Perturbation theory refers to methods for understanding physical systems by treating them as small modifications to exactly solvable systems.

learn more… | top users | synonyms

0
votes
0answers
24 views

First order time-independent perturbation theory: How to practically calculate the perturbed wave-function

This is one of the problems that draws the line between academically learning something, and having to use it. While I learned the formulas relevant to this, I just want to make sure I'm using them ...
7
votes
0answers
114 views

How to properly use Perturbation Theory in classical systems?

Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom: $$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$ We can solve it ...
3
votes
1answer
62 views

Relativistic correction to Hydrogen atom - Perturbation theory

Given the relativistic correction $$ H_1' = - \frac{p^4}{8m^3 c^2} $$ to the Hamiltonian (i.e. a perturbation), what does it mean when $[H_1', \mathbf{L}] = 0$? The book I'm reading says this implies ...
0
votes
1answer
36 views

Energy gap in phonons and violation of perturbation theory

In a 1 dimensional chain of similar ions which are connected to each other with similar springs there is just one corresponding frequency for each wave vector. But solving the problem of one ...
0
votes
1answer
46 views

Time independent perturbation theoery

Why do we talk of transitions only in time dependent perturbation theory when the eigen states are corrected even in time independent perturbation theory? If we can,for sake of argument,say : eigen ...
0
votes
2answers
56 views

Bending of Light in General Relativity using Perturbation

It is standard textbook calculation (e.g. Schutz's First Course in General Relativity page 294) that we can find a total angular change in light deflection due to gravity to be ...
0
votes
2answers
58 views

Time-evolution operator of a perturbed system

How do we evaluate the time-evolution operator of a perturbed system with time-independent perturbation ? For example: In a two state system acted up on by a time-independent perturbation, let's say ...
1
vote
0answers
49 views

The Hilbert space that contains the first order correction to the state vector in Time-independent Perturbation Theory

When deriving the expression for the first order correction to the state vector of the new hamiltonian( H = H0 + H' ) we assume that $|\psi$n1> = $\sum_{m \neq n}$ C$_m$(n) $|\psi ^0 _m>$ ...
11
votes
5answers
330 views

Can Feynman diagrams be used to represent any perturbation theory?

In Quantum Field Theory and Particle Physics we use Feynman diagrams. But e.g. in Schwartz's textbook and here it is shown that it applies to more general cases like general perturbation theory for ...
3
votes
1answer
96 views

Fermi's golden rule and infinite probablity?

I am slightly confused about the application of Fermi's golden rule. Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: ...
7
votes
2answers
161 views

Perturbation theory for a particle in a weak potential

I have a basic question about quantum mechanics, maybe it has a basic answer. Take a free particle in a quartic potential, $L=\frac{1}{2}\dot{x}^2-\lambda x^4$ This is massless $\phi^4$ theory in ...
0
votes
0answers
53 views

Perturbation Theory - 1D potential well

Consider an electron in a one-dimensional potential well of width Lz, with infinitely high barriers on either side, and in which the potential energy inside the potential well is parabolic, of the ...
0
votes
0answers
23 views

Nonequilibrium Green's functions weakly interacting two-component Bose gas

I am planing to describe time evolution of two-component BEC. I was thinking about non-equilibrium Green's functions, but I don't if the method can be applied to the problem describe below. I know ...
2
votes
1answer
101 views

Normal Modes for Standing Waves in 1-D Acoustic Ducts with Arbitrary (but real) Impedance Jumps

Let's say we have a 1-D duct, such as this: Where $Z_i \equiv \frac{P}{US}$ is the acoustic impedance, L is the length of the duct in question, and S is the area of the cross-section. In general, ...
0
votes
0answers
53 views

Particle creation through a time dependent Hamiltonian

We know that a time dependent Hamiltonian can create particles. We know this for instance from field theory in curved spacetime, where for instance in an expanding or contracting universe creation and ...
1
vote
1answer
61 views

Perturbation by electrical field in infinite potential well: difference in first energy corrections because of difference in the limits of the well

In time independent perturbation theory we can calculate the first and second energy corrections resulted by a potential V in the Hamiltonian $ H=H_o + λV , $ , λ<<, by the expressions: $$ε_1 = ...
1
vote
1answer
56 views

What are “the background equations” in cosmology?

We're currently working on perturbations within cosmology. There is something I have not heard before which has cropped up, that is: a reference to the term "the background equations". Are these just ...
0
votes
0answers
28 views

Book Recommendations on Perturbation Theory

I am interested in studying Quantum Electrodynamics and figure I should begin by learning Perturbation theory and Asymptotic expansions. If anyone could recommend some books, or starting points for ...
3
votes
1answer
68 views

Griffiths Intro to QM Section 9.1.2: What type of approximation is he using here and what is the justification for it?

