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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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72 views

Zeeman effect in a magnetic field with multiple components

For calculating weak-field Zeeman energy shifts, the literature tends to assume the perturbing magnetic field is in the Z-direction; they choose their alignment such that this is true. But what ...
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79 views

A question from cosmological perturbation theory

We consider the following scalar perturbation on the FRW metric $$ds^2=-(1+2\Phi)dt^2+2a(\partial_iB)dx^idt+a^2[(1-2\Psi)\delta_{ij}+2\partial_{ij}E]dx^idx^j,$$ where $\Phi$, $B$, $\Psi$ and $E$ are ...
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124 views

Symmetries in degenerate perturbation problems

In Griffiths' degenerate perturbation chapter, he mentions how finding symmetries of the original and perturbing Hamiltonians can simplify the process of first order degenerate perturbations. The ...
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1answer
294 views

Griffiths's take on degenerate perturbation theory [duplicate]

I've searched around here and it seems I'm not alone with my frustration towards Griffith's explanation of degenerate perturbation theory. Right from the start, Griffiths claims that some perturbation ...
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1answer
123 views

Is it possible to tell whether a potential is unbounded using only perturbation theory?

A very common inverse problem in mathematical physics is trying to understand the potential of a quantum mechanical system given its scattering data. Such problems, although very interesting, are very ...
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1answer
156 views

How to path-integrate over the half-line?

Consider the path-integral over a scalar field $\varphi$: $$ Z=\int_{\mathcal S}\ \mathrm e^{iS[\varphi]}\mathrm d\varphi $$ where $\mathcal S$ is some function space (say, Schwartz or its dual). How ...
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62 views

Screened Coulomb Potential effect on Hydrogen [duplicate]

How does a screened coulomb potential affect the ground state energy of hydrogen atom? Just by looking at the perturbed potential, can we make a prediction about the energy shift? The screened ...
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4answers
481 views

Perturbation Theory and Thermodynamic Limit

Suppose we have a classical Hamiltonian that can be divided into an “easy” part $H_0$ and a “difficult” part $\Delta H$ that depends on a parameter $g$: \begin{equation} H = H_0 + g \Delta H ~. \end{...
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1answer
454 views

Why do we care about old-style, counterterm renormalizability?

There are a few different definitions of renormalizability that are standard in quantum field theory textbooks. They're all called the same thing, but I'll make up names to make the distinctions clear....
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146 views

Time dependent perturbation theory

I am given that a system is defined by two independent states |+> defined as: \begin{bmatrix}1 \\ 0\end{bmatrix} and |-> defined as: \begin{bmatrix}0 \\1\end{bmatrix} With respect to these ...
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143 views

Energy density of primordial gravitational waves- two different results for the same quantity?

I want to calculate the energy density of primordial gravitational waves, $Ω_{gw}$ in a specific time. The latter is in general given by the relation : $Ω_{gw}(k,τ)= \frac{1}{\rho_{c}(τ)} \frac{d\rho_{...
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62 views

Perturbation theory around solvable but non-free hamiltonian

I have a Hamiltonian $H=H_0+H_1.$ $H_0$ is completely solvable; I know its ground state wave-funtion, Green function,etc. If $H_0$ was furthermore a free hamiltonian, standar diagrammatic ...
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1answer
225 views

Time-dependent perturbation theory derivation with 2-level system (Griffiths)

I'm reviewing time-dependent perturbation theory (TDPT) via Griffiths QM book. I'm looking at section 9.12: he reviews the results for a two-level system where the system starts in one eigenstate, ie. ...
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1answer
57 views

Why is it intuitively unreasonable for this transition probability to grow quadratically in $t$?

In Sakurai's "Modern Quantum Mechanics" section 5.6, there is a seemingly simple statement made that I do not understand the logic of. The author is considering a physical situation in which we "turn-...
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2answers
446 views

Perturbation theory with infinite potential

I'm trying to solve an excercise that involves first order perturbation theory and an infinite potential. To ease the problem, I tried to consider an easier one dimensional model. Consider an infinite ...
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1answer
62 views

Sinusoidal drive of two level system: why can we ignore one of the two terms?

I am studying the time-dependent perturbation theory from Griffith's Introduction to Quantum Mechanics. The context is a two-level system under a sinusoidal perturbation: $H'(\textbf{r}, t) = V(\...
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1answer
249 views

Loop counting — what if the graph is not planar?

