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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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Supersymmetry Perturbation Theory

Source:Mirror Symmetry p.198 I have the Hamiltonian $$H = \lambda\bigg( \frac{1}{2} \tilde{p} + \frac{1}{2}h''(x_i)^2(\tilde{x}-\tilde{x_i})^2 + \frac{1}{2}h''(x_i)[\overline{\psi}, \psi] \bigg) + \...
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Why is the “fine structure” correction called that way?

I'm working on the fine structure correction to the Hydrogen atom. I have more of a conceptal, maybe historical question, why is this correction called this way? and why is the fine structure constant ...
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Estimating error in perturbation theory

Is there a simple way to estimate the error in the eigenvalues when approximating a hamiltonian by its $n^{th}$ order perturbation expansion?
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How to ascertain that the Rayleigh-Ritz variational method gives the exact value of the ground state energy?

So the Rayleigh-Ritz variational method can be used to calculate the ground state energy of a quantum system. If $\phi(x)$ is a suitable (square integrable) and normalised function of the coordinates ...
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Where does this Differential Equation comes from?

Im studying Stark Effect and im trying to prove that the second order correction to the ground state of hydrogen like atoms goes like \begin{equation} \delta E^{(2)}_{100}= -\frac{1}{4}a_o^3 E^2(4+5Z^...
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For degenerate perturbation theory, how do we interpret the eigenvectors and eigenvalues of $\hat V$?

For the eigenvectors that are unmixed by the matrix $\hat V$, the eigenvalues are the energy corrections of this eigenbasis. However, the eigenbasis tends to always be (as far as I'm aware) a linear ...
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QFT perturbation theory

I would like to clarify the following statement: Perturbation theory (PT) in QFT is derived with several assumptions such as: adiabatic interaction, spectrum is bounded downward... This statement ...
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Different purposes for using the Large-$N$ Expansion

I've started studying the Large-$N$ expansion and there seems to be several different reasons for using it. In the context of the SYK model, the limit is useful because it reorganizes the Feynman ...
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Horizon entry, Meszaros suppression and start of perturbation growth

I thought that the onset of perturbation growth was determined by horizon entry of the perturbation (because there won't be a gravitational collapse of an over dense region not causally connected to ...
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Expression of proper time with light perturbation on Minkowski metric

In my lecture on General relativity, it is said that, by taking the following metric : $$g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x),~ {\rm with}\ h_{\mu\nu}\ll1$$ one has the definition below of ...
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What is Wick's theorem and what this is use for? [closed]

I am reading Wick's theorem but although I look for it to clearly understand in some textbooks and youtube videos but still it is unclear to me. I cannot get my head over what is normal ordering ...
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Quantum field theory: corrections to excited state correlation functions

I want to know how to calculate the lowest-order-in-the-coupling-constant correction to $$M(x, y,k,p)=\langle k|\phi(x)\phi(y)|p\rangle$$ in $\phi^4$ scalar field theory in a relatively general ...
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Dot product of small perturbation of wave function

I have problem with this when I am doing my assignment. ∇(Ψ+δΨ)⋅∇(Ψ+δΨ) Can anyone help me explain this and how to get the expansion? Thank you.
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Solving the problem using Many Body Perturbation Theory

I am trying to solve the following Hamiltonian using Many body perturbation Theory. $$H=\sum_{i=1}^{N}\Bigg[\frac{P_{i}^{2}}{2m} -\sum_{i,j}\frac{1}{|\vec{r}_{i}-\vec{R}_{j}|}\Bigg]$$. I split this ...
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Relationship between time derivatives at fixed physical r and at fixed comoving x coordinates

I am currently studying Newtonian Perturbation Theory in cosmology. We have introduced the relation between the physical coordinates r and the comoving coordinates x in an expanding universe: $\bf r$...
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What is the symmetry behind this degeneracy?