I really do not understand Griffiths logic in this section and was wondering if someone could help. This is basically a 1st order coupled system of ordinary differential equations but I haven't seen ...
1
vote
1answer
48 views

Degenerate perturbation theory

I don't understand the part about turning off the perturbation. What is meant by or what is he referring to when he says "upper" and "lower" states. Why must the "upper" state reduce to a ...
3
votes
0answers
36 views

Why is stat mech involved in mean field theory?

Mean field theory involves approximating an interacting system by a non-interacting one, by replacing some operators with their expectation value. However, to my surprise, free energy is involved in ...
1
vote
1answer
34 views

Why are the perturbation term in Zeeman effect not diagonalized?

In the case of weak field Zeeman effect (anomalous Zeeman effect) in hydrogen atom, the unperturbed Hamiltonian reads as $$ H_0 = \frac{\hat{p}^2}{2m} + \frac{C_1}{r} + f(r)\mathbf{L}\cdot\mathbf{S} ...
2
votes
2answers
151 views

What are non-perturbative effects and how do we handle them?

Schwartz's QFT book contains the following passage. To be precise, total derivatives do not contribute to matrix elements in perturbation theory. The term $$\epsilon^{\mu\nu\alpha\beta} ...
2
votes
2answers
107 views

Higher orders in perturbation theory

I would like to compute an energy level up to many orders in perturbation theory. My difficulty right now is not in the calculation itself but in understanding the algebraic structure of the higher ...
0
votes
0answers
51 views

Is the Fermi golden rule really accurate for calculating the life time of an atomic level?

In my impression, Fermi golden rule is routinely used in calculating the life time of an excited atomic level. But it is based on the first order perturbation theory, so it is not expected to be ...
3
votes
2answers
446 views

Why are many physicists trying to develop non-perturbative quantum theories? [closed]

I would like to briefly know where (and why) does perturbation theory fail and why are physicists so desperate looking for non-perturbative theories.
1
vote
1answer
43 views

1st order perturbation of energy for quantum harmonic oscillator [closed]

I am trying to do part B of Griffiths QM 2nd edition problem 6.2 It asks to find the 1st order correction to the energy for a perturbation of a quantum harmonic oscillator where the new spring ...
2
votes
0answers
76 views

Relation between the reduced Green's function and the full Green's function

Let us assume that we have some Hamiltonian and we know its spectrum $$H_0 \psi_n = E_n \psi_n .$$ We define the Green's function in as $$ G(x,y,E) =\sum_m \frac{\psi_m^*(x)\psi_m(y)}{E-E_m}, $$ and ...
0
votes
0answers
40 views

Why drop the kinetic energy operator in Griffiths QM 2nd edition Example 6.1?

In Griffiths Quantum Mechanics Example 6.1 on page 252, the problem is just a simple square well where the potential floor is raised from zero to $V_0$. Thus he states that the Hamiltonian for this ...
0
votes
1answer
39 views

Expansion of wave function and energy in terms of small parameter

In time-independent perturbation theory, the Hamiltonian is perturbed with a perturbation of the form $\lambda V$, and the eigen-energies and wave-functions of the unperturbed Hamiltonian are ...
1
vote
1answer
64 views

Derivative with respect to perturbation in QM

The book "Introduction to Computational Chemistry" by Frank Jensen claims the following (Eq. 10.35): Let $H(\lambda) = H_0 + \lambda V$ be a Hamiltonian parametrized by a perturbation strength of ...
0
votes
1answer
27 views

Quadratic versus linear Stark shift

I'm trying to understand why the Stark shift changes from quadratic to linear as the applied electric field increases. I think there is some kind of connection to whether the induced dipole moment is ...
3
votes
1answer
93 views

Doubt in Dyson's argument about the divergent nature of the perturbative expansion in QED

I am trying to understand Dyson's argument about the divergent nature of the perturbative expansion in QED. Quoting his own words [...] let $$F(e^2)=a_0+a_1e^2+a_2e^4+\ldots$$ be a physical ...
7
votes
2answers
260 views

Why do we still use perturbation theory, when we have advanced numerical methods and fast computers?

