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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Sound speed in cosmology: non-adiabatic perturbations

The definition of sound speed is given by: $$c_{s}^{2}\equiv \frac{\partial P}{\partial \rho}.$$ In some books of cosmology to calculate the expresion for the sound speed in a baryon-photon fluid they ...
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Why do we ignore the proton when computing relativistic correction in the Hydrogen atom?

I have read a number of textbooks (e.g. Sakurai), and they all seem to say that the unperturbed Hamiltonian of hydrogen is: $$ H_0 = \frac{p^2 }{2m_e} - \frac{e^2}{r} \tag{1} $$ and the relativistic ...
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Time-Depedent Pertubation Theory - Ionization Rate of a Hydrogen Atom

So, i was studying Time-Depedent Pertubation Theory, using the book "Lectures on Quantum Mechanics" by Steven Weinberg when i ran across this problem: "Calculate the rate of ionization ...
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Validity of weak gravitational field approximation (Schutz's First course in GR)

I'm studying GR with Schutz' First Course in General Relativity and I have some trouble. When field is weak enough, we can take such coordinate system that our metric is written as $$ g_{\alpha\beta} =...
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Perturbation Theory, Infinite Square Well, Kronig Penney [closed]

This is a two part question from a paper I did, I couldn't answer it at the time and can't find sufficient help in my notes. I've been trying to solve it since but it's a lot harder than normal to get ...
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Derivation of PT for correlation functions (Peskin & Schroeder, Ch. 4.2)

I am trying to understand the basics of PT in QFT with Peskin & Schroder text, Chapter 4. The discussion in the book starts with the perturbation expansion of the two-point correlation function in ...
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37 views

Perturbation series of linear anharmonic Oscillator

How do I show that in this eigenvalue problem, the perturbation series for $E(a)$ has vanishing terms of order $a^n$ for $n>=3$, also that perturbation series for eigenfunction $y(x)$ is convergent ...
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Treat stochastically non-Hamiltonian perturbations

Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} $f(q)$ is a non-...
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Significance of Diagonalization in Degenerate perturbation Theory

I am studying Degenerate perturbation Theory from Quantum Mechanics by Zettili and i'm trying to understand the significance of diagonalizing the perturbed Hamiltonian. He uses the stark effect on the ...
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1answer
33 views

Choosing a global phase in time-independent perturbation theory

I have been trying for quite some time now to find a satisfactory justification of a standard assumption we usually do in time-independent perturbation theory, but I am still puzzled. Here is the ...
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Order of three-point function in cosmology using in-in formalism

In early universe cosmology, the three-point correlation function of the curvature perturbation can be solved using \begin{equation}\langle \zeta_{k_1}(t)\zeta_{k_2}(t)\zeta_{k_3}(t)\rangle = -i \int_{...
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Kubo formula to compute the energy dissipated in heat bath

In the case of a perturbed Hamiltonian $H_0$ \begin{equation} H=H_0 +\theta(t-t_0)W(t) \end{equation} at $t=t_0$ the Hamiltonian admits eigenvalues $E_n(t_0)$ and for positive $t-t_0$ then the ...
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How to fix the quantization axis of an atom

Suppose I send linearly polarized light onto a hydrogen atom. Using first order perturbation theory one can show that, depending on the relative polarization of the light to the quantization axis of ...
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Reference request: upper-level undergraduate treatment of time-dependent perturbation theory using the interaction picture approach

I find the treatment of time-dependent perturbation theory in Griffiths to be quite difficult/intensive. Our professor introduced us to the idea of using the interaction picture and this feels much ...
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Can someone explain how the following equations come to be in the Cosmological Pertrubation Theory?

I'm just starting out studying the Cosmological Perturbation Theory and most of the things do NOT make sense to me. For example, how these two equations came to be. Please, also mention some starting ...
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Perturbing PDF with spatial dependent perturbation

Let's consider a PDF $\rho(x)$, with normalization 1. Let's perturb it in the following way: $$ \rho(x+\varepsilon F(x) ), $$ with $\varepsilon$ small. I impose that the perturbed PDF is again a PDF ...
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Having trouble with a statement in sakurai modern qm [duplicate]

I am facing trouble with a statement made after eq 5.2.4 in sakurai modern qm and the statement is $P_1(E-H_0-\lambda P_1VP_1)$ is singular in $P_1$ subspace. How one can we show the validity of the ...
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1answer
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Time dependent perturbation, particle on a segment

Consider a particle of mass $M$ moving alongside a segment of length $a$. Wavefunction: $$\psi_n(x)=\sqrt\frac{2}{a} \sin(\frac{n\pi}{a}x)$$ Energy: $$E_n=\frac{\hbar^2\pi^2 n^2}{2Ma^2}$$ Time-...
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How to obtain large order perturbation series for cubic anharmonic oscillator?

