Rule tells that for a perturbing Hamiltonian $\hat{V}$ that couples initial state $\left| i \right\rangle$ to final state $\left| f \right\rangle$, the transition probability per unit ...
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Perturbations of an integrable system with no resonant tori
Suppose I have a Hamiltonian $H_0$ which is just a collection of $N$ non-interacting harmonic oscillators. Written in action-angle coordinates $(J_i, \theta_i)$ we have $H_0 = \sum_{i=1}^N \omega_i ...
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Metric perturbation and its properties as scalar, vector and tensor
I am learning the black hole perturbation theory through the original work of Regge and Wheeler, "Stability of a Schwarzschild Singularity" (1957). And in its second page, it claims that the ...
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Perturbation theory: Why is the inner product expressed as $\langle\psi_m^0|H'|\psi_n^0\rangle=-2V_0(-1)^{\frac{m+1}{2}}$? [closed]
Introducing a perturbation to an unperturbed state, say $\psi_n^0$, with eigenenergy $E_n^0$ yields the first degenerate state as:
$$\psi_n^1=\sum_{m\neq n}\frac{\langle\psi_m^0|H'|\psi_n^0\rangle}{(...
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How are Lie series used as canonical transformations in perturbation theory?
I have a few questions on how to use Lie series as a canonical transformation, which are widely used in perturbation theory (celestial mechanics).
I know that these series are related to a Taylor ...
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Solving differential equation in perturbation theory
The differential equation of an anharmonic Oscillator with Newtonian friction is
$$
\ddot{x}+\varepsilon \dot{x}^2+x=0
.$$
The initial conditions of the System are
$$
\begin{align*}
x(0)&=1\\
\...
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Why does the sign change here in a perturbation derivation?
I am studying Lindstedt's method in Gregory classical mechanics (great book). I don't understand a perhaps simple algebraic simplification.
Why does the sign change here? Is this an approximation as ...
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Perturbation theory for optical lattice
This afternoon, I wondered about the following problem, but I cannot find the continuation. Can you help me ?
We consider the following unperturbed 1D optical lattice Hamiltonian
$$\hat{H}_0=\frac{\...
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Power series of quantum mechanical eigenfunctions
In Griffith's Quantum Mechanics book, when he tackles time-independent perturbation theory, he states that one can write the eigenfunctions of the perturbed system,
$$
H = H^0 + \lambda H',
$$
as a ...
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Poisson equation in Cosmology at first order in perturbation theory
The book Cosmology by Daniel Baumann states that the Poisson equation for a universe where we consider the effects of both gravity and expansion, expressed in physical coordinates $\vec{r}=a\vec{x}$, ...
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How did the Higgs perturbatively give us the mass of all other particles?
In a recent conversation with my professor, he explained to me a misconception I had, in that given how physicists says that the Higgs "gives" the mass of all other particles, I was under ...
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Coupled degeneracies vs uncoupled degeneracies
I am reading about degenerate perturbation theory. I am specifically concerned with this problem $\frac{-h^2}{2I} \frac{d^2}{d\phi^2} \psi = E \psi$.
Here, we get eigenstates of the form $ e^{\pm ni\...
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Interpretation of second-order term in time-dependent perturbation series (Dyson series)
$\newcommand{\ket}[1]{\left \lvert #1 \right \rangle}$
Context
Consider a system described by
$$H(t) = H_0 + V_0 v(t) \mathcal{O}$$
where $V_0$ defines the strength of a time dependent perturbation, $...
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Indices of $(\text{Riem})^3$?
This question relates to writing higher curvature terms in momentum space with respect to GR as an effective field theory.
I know that $R_{\alpha\beta\mu\nu} \sim \partial_\beta\partial_\mu h_{\alpha\...
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Constant perturbation in Time dependent perturbation theory
I was reading the topic of Time dependent perturbation theory by Nouredine Zettili book. In that for a constant perturbation and after evaluating the probability from initial to final state a factor ...
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If the metric tensor is unitless, why do its perturbations pick up units of Newton's constant?
If the metric tensor is unitless, why do its perturbation terms pick up units of Newton's constant?
In the following expansion, metric perturbations pick up a factor of $\kappa\propto\sqrt{G}$
\begin{...
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How to calculate energy correction we get from time-independent perturbation theory using Feynman diagram?
I am dealing with scalar field theory in $0+1$ dimension with the following free theory Hamiltonian,
$$
\mathcal{H}_0 =\frac{1}{2}\big[\pi^2+m^2\phi^2 \big]\tag{1}
$$
with a quartic interaction of the ...