If my question sounds ignorant or even insulting, I apologise. I may be completely wrong, since I'm not a theoretical physicist. So, I understand why perturbation theory was originally used in ...
0
votes
1answer
48 views

Time independent perturbation theory for a 1D simple harmonic oscillator system

I have been looking through my notes and it says in a footnote that the approximation of energy levels using perturbation theory is more accurate when the energy shift of the energy levels due to the ...
1
vote
2answers
158 views

Finite temperature correlation functions in QFT

Suppose that we want to calculate this imaginary time-ordered correlation function for an interacting system (in Heisenberg picture) : $$\langle \mathscr{T} A(\tau_A)B(\tau_B) \rangle =\frac{1}{Z} ...
0
votes
0answers
40 views

Radiative corrections and stability

What is meant by the terms radiatively stable and radiatively unstable? I know that when calculating physical observables in quantum field theory, such as the mass of the electron, to obtain a more ...
3
votes
1answer
45 views

Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
0
votes
1answer
64 views

Derivation of Hartree-Fock equations using 2nd quantization [closed]

I derived the following effective Hamiltonian: $$ H_{eff} = \sum_k{ \left( \, \epsilon_k + \sum_{k_2}{\left(<k \, |<k_2 \, |\,u\,| \, k_2>|\, k> - ...
0
votes
2answers
67 views

How does one get the first few terms of the S-matrix expansion?

According to a set of notes I'm reading $$\langle p_f | S | p_i \rangle = \delta(p_f-p_i) + 2 \pi \delta(E_f-E_i) \bigg[\langle p_f | V | p_i \rangle + \cdots\bigg] \tag{1.29}$$ I don't understand ...
0
votes
0answers
14 views

Rayleigh-Schrodinger perturbation of double-well

Might be a silly question but anyway. I know how to use the Rayleigh-Schrodinger method when the total Hamiltonian as $H=H_0 +H'$ where the first term is known and the second term is proportional to a ...
4
votes
0answers
65 views

Symmetry of interaction lagrangian and symmetry of full lagrangian

Suppose we have lagrangian $$ \tag 1 L = \frac{\theta}{f_{\gamma}}F_{EM}\tilde{F}_{EM} +\frac{1}{2}(\partial_{\mu}\theta)^2 - \frac{1}{2}m_{\theta}^2\theta^2 + L_{SM}, $$ where $\tilde{F}_{EM}$ ...
0
votes
0answers
150 views

Perturbation Theory Question

How can you work out the average perturbation, from a normal hamiltonian, of all states that rely on the quantum numbers of s = __ and l = __, with the perturbation being proportional to the product ...
2
votes
0answers
134 views

Number of distinct Feynman Diagrams for different orders of $\phi^4$ theory for 2 point function

There is 1 distinct Feynman diagram for zeroth order and 2 distinct diagrams for first order in $\phi^4$ theory for two point function. I want to know is there a way to predict the number of distinct ...
1
vote
2answers
77 views

Operators for a Perturbed Hamiltonian: Heisenberg Picture ($\hat{x}$, $\hat{p}$)

Problem I am trying to calculate the Equations of Motion in the Heisenberg picture for $\hat{x}$ and $\hat{p}$ in a perturbed Hamiltonian, $$ \tag{1} \hat{H} = \hat{H}_0 + \hat{H}' $$ Assume ...
0
votes
0answers
62 views

Perturbation theory of $\lambda q^4$ perturbed harmonic oscillator

For a perturbed Hamiltonian $$ H = H^{(0)} + H' $$ the perturbation theory approach $$ \Psi = \Psi^{(0)} + \lambda \Psi^{(1)} + ... \\ E = E^{(0)} + \lambda E^{(1)} + ... $$ leads to the equations ...
8
votes
1answer
425 views

Renormalization group resummation

I'm having trouble in understeanding a mathematical feature of RG, namely how it provides a way to resum the perturbation series and how that's defined mathematically. From a conceptual point of view ...
2
votes
2answers
211 views

What is the advantage of the canonical transformation when getting the effective hamiltonian?

Assume we have hamiltonian $$ H = H_0 + \lambda V$$ where $ H_0 $ is unperturbed hamiltonian which we know the eigenstastes, and $ V$ is a perturbation. In the effective hamiltonian approach using ...
2
votes
1answer
96 views

Perturbation Method in Mechanics: Average velocity of a small mass on a vibrating inclined plane [closed]

I've stumbled across this delightful and difficult collection of problems, by Jaan Kalda. The following problem has stumped me. (It's probem 16 on the sheet, which I have provided as a link) ...
1
vote
0answers
79 views

What is the exact meaning that QED perturbative series is only asymptotic and eventually diverges at very high orders?

When I read paper PRB89, 235431 about the effective field theory of graphene, there is a statement that QED perturbative series is only asymptotic and eventually diverges at very high orders (e. g. ...