I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well: Suppose we perturb the infinite cubical well \begin{equation} V(x,y,z)=\begin{cases} 0, \...
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172 views

First-Order Perturbation of Energy Eigenfunction

I have a homework questions where I'm struggling to understand the methodology to use. We derive first the energy functional for the energy eigenfunction equation (this is fine, I used some vector ...
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Positive and negative powers of small parameter in perturbation problem

I'd like to perturbatively handle an eigenvalue problem similar to this: $$ \lambda f = (\hat{H} + (1/\epsilon^2) \hat{V} + \epsilon {W}) f, $$ where $f$ is a function, $\lambda$ is an eigenvalue, $\...
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Zee's explanation of expressing bare coupling by physical coupling

In terms of bare parameter $\lambda$, the $\phi\phi\to\phi\phi$ scattering amplitude is $\lambda\phi^4$ theory is given by $$\mathcal{M}=-i\lambda+iC\lambda^2\Big[\ln\Big(\frac{\Lambda^2}{s}\Big)+\ln\...
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Virtual terms in the Dyson series (time dependent perturbation theory)

Let the interaction evolution operator in the interaction picture be $$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$ where $T$ is the time order operator and $H_I=H-H_0$ is the ...
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Reference suggestion for degenerate canonical perturbation theory in classical mechanics

Please suggest a good book for degenerate canonical perturbation theory in classical mechanics (not quantum mechanics).
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Time-independent and time-dependent perturbation theory yield different results

First, here's the problem statement. Suppose you have an infinite square well of length $a$, where the box extend from $x=0$ to $x=a$. At $t=0$, you add a perturbation $H'$ of the form: \begin{...
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Perturbation theory for molecules, dipole approximation, chromophore

I am interesting in chromophore group and dipole approximation. For example, i have a molecule (acetone or any other ketone/enol) which is belongs to some symmetry group. Because of the symmetry ...
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Perturbed Ricci tensor due to metric perturbation i.e. $R^{(2)}_{\mu\nu}[h]$ in Linearized theory of Einstein field equation

This is an equation (7.153) from Chapter-7 of Sean Carroll's An introduction to General Relativity: Spacetime and Geometry book. I think all of you who studied GR and went thorugh Carroll's book have ...
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Exact solution for the perturbation of the inverse metric

So when we usually linearize general relativity with respect to metric perturbations $g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}$, we compute the correction to the inverse of the metric to first ...
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Linear response treatment of the magnetization of a system of noninteracting fermions

While trying to solve an exercise, I ran into what looks like a contradiction. I'm sure I'm making some kind of mistake, but I couldn't spot it. I'm not asking for help in solving the exercise, which ...
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1answer
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How to calculate second-order correction to the energy from matrix elements of perturbation?

A particle is in the one dimensional harmonic potential $V(x)=\frac{1}{2}m\omega^2x^2$ with a small perturbation $V'$. I want to calculate the first- and second order correction to the ground state ...
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Perturbative series in physics: why are coeffcieints of Gevrey-1 type (i.e. bounded by $\alpha C^n(n!)^1$

I have only been able to find this explicitly mentioned in this paper on resurgence techniques in physics. And have chased up the hints it gives, but they are not very explanatory. Essentially, the ...
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Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator

In following Prof. Toyer's Computational Quantum Physics lecture notes, I came across the following: In computing the Schrödinger equation in real space, one can make a "split operator" Ansatz, for ...
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How to pick a boundary layer coordinate or stretching transformation

I am following Introduction to Perturbation Methods by Holmes and am unsure how I to pick the power in my boundary layer coordinate if my governing equation is the Laplace equation given by \begin{...
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Reference for Feynman diagram technique(position space) in Thermal Field Theory

I am trying to study perturbative expansion of Sachdev-Ye-Kitaev model, where I know that the dominant terms are the Melonic diagrams. I am interested in seeing how perturbative corrections affect the ...
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What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. ...
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Free energy: comparison between exact solution and perturbation theory

I'm studying a system in condensed matter for which there is an exact solution for the free energy $F(\lambda)$, where $\lambda$ is the parameter of the interaction. I can expand $F(\lambda)$ at some ...
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Perturbation theory with a continuous degenerate spectrum

Let's assume that the unperturbated system $H_0$ is a free particle . It has the following energy spectrum $$ E = \frac{p^2}{2m} $$ and the set $\{ \vert k \rangle \} $ forms a complete basis for $H_0$...
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Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields $$ \langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\...
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What if $\omega =0$, which is the frequency of the perturbation term?

