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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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46 views

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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39 views

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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123 views

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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60 views

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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2answers
211 views

Why this self-loop diagram is not included in $\phi^4$-theory of Peskin & Schroeder?

Consider a $2\rightarrow2$ scattering process in $\phi^4$-theory. On p. 326 in the book of Peskin & Schroeder, they consider the 3 1-loop corrections in the parenthesis: My question is: Why don't ...
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1answer
33 views

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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2answers
715 views

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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104 views

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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1answer
52 views

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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1answer
902 views

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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81 views

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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1answer
133 views

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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3answers
324 views

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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1answer
123 views

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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1answer
100 views

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 ...
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1answer
79 views

First Order Approximation of the Navier-Stokes Equation: Order of Magnitude of the Gradients of First-Order Fields

I am currently working on a project in acoustics and I am studying first and second-order approximations to the Navier-Stokes equation. I have been reading the book 'Theoretical Microfluidics' by ...
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0answers
123 views

Rotating wave approximation : Why do we need both weak coupling and one of the frequency higher with respect to the other one

I have read Rigorous justification for rotating wave approximation to have an idea of a rigorous proof of RWA approximation. The main idea I have from this is that you can see it if you write a ...
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24 views

The fine structure constant and the strength of interaction between two particles

In my notes, the following is mentioned: We consider the scattering of a beam particle with energy $E$, momentum $p$ and charge $ze$ off a charge distribution $\rho (x)$ of total charge $Ze$. We ...
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1answer
40 views

Question about the perturbation solution involved Rashba spin-orbit coupling?

Currently, I am reading the original paper about spin field effect transistor proposed by Supriyo Datta and Biswajit Das. In the last part of this paper, to obtain a larger overall current modulation ...
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1answer
93 views

Consistency of time-dependent and time-independent perturbation theory

I am confused as to how time-independent and time-dependent perturbation theories in QM give consistent results at the instant the perturbation is switched on. Suppose I have a two-level system which ...
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1answer
36 views

Perturbation of Diracs equation (first order)

I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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0answers
83 views

Constant Density of States and Perturbation theory

Given a constant Density of States (DOS) corresponding to a one-electron hamiltonian, $\text{DOS}(\omega)=\dfrac{1}{2D} \chi_{[-D,D ]}(\omega),$ where $\chi$ is the indicator function, I want to know ...
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1answer
171 views

Non-dimensionalization and perturbative expansion

I need to expand an equation, of the form $$\dot{r} = \gamma(a,\mu) F_1 + g(\mu,\ell,h,R) F_2$$ in powers of $\epsilon = a/\ell$. So I thoughts I would non-dimensionalize it first. I know that $$\...
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0answers
63 views

Contribution from $u$-channel and $t$-channel processes in OPE analysis for deep inelastic scattering

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.633 the moment sum rules for the deep inelastic form factors are discussed $$\int_0^1 dx x^{n-1}f_f^+(x,...
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73 views

Spring with oscillating support (Goldstein chapter 11, problem 2)

The problem: A point mass m hangs at one end of a vertically hung hooke-like spring of force constant k. The other end of the spring is oscillated up and down according to $z=a\cos(w_1t)$. By ...
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4answers
312 views

Why can we add counterterms?

I'm having a hard time understanding why renormalized perturbation theory works. Why is it permissible to add counterterms to the Lagrangian? Terms which are often divergent themselves and carry ...
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0answers
56 views

Time-dependent perturbation theory in a degenerate system

In the derivation of probability transition of time-dependent perturbation theory (see for example these notes, from Ben Simons from Cambridge University), I have only encountered treatments of non-...
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1answer
71 views

Second Order Correction to the Perturbative Approximation of the Transition Amplitudes in RQM

I am studying Relativistic Quantum Mechanics from my professor's notes. When calculating the second order perturbative correction to the transition coefficient $T_{fi}$* in a scattering process by a ...
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1answer
174 views

Proving the collapse of a many body system (Fetter and Walecka problem 1.2)

I was trying to solve the problem 1.2 from Quantum theory of many-body systems by A. Fetter and J. D. Walecka. I succeeded in the first part, obtaining the suggested formulation for the expectation ...
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1answer
117 views

The stark effect on ground state [closed]

When considering the Stark Effect, we consider the effect of an external uniform weak electric field which is directed along the positive $z$-axis, $\vec E = E \hat z$, on the ground state of a ...
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1answer
368 views

Why first-order Born Approximation doesn't satisfy optical theorem?

First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential: $$ f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
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2answers
296 views

Derive Effective Hamiltonian Directly Using Perturbation Theory?