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Finding initial conditions from the temperature autocorrelation function
So I was reading Mukhanov and in section 9.4 titled Correlation functions and multipoles, he talks about obtaining the auto-correlation function
\begin{equation}
C(\theta) = \bigg\langle \frac{\delta ...
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Derivations about Time-Dependent Perturbation Theory (Griffiths)
I've been reading the Time-Dependent Perturbation Theory in Griffiths' book. There are many places that puzzle me a lot.
The system is two-level($\psi_a,\psi_b,E_a<E_b$). Suppose the particle ...
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Quantum Mechanics - does a paired transition need a perturbation?
I'm doing some theory (to model a device) which, with a bunch of simplifications, wants to consider carrier transitions between extended states in a generalised semiconductor. In particular I'm after ...
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Two charged free particles with the same energy coming toward each other. What will be the time-dependent perturbation in this case?
Let's assume we have two charged free particles with the same energy coming toward each other. What will be the time-dependent perturbation in this case? More precisely, I am asking what will be the ...
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Calculation of Einstein coefficient for spontaneous emission [closed]
I want to calculate the rate of spontaneous emission for the transition: $|3,2,2\rangle$ to $|2,1,1\rangle$ in a hydrogen atom.
Corresponding to the formula of $A_{ab}$ the only thing to derive is the ...
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Born-Oppenheimer approximation in many-body perturbation theory
In order to obtain phonon spectrum, we usually do Born-Oppenheimer approximation and assume that the electrons are always at the ground state when the atoms move, and by calculating the force on each ...
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Why perturbational approach fails to describe the bands?
I'm trying to find the band structure of Graphene using tight-binding for a unit cell with 6 carbon atoms(this is a toy model for my own research). The hamiltonian is as below:
$$H=
\left[\begin{...
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Raising and lowering indices to second order
If I consider a metric perturbed to second order, $$g_{\mu\nu}= \eta_{\mu\nu} + \lambda h_{\mu\nu}^{(1)} + \lambda^2 h_{\mu\nu}^{(2)},\tag{1}$$ how should I raise and lower indices for a generic ...
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What's the correspondence between Feynman diagrams and field configurations?
If I understand correctly, a Feynman diagram represents a finite set of "interactions", such as the exchange of a photon between two electrons. You can think of it as a graph in which the ...
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Vanishing expectation of commutator in the Kubo formula
In linear response theory, the Kubo formula describes the change in $\langle A \rangle$ for some observable $A$, due to applying a time-dependent perturbation $V(t)$ to the Hamiltonian $H_0$. It is ...
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For a harmonic, time dependent perturbation in QM, how is energy conservation imposed?
I am currently reading Sakurai's Modern Quantum Mechanics, and in the section on time dependent perturbation theory, he derives the first order coefficient for an energy state n at time t in the ...
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Gauge invariance or diffeomorphism invariance in GR of observables?
I am confused by the definitions of a gauge transformation, a coordinate transformation and a diffeomorphism. In particular, should observables in GR be fundamentally invariant under gauge ...
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How quasinormal modes of black holes related to black hole temperature?
I have heard that quasinormal modes of black holes are related to black hole temperature. QNMs describe how a black hole reacts to perturbation. Forex, when a black hole is perturbed, it will emit ...
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Why does square of Planck length come as coupling constant when quantizing gravity in 3+1D?
In Birrell and Davies, the author says in the Introduction that
If the gravitational field is treated as a small perturbation, and attempts are made to quantize it along the lines of quantum ...
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Recover perturbation theory as appropriate limit of the lattice theory
I remember hearing at some point that pertubation QFT using Feynman diagrams can be thought of as certain limits of lattice QFT. Is there a precise statement of this fact? Or is it just a heuristic ...
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Interacting QFTs and Virtual Particles
Short introduction to my understanding:
As far as i understand, virtual particles are usually defined to be the internal lines in Feynman Diagrams. But we know that those are just useful tools to ...
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Small knock ($\delta$) on rotating Pendulum in Lagrangian [closed]
Following Setup:
There is a Pendulum swinging around a vertical axis, with angles $\varphi$ for rotation, and $\theta$ for the incline compared to the horizontal axis. The mass is $m$ and the length ...
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Do excited states of hydrogen give cosmologically relevant hyperfine transitions?
In Griffiths quantum mechanics, there's a discussion of hyperfine splitting for the ground state of hydrogen - this gives rise to a small level splitting corresponding to an emitted wavelength of $21$ ...
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Where is the Darwin term in Griffiths quantum mechanics?
I was reading the Wikipedia page on fine structure, and it had the following expression for the fine structure of Hydrogen:
To recap the image, Wikipedia includes the spin-orbit interaction, the ...
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