Consider the potential $$V(x)= \frac{x^2}{2} + gx^3.\tag{1}$$ Then the time-independent Schrödinger equation becomes $$\left(-\frac{1}{2}\frac{d^2}{dx^2} + \frac{x^2}{2} + gx^3 \right)\psi = E(g) \...
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1answer
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What are the 'good' quantum numbers for the weak- and strong-field Zeeman effect?

I'm quite confused on the 'good' quantum numbers. I thought the good quantum numbers could be defined as the quantum numbers which corresponding operators commute with each other and the Hamiltonian. ...
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Question about perturbation theory and even and odd wavefunctions

I was solving a question about perturbation theory and I came across something my teacher didn't mention and I can't seem to understand it. In the question there is an external electric field on a H-...
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Perturbation theory with degenerate and non-degenerate states (at the same time)

In the time independent perturbation theory, the perturbed wavefunction is [1]: $$ \left|n^{(1)}\right> = \sum_{k\neq n} \frac{\left<k^{(0)}\right|V\left|n^{(0)}\right>}{E_n^{(0)}-E_k^{(0)}}\...
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Perturbation Theory Applied to the Quantum Harmonic Oscillator [closed]

I am trying to compare the wave function obtained by exact method and by approximated method. The potential is \begin{equation} V(x)=\frac{1}{2}m\omega^2+Ax\end{equation} I found a solution but I am ...
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Text reference for linear Stark Effect

I was trying to study the linear stark effect and an example of the splitting of $2S\pm2P_z$ orbitals, described here. Can someone suggest to me a reference for the similar material discussed but in ...
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1answer
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Planetary motion considering natural satellite

Using Kepler's laws of gravitation we can determine the time period of a planet revolving around the sun. However, this excludes the gravitational effect due to satelite(s) orbiting around the planet. ...
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1answer
58 views

Perturbations of the Metric in GR

I have been studying a bit of GR and am still a bit confused as how it works with perturbations of the metric or even in what norm the perturbation is meant to be understood. For example, what ...
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28 views

Hellmann-Feynman theorem for nearly degenerate states

If we have: $$\tag{1} E(P)=\int \psi_a(P)^{*} H(P) \psi_b(P)\ \mathrm{d} \tau $$ Then taking the derivative wrt. the parameter P yields: $$ \begin{aligned} \frac{\mathrm{d} E}{\mathrm{d} P} &=\int\...
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Why does $q(t) \to-i\hbar \frac{\delta}{\delta J(t) }$ for the generating functional of a perturbed harmonic oscillator?

When computing a generating functional, $Z[J]$, in terms of the generating functional of Green functions, $Z[0]$, in my lecturer's notes we reach the following terms: $$Z[J]= \mathcal{N} \int Dq \...
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2answers
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Perturbation in 2D oscillator

2D oscillator $H_0=\frac{P_x^2}{2m}+\frac{P_y^2}{2m}+\frac{1}{2}m\omega^2\left(x^2+y^2\right)$ with perturbation $H_1=h\omega \left(\frac{L_z^2}{h}-2\right)$ How to write the perturbation in terms ...
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Perturbation theory - integrals over different wavefunctions

I am reading a research paper on how to calculate g-tensors* computationally (Correlated four-component EPR g-tensors for doublet molecules). In the theory section, they write "[...]from (quasi-)...
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$\phi^3$ theories in 2+1 dimensions

I quite often see papers considering a $\phi^4$ theory in three spacetime dimensions, but rarely do I see papers with $\phi^3$ terms. I understand that these kinds of interactions terms can have ...
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2answers
42 views

Adiabatic Pertubation in an Infinite Potential Well

I'm working on a problem of my quantum mechanics homework set. The problem is as follows: A particle is in the ground state of infinite potential well (the well is determined by the region 0 < x &...
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In Brillouin-Wigner perturbation theory, how to choose the correct energy from all the solution?