In analytic mechanics, when we found a equilibrium position of the system, to determine the stability of that configuration, we apply $q \to q_0 + \epsilon \eta$ with $|\eta| \ll 1$ s.t $q_0$ is the ...
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Is there such a thing as complex conjugate of the first order correction of a wave function?

If I am given the first-order correction to a wave function as $$\left|\psi_n^{(1)}\right\rangle = \sum_{m \neq n} \frac{ \left\langle\psi_m^0\right|H'\left|\psi_n^0\right\rangle }{E_n^0 - E_m^0}\...
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The form of the wave function in time-dependent perturbation theory

I'm trying to understand why the wave function is expressed as follows in a time dependent perturbation. I understand that since the $c(t)$ are unspecified functions, it is mathematically reasonable ...
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Perturbation theory development for the ground state of the QM particle in the box with a centered dirac-delta spike

In the course of a discussion in the chat there emerged an interesting problem, namely a particle in an infinite well with an additional Dirac-delta function spike of scalable hight: $$ H = -\frac{\...
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Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]

I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form $$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$ where the interest is in the ...
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How do one find the first correction order of $\lambda$ for scattering cross section?

I have a lagrangian $\mathcal{L}=\mathcal{L_0+\mathcal{L_1}}$, where $\mathcal{L_1}$ is a perturbation given by:$$\mathcal{L_1}=-(1/3!) \lambda \phi^3-(1/4!) \lambda \phi^4 $$ and$$\mathcal{L_0}=-\...
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Rigorous justification for non-relativistic QM perturbation theory assumptions?

In perturbation theory for non-relativistic quantum mechanics, you begin with a Hamiltonian of the form $$H=H_0+\lambda H'$$ and assume that the perturbed eigenstates and eigenvalues can be written as ...
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Perturbation to the flat space metric

from the geodesic equation for non-relativistic case where $$v_i\ll c$$ $$\frac{dx^i}{dt}\ll1,{\rm for }\ c =1$$ $$\frac{dx^i}{d\tau}\ll\frac{dt}{d\tau}$$using this the geodesic equation for proper ...
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Dirac solution with coulomb-field (perturbation theory)

The dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$ The solution up to first order is $...
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Why is perturbation theory used in quantum mechanics?

I don't seem to understand what perturbation theory really is, and what it is needed for. Can someone please provide an explanation for what it is, and why it is needed in QM?
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Gauge invariance in GR perturbation theory

I have been following this video lecture on how to find gauge invariance when studying the perturbation of the metric. Something is unclear when we try to find fake vs. real perturbation of the ...
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Optical theorem in $\phi^4$: which poles contribute to discontinuity in Feynman amplitude?

Section 7.3 ("The Optical Theorem") in Peskin and Schroeder's QFT text contains a leading order verification of the optical theorem in $\phi^4$ theory by calculating the (discontinuity across the ...
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Why don't very high order Feynman diagrams contribute significantly?

In a particle physics lecture I had today it was stated that the magnetic moment, $g$, is not quite equal to 2, and the difference is accounted for by QED. Later it was stated that we can see this ...
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Solution of the coupled non-linear oscillators by using perturbation theory [closed]

The integration shown here, $$∫_{-\infty}^{+∞}x^r\mathrm{Exp}[−x^2]\mathrm{H_n}^2[x]\mathrm{d}x,$$ appears when we try to calculate the spectrum of the perturbed non-linear oscillators by using ...
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Hydrodynamic interaction between two spheres in $Re\ll 1$ flow

I am studying the interaction between two spherical particles of radius $a$ in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces ...