Perturbation theory Definition and 267 Threads

Hi everyone, I am doing a time dependent perturbation theory, in a case when the electron is prepared in a state of the continuous part of the energy spectrum. Existence of the discrete part and the degeneracy of the continuous part is irrelevant at the moment and will not be considered...

Homework Statement Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential $$ \phi(r) = \begin{cases} \frac{Ze}{R}(\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}) &\quad r<R\\ \frac{Ze}{r} &\quad r>R \\...

Hello guys, I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory. I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski...

Hello! I am reading Griffiths and I reached the Degenerate Time Independent Perturbation Theory. When calculating the first correction to the energy, he talks about "good" states, which are the orthogonal degenerate states to which the system returns, once the perturbation is gone. I understand...

What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.

Homework Statement The photon is normally assumed to have zero rest mass. If the photon did have a tiny mass, this would alter the potential energy the electron feels in the hydrogen atom (due to the Coulomb interaction with the proton). The potential then becomes yukawa potential...

Homework Statement I am identifying equations on the final exam equation sheet for my quantum II class. I've identified them all except this one, what I am guessing is a transition rate for some kind of emission or absorption of radiation case. Please help me identify the physical situation...

Hi I was hoping someone could advise me on a textbook/platform where I can learn more about the perturbation theory applied to helium and the perturbation theory time depedant. Thanks

As I understand it, in the context of cosmological perturbation theory, one expands the metric tensor around a background metric (in this case Minkowski spacetime) as $$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$$ where ##h_{\mu\nu}## is a metric tensor and ##\kappa <<1##. My question is, how...

Homework Statement Consider two real scalar fields \phi,\psi with masses m and \mu respectively interacting via the Hamiltonian \mathcal{H}_{\mathrm{int}}(x)=\dfrac{\lambda}{4}\phi^2(x)\psi^2(x). Using the definition of the S-matrix and Wick's contraction find the O(\lambda) contribution to...

I'm fairly new to QFT and I'm currently trying to understand perturbation theory on this context. As I understand it, when one does a perturbative expansion of the S-matrix and subsequently calculates the transition amplitude between two asymptotic states, each order in the perturbative...

Homework Statement The problem consists of 2 parts,the first one(I have done it) is on the following website: https://www.physicsforums.com/threads/transition-probability-from-two-states.804343/ Q1: I calculated the desired result p(t) = sin^2(Ut/h). However,I don't understand why <1,t | 2 >...

Homework Statement Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##. Homework Equations First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...

Homework Statement I have ##H'=ax^3+bx^4##, and wish to find the general perturbed wave-functions. Homework Equations First-order correction to the wave-function is given by, $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$ The...

Hello everyone, I am currently trying to understand how we can use feynman diagrams to estimate the matrix element of a process to be used in fermi's golden rule so that we can estimate decay rates. I am trying to learn by going through solved examples, but I am struggling to follow the logic...

Homework Statement A Hydrogen atom is interacting with an EM plane wave with vector potential $$\bar A(r,t)=A_0\hat e e^{i(\bar k \cdot \bar r -\omega t)} + c.c.$$ The perurbation to the Hamiltonian can be written considering the proton and electron separately as...

Why do we need perturbation theory in quantum mechanics?

Note this isn't actually a homework problem, I am working through my textbook making sure I understand the derivation of certain equations and have become stuck on one part of a derivation. 1. Homework Statement I am working through my text (Quantum Mechanics 2nd Edition by B.H Bransden & C.J...

I'm struggling to understand degenerate perturbation theory. It's clear that in this case the 'normal' approximation method fails completely seeing as you get a divide by zero. I follow the example for a two state system given in e.g D.J Griffiths "Introduction to Quantum Mechanics" However...

Homework Statement The ground state energy of the 1D harmonic oscillator with angular frequency ##\omega## is ##E_0 = \frac{\hbar \omega}{2}##. The angular frequency is perturbed by a small amount ##\delta \omega##. Use first order perturbation theory to estimate the ground state energy of the...

Homework Statement I have the particle in the infinite square well and need to calculate the first order correction energy and the wave function. L is the width and the potential is: 1/2 mw2x2 in the -L/2 < x < L/2 and infinity in x <= -L/2 and x>=L/2 Homework Equations H'=H-H0[/B] The...

Hi. I have been looking at some notes for time dependent perturbation theory. The equation for the transition probability involves the matrix element < f | H | i > where f is the final state , i is the initial state and H is the perturbation switched on at t=0. If H is a constant , ie. just a...

Homework Statement I am working on a physics project for which I need to use perturbation theory to calculate the first- and second-order corrections to the eigenvalues and eigenvectors of a perturbed matrix. The unperturbed matrix is real and symmetric, and the eigenvalues and eigenvectors are...

I'm reading section 5.2 "Time-Independent Perturbation Theory: The Degenerate Case" of the book "Modern Quantum Mechanics" by Sakurai and Napolitano and I have trouble with some parts of the calculations. At firsts he explains that there is a g-dimensional subspace(which he calls D) of...

Homework Statement Consider a quantum particle of mass m in one dimension in an infinite potential well , i.e V(x) = 0 for -a/2 < x < a/2 , and V(x) =∞ for |x| ≥ a/2 . A small perturbation V'(x) =2ε|x|/a , is added. The change in the ground state energy to O(ε) is: Homework Equations The...