I'm currently studying the Brillouin-Wigner theory from the book "Atomic many-body theory" of I. Lindgren & J. Morrison. The energy $E$ of the perturbated state $|\alpha>$ is linked to the ...
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Wilson's approach to renormalisation according to Peskin & Schroeder

Although Peskin & Schroeder treats Wilson's approach to renormalisation theory in some depth, I don't get one of its main points. According to P&S (p.401): Imagine that we wish to compute ...
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2answers
108 views

Understanding electrons in a weak periodic potential fourier analysis

I have been trying to understand Ashcroft's take on electrons in a weak periodic potential, and his approach by Fourier analysis is proving to be extremely challenging. I understand how to get to the ...
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38 views

Is there such thing as degenerate time-dependent perturbation theory?

Is there such thing as degenerate time-dependent perturbation theory? So far I haven't found any reference to this concept online, and I can see why since there is no denominator that contains $E_f - ...
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1answer
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Reduced Green's function for a quantum harmonic oscillator

Is there a known formula for the reduced Green's function for the (1D) harmonic oscillator? As far as I am aware, the reduced Green's function could be used as a tool for working with time-...
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1answer
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Photon scattering final state

I'm looking through the linked paper on x-ray scattering. The Section $5$ says that the expression $(67)$ (which contains an error but the editor mentions the correct variant at the beginning of the ...
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Arguments for Feynman diagrams (or related) as Lagrangian tools, or “vice-versa”?

I know that the Feynman diagrams and perturbation series are used as computational tools to evaluate a specified Lagrangian for a QFT. This is usually used as an argument in itself to discourage ...
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How can we still ignore degenerate terms after changing basis in perturbation theory, isn't the denominator still zero?

I understand that for degenerate eigenfunctions of a Hamiltonian, one cannot use the non-degenerate perturbation theory to find the first order changes to the eigenfunctions since it includes terms of ...
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perturbation of the form $W(x) = a\omega_0\delta(x− \frac{a} {2})$ calculate the changes in the energy level of the particle

Consider a particle of mass $m$ in one-dimensional infinite potential well of width $a$, i.e. $V (x) = 0$ for $0 ≤ x ≤ a$, and $V (x) = \infty$ for $x < 0$ and $x > a$. The particle is subject ...
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1answer
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Central Field In Many Electron Atoms

I was reading Hartree Self Consistent Field and came across "Atomic Physics by P. Ewart" (PDF). In Central Field Approximation page 8 it gives the formula for the Hamiltonian in two terms, the ...
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Why particle in a bended box seeemd to be resilient?

Suppose a particle in a box confined in a 1 dimensional segment or length $L$. However, let's put the box on 2 D surface. Further, let's bend the box into a semi circle, the radius was thus $\...
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2answers
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Transcendental equations solution using perturbation expansion

How to solve an equation of the following form: $$ e^{ik} = -1 + ik\epsilon $$ where, $\epsilon$ is a small number. As $\epsilon$ goes to zero, the value of $k$ goes to $\pi$ (choosing only the lowest ...
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Perturbation expansion in bound scattering states for double Dirac barriers [closed]

I was working my way through scattering theory notes by David Tong.In there,he discusses the analytical property of the $S$-matrix and uses it for the resonance states for the Double - Dirac potential ...
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1answer
122 views

Complex energy interpretation in perturbation theory

I'm working on time-independent degenerate perturbation theory for the Hydrogen first excited state. I have the following perturbation $H$: $H = \lambda V_0 \sin^2 \theta \sin 2\phi = \lambda V$. We ...
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1answer
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General Condition for 1st order correction of perturbation theory to be ZERO? [closed]

I need some help on the following question regarding perturbation theory of a quantum system: Give an example of a perturbation to a system for which the first order correction is zero, and ...
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105 views

3D harmonic oscillator in spherical coordinates

I have a conceptional question about a variant of the 3-dimensional q.m. harmonic oscillator which serves both, deeper and better personal understanding, but also discussion and potential missing ...
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1answer
46 views

Integral of two radial wave functions

I am wondering how to evaluate the following integral of two radial hydrogenic wavefunctions: $$ \int r^{3}R^{*}_{nl} R _{10} dr $$ The whole problem is related to the calculation of the Stark ...
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1answer
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Is there a theoretical limit to the splitting of atomic energy levels?

We know that the hyper fine interaction is due to interactions between the nucleus and the electron and Zeeman splitting induces further quantization but what I am wondering is, are there any higher ...

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