It is typically claimed that $\hbar$ counts the number of loops in a connected diagram. E.g., Weinberg's QFT, Vol.II, equation 16.1.10. This rests on the fact that for a diagram with $I$ internal ...
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42 views

Are there situations where a Hamiltonian matrix is ill-conditioned?

By ill-conditioned I mean it in the mathematical sense that the condition number of the Hamiltonian is large. One interesting note is that the condition number of a matrix can be written in terms of ...
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1answer
422 views

Beta function in QFT renormalization group

In order to know how the coupling constant depends on the energy scales,it is necessary to know the Beta function.Normally the Beta function is is calculated perturbatively. Now, my question is this: ...
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1answer
52 views

Why does perturbating spins in the x axis not remove a degeneracy?

Suppose we have a Hamiltonian proportional to two spin operators in the z axis: $$ H_0 = (\vec{s}_{1} +\vec{s}_{2})^2 $$ Now suppose I have a perturbation proportional to a different component of ...
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80 views

Relationship between Keldysh formalism and time-dependent perturbation theory

When treating systems in weak external time-dependent electromagnetic field we can use usual time-dependent perturbation theory or the Keldysh formalism which is tailored for such non-equilibrium ...
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1answer
111 views

Linear response correlation functions

I'm working my way through Methods of Molecular Quantum Mechanics by R. McWeeny and have run into a derivation I can't seem to figure out. So in chapter 12, he obtains an expression for the first ...
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114 views

Electric field perturbation question

I've been practising some nice examples of electric field perturbations (aka the numbers always come out nice), so I created my own variation. This created some problems, and I would love some help. ...
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166 views

Spectral decomposition and a harmonic osillator [closed]

A system is described by a Hamiltonian $$H^0=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2.$$ A perturbation in the form of $$H'=\lambda \frac{4m^2\omega^2}{\hbar}x^4$$ is applied. I showed that $H'=\hbar\...
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1answer
120 views

How can I go into the interaction picture when it is difficult/impossible to exponentiate the operator?

I have a Hamiltonian of the form $H = H_0(t) + V(t)$ and wish to go into the interaction picture with $H_0(t)$. Normally this would be done by defining an evolution operator $U(t) = e^{-iH_0t/\hbar}$ ...
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Basic questions on the PPN formalism in General Relativity

I'm trying to learn about testing modified gravity using the PPN formalism. I have several textbooks that I am reading through (including Clifford Will's book), and have some basic questions on the ...
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1answer
432 views

Degenerate eigenstates and related eigenvalues in perturbation theory

I am studying time-independent perturbation theory in quantum mechanics and I found myself not understanding how to associate degenerate eigenstates their specific Hamiltonian eigenvalue correction. ...
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1answer
248 views

Some questions about perturbation theory in QM

If I want to calculate perturbed energy states in 2-fold degenerate case in Quantum Mechanics. Assume the Hamiltonian is $ H=H_0+H^\prime$ Then we can calculate matrix elements of W : $ \langle a|H^...
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268 views

Perturbations around backgrounds with no symmetries

It is well-known that in perturbation theory around an FLRW spacetime, one can decompose any perturbation in terms of scalars, divergence-free vectors and a traceless symmetric tensor, known as the ...
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Renormalization group approach to renormalization

Given a $n$-point bare Green function in a massless asymptotically free theory, we have that the following limit exists and is finite \begin{equation} \lim_{\Lambda\rightarrow\infty} Z^{-n/2}(g_0,\...
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1answer
107 views

Degeneracy split in $n=2$ hydrogen due to uniform electric field. Why is main diagonal zero?

I was checking about the stark effect in the hdyrogen in this site for $n=2$. Now, there are four states to deal with, $\psi_{200}, \psi_{21-1}, \psi_{210}, \psi_{211}$. The perturbation hamiltonian $...
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2answers
732 views

Why are derivatives in interaction terms treated differently from derivatives in the kinetic term?

I know that derivative couplings in a Lagrangian interaction, such as $$\mathcal{L}_{int} = \lambda \phi (\partial_{\mu}\phi)(\partial^{\mu}\phi)$$ bring down two momentum factors into the matrix ...
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42 views

In an acoustic wave, what are the first order and second order perturbations?