I am struggling with the concept of deriving an effective Hamiltonian using perturbation theory. Say we have $$ \hat{H} = \hat{H}_0 + \hat{V} $$ and suppose we know the energies $E_n^{(0)}$ and ...
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1answer
36 views

How to choose a boundary layer coordinate or stretching transformation in matched asymptotic expansion

In matched asymptotic expansions how should one properly chose a boundary layer coordinate or stretching transformation. At the moment I am following example 2.3 from Introduction to Perturbation ...
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1answer
159 views

Need for normalization in non-degenerate perturbation theory

I'm currently taking a class in QM and we came across the topic of non-degenerate perturbation theory. Let us for further discussions assume that $H_0$ is the unperturbed Hamiltonian with solutions of ...
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1answer
80 views

Why should the $\phi^4$ term necessarily cause scattering while a $x^4$ term in anharmonic oscillator only causes correction of energy levels?

Consider an anharmonic oscillator in quantum mechanics, described by the Hamiltonian $$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2+bx^4.$$ The $bx^4$ term doesn't cause scattering. The effect of this ...
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1answer
104 views

Does the vanishing of the one-loop beta-function imply no running to all orders?

This question sounds ridiculous, but bear with me. I am having a hard time reconciling the following two facts: Classical global symmetries can become anomalous upon quantization, and the anomalous ...
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0answers
93 views

Problems on perturbation theory with spin

Problems on perturbation theory with spin (solve all perturbations up to only its first-order approximation): Taking into account the relativistic correction of electron kinetic energy and spin-orbit ...
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1answer
169 views

Why is the Fermi Golden rule called so?

I was studying time dependent perturbation theory and this rule came under the case of constant (weak) perturbations. I understood the rule and the derivation but what I cannot understand is that is ...
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0answers
51 views

Diffraction of electron from single slit using time dependent perturbation theory

While reading about time dependent perturbation theory, I began wondering if it was possible to obtain the diffraction pattern by treating the walls as an infinite potential barrier and the slit as a ...
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3answers
177 views

What does it mean for a perturbation expansion to break down?

For an anharmonic oscillator with Hamiltonian $H = {\hat p_x^2\over 2m}+{1\over 2}m\omega^2\hat x^2+b\hat x^4$ I found the first order shift in energy is: $$E_n^{(1)}={3 \hbar^2 b\over 4m^2\omega^2}\...
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2answers
81 views

In perturbation theory, why can perturbed eigenfunctions be expanded into the basis set of the unperturbed eigenfunctions?

So I am studying non-degenerate time independent perturbation theory and I came across the derivation of the first order correction to the wavefunction. So the notes given to me affirm that the each "...
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0answers
44 views

Why don't the corrections to energy disappear in perturbation theory?

The corrections to the wavefunctions and energies depend on $<\psi_m^0\,| \,H'|\psi_n^0>$ to some order. I would've thought that $<\psi_m^0\,| \,H'|\psi_n^0> \, =\, <H' \psi_m^0\,|\...
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1answer
424 views

Linearizing the Einstein-Hilbert action; shortcuts?

I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the ...
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2answers
120 views

Solving a 2x2 Perturbed Hamiltonian Exactly

Problem Consider Hamiltonian $H = H_0 + \lambda H'$ with $$ H_0 = \Bigg(\begin{matrix}E_+ & 0 \\ 0 & E_-\end{matrix}\Bigg) $$ $$ H' = \vec{n}\cdot\vec{\sigma} $$ for 3D Cartesian vector $\...
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Using relativistic QFT to compute energy levels

I've taken a year of QFT so far, and although there seems to be a lot of attention paid to scattering amplitudes and decay rates and perhaps bound states, I view computing energy spectra as certainly ...
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2answers
239 views

Can we show that the ground state of the He atom is a spin singlet rather than triplet?

The ground state of Helium atom is a state in which the space part of the wavefunction is symmetric and the spin part is antisymmetric under the exchange of the electrons. Therefore, the ground state ...
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1answer
114 views

Path integral formulation of the density matrix ρ

In Feynman's Statistical Mechanics - A Set of Lectures, upon the introduction of the path integral, a series of approximations are made in order to calculate integrals. I am unsure how exactly to get ...
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1answer
114 views

Gravitational attraction between quantum particles [closed]

Let's say we have a quantum particle with mass $m$ in a 1-Dimensional box. The potential outside the box is infinite. Say that $n=2$, so that $|\psi|^2$ will have two maxima. How would the ...
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0answers
33 views

How do we calculate the first order energy correction with total spin?

We're trying to calculate matrix elements for a perturbation theory problem. One element looks like (I've left off the $B_z$ field and some $ \hbar $ factors): $W_{ab} = <1 1 | (1 - (S_2)^2 + (S_1)...
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0answers
173 views

Feynman propagator for Dirac fields and $i\epsilon$ prescription for analytic continuation

The analytic continuation from Euclidean space to Minkowski spacetime is perturbatively well (uniquely) defined to all orders for the Feynman propagator for Dirac fields with the so called "$i\epsilon$...