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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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### Where can I find relativistic corrections to 2s and 2p levels of Hydrogen Atoms?

I am currently studying for a Quantum Mechanics test, and I want to calculate the 2p and 2s hydrogen atom corrections for the relativistic, spin-orbit and darwin corrections, using perturbation theory....
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### Resonant Hamiltonian Mechanics

My question is regarding applying averaging theory to a perturbed Hamiltonian. Now, my Hamiltonian is of the form $$H=H_0 + R(q_i,p_i)$$ Where R is the disturbing potential which is a function of the ...
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### Calculating exact energy levels of perturbed Hamiltonian

I wish to find the exact energy levels of the following perturbed hamiltonian. $$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$ I believe that it can be solved by using the ...
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### Deriving the continuity equation for a perturbation from the continuity equation for the full density

I am attempting to derive the continuity equation for a density perturbation $\delta$, given the continuity equation for the full density $\rho(\mathbb{x}, t)$. This is in the context of cosmological ...
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### Help with calculating the Ricci tensor for the PPN formalism

I'm trying to follow the calculation done by Will in his book Theory and experiment in gravitational physics, and I was hoping for some help in calculating the Ricci tensor components in Section 5.2 (...
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### Learning resources for the PPN formalism

I'm trying to learn more about the PPN formalism, and I was hoping to find some papers that show how the various parameters are calculated. Basically, I want to go through the usual calculations of ...
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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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### 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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### 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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### 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 $H_0$...
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\...
### 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 ...