From a reading of several articles like the one by Settness et al [1], there is an equation describing pressure and velocity fields to be combination of first order and second order terms. What do ...
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2answers
506 views

Asymptotic series in QFT

In QFT is said that the renormalized Dyson series is only asymptotic. But to be able to say it is necessary to know to what function of $g$ (the coupling constant) the Dyson series is asymptotic. ...
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48 views

Density of States and Quantum Jumps

The specific question that I'm working on is "If I have a particle in the bound state of a 1-D delta function potential at $t = - \infty$, and I apply a harmonic perturbation $V(x,t) = V_0xcos(\omega ...
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1answer
380 views

A small error in Landau & Lifschitz “Mechanics” (3rd ed.)?

I think I found a small error in Landau & Lifschitz "Mechanics" (3rd ed.). In section 28 (Anharmonic oscillations), they are discussing how to solve the following anharmonic oscillator problem: ...
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1answer
292 views

Why perturbation theory in Fourier space works?

I am trying to learn some statistical field theory topics having pure math background and I cannot understand some things. I would formulate the question for time-dependent fields but I guess it makes ...
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1answer
419 views

Perturbations in FRW metric

Considering the FRW metric with perturbations; how can I calculate the Einstein tensor (without a very very disgusting expression which comes from the variation of the difference between the Ricci ...
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1answer
54 views

What is the form of the $n$-th order term of the perturbation series of an eigenvalue?

Suppose I have a matrix given by a sum $A=D+\epsilon B$, where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as power series in $\epsilon$. The leading order is just the ...
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1answer
137 views

Feynman diagrams for eigenvalue perturbation theory

I posted this question in MathOverflow but was not lucky with the answers, so wil try here. Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, ...
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1answer
155 views

Degenerate perturbation theory: typo in Sakurai: Eq. 5.2.15

Book: Modern Quantum Mechanics (Revised edition): J J Sakurai In the equation 5.2.15 on the RHS, should the symbol $\epsilon$ be replaced with the 'not an element of' symbol?
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427 views

Degenerate perturbation theory: Sakurai's statement does not make sense

Book: Modern Quantum Mechanics (Revised edition): J J Sakurai After Eq. 5.2.15, Sakurai summarizes the recipe to treat degenerate perturbation theory in four bullet points. The last two bullet points ...
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1answer
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How to find energy levels with approx up to $\lambda^2$?

The problem is from stationary perturbation theory. Given a Hamiltonian $H = H_0 + \lambda V$ where $\lambda \ll 1$, $H_0 = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$, $a = b$, $V = \begin{...
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2answers
434 views

Non-integrable Wavefunctions

Suppose first-order perturbation yields a credible correction to the energy, but a correction to the wave function that's not square-integrable. That can happen, I see no reason why it couldn't. ...
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3answers
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Does QED really break down at the Landau pole?

In QED, the fine structure constant $\alpha$ runs upwards in the UV, with a loop calculation (involving a geometric series of the vacuum polarisation diagram) indicating a divergence in $\alpha$ at $\...
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Projection operators in a problem about the Hubbard model

I am attending a course in Statistical Field theory. So far, we have covered little more than field operators and second quantization. Exercise series are used to introduce topics that we do not ...
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137 views

How to perturb a spacetime?

Suppose we have some vacuum spacetime, described by the metric $g_{\mu \nu}$ which is a solution to Einstein's field equation $G_{\mu \nu} = 0$. For concreteness, let's just suppose I have some M = ...
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762 views

Question about convergence of Dyson Series. Why Dyson series is in general divergent?

Given operator equation like: $$i{\frac d{dt}}U(t,t_{0}) =V_I(t)U(t,t_{0})\tag{1} $$ The Dyson series solution is \begin{array}{lcl}U(t,t_{0})&=&1-i\int _{{t_{0}}}^{{t}}{dt_{1}V_I(t_{1})}+(-...
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1answer
147 views

Analytical continuation of 2,3,4-point integrals

I was reading a paper that gives a nice collection of all scalar integrals that crop up in QCD loop calculations. Such integrals are computed in some kinematic region and then the authors say the ...
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1answer
2k views

Transition probability: Sudden approximation if the perturbation is large

I am trying to solve a problem where a system (a quantum harmonic oscillator) is suddenly perturbed by a large field of strength $E$. I want to calculate the transition probability for it to go from ...