Homework Statement I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive...

The following is taken from page 13 of Peskin and Schroeder. Any relativistic process cannot be assumed to be explained in terms of a single particle, since ##E=mc^{2}## allows for the creation of particle-antiparticle pairs. Even when there is not enough energy for pair creation, multiparticle...

In my course notes for atomic physics, looking at time independent perturbation for the non-degenerate case, we have the following: http://i.imgur.com/ao4ughk.png However I am confused about the equation 5.1.6. We know that < phi n | phi m > = 0 for n =/= m, so shouldn't this mean that < phi n...

Homework Statement Derive the transformations ##x \rightarrow \frac{x+vt}{\sqrt{1-v^{2}}}## and ##t \rightarrow \frac{t+vx}{\sqrt{1-v^{2}}}## in perturbation theory. Start with the Galilean transformation ##x \rightarrow x+vt##. Add a transformation ##t \rightarrow t + \delta t## and solve for...

Homework Statement "Suppose that a hydrogen atom, initially in its ground state, is placed in an oscillating electric field ##\mathcal{E}_0 \cos(\omega t) \mathbf{\hat{z}}##, with ##\hbar \omega \gg -13.6\text{eV}##. Calculate the rate of transitions to the continuum." Homework Equations ##R =...

Hey guys, I signed up here because I needed some information on some quantum physics problems. My question is related to quantum physics, and more precisely the derivation of time dependent perturbation theory. First of all, I am not able to understand all the maths structures and formulas...

Today, in my advanced particle physics class, the professor reminded the time-dependent perturbation theory in NRQM and derived the formula: ##\displaystyle \frac{da_m(t)}{dt}=-i \sum_n e^{-i(E_n-E_m)} \int_{\mathbb R^3}d^3 x \phi^*_m (\vec x) V(\vec x,t) \phi_n(\vec x)##. Then he said that...

I came across a technique called "multiple-scale analysis" https://en.wikipedia.org/wiki/Multiple-scale_analysis where the equation of motion involves a small parameter and it is possible to obtain an approximate solution in the time scale of $$\epsilon t$$. I am wondering if it is possible to...

I have solved this exercise, but I'm not sure that it is good. Please, can you check it? A lot of thanks! 1. Homework Statement The hamiltonian is ##H_0=\epsilon |1><1|+5/2 \epsilon (|2><2|+|3><3|)## The perturbation is given by ##\Delta(|2><3|+|3><2|)## Discuss the degeneration of H0. Using...

Hello everyone, thanks for reading I'll explain my question. At first, light was described as electromagnetic waves, until Einstein proposed the photoelectric effect and thus creating the concept of photon, a particle of light with momentum and energy, but no mass. It could explain why the...

Homework Statement I'm trying to understand how we can find - at the first order - the energy-shift and the eigenstates in a degenerate case. My notes aren't clear, so I have searched in the Sakurai, but the notation is different, I have read other notes but their notation is different...

Homework Statement Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you...

A. Neumaier submitted a new PF Insights post Causal Perturbation Theory Continue reading the Original PF Insights Post.

Hey there, i have a question regarding basic inflation and structure formation via linear first order perturbation theory in cosmology. I read through different material (Baumann lecture notes, wikipedia articles, Mukhanov, ...), but at this point i am just confused and find it hard to get an...

Hello! I just want to make sure that I have understood the following argument the correct way: For a given quantum system we take the hamiltonian to be a time-independent (and soluble) part, and a time-dependent part. ## \hat{H} = \hat{H_0} + H'(t) ## Now, the solutions to the unperturbed...

In QFT, we can expand the propagator and obtain the diagrammatic expansion to build up the Green's function. If we have a hamiltonian of the type H = H_{0}+V, where V is the perturbation, we can build up the Feynman diagrams,and if we could build up all of them to infinite order, we would...

Nonlinear sigma models are particular field theories in which the fields take values in some nontrivial manifold. In the simplest cases this is equivalent to saying that the fields appearing in the lagrangian are subject to a number of constraints. Since the lagrangian fields are not independent...

Homework Statement Hello, I'm just curious as to whether I'm going about solving the following problem correctly... Problem Statement: A particle mass m and charge q is in the ground state of a one -dimensional harmonic oscillator, the oscillator frequency is ω_o. An electric field ε_o is...

Consider some system in some initial state ##|k^{(0)}\rangle##. The probability that such a state makes a transition to some other state ##|m^{(0)}\rangle## can be computed to various orders in time dependent perturbation theory. E.g the total first order probability that the system has made a...

Homework Statement A 1-d harmonic oscillator of charge ##q## is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form ##E(t) = \frac{K }{\sqrt{\pi} \tau} \exp (−(t/\tau)^2) ##. Assuming that when ##t = -\infty##, the...

I was trying to learn renormalization in the context of ChPT using momentum-space cut-off regularization procedure at one-loop order using order of p^2 Lagrangian. So, 1. There are counter terms in ChPT of order of p^4 when calculating in one-loop order using Lagrangian of order p^2 . 2...

Homework Statement [/B] The isotropic harmonic oscillator in 2 dimensions is described by the Hamiltonian $$\hat H_0 = \sum_i \left\{\frac{\hat{p_i}^2}{ 2m} + \frac{1}{2} m\omega^2 \hat{q_i}^2 \right\} ,$$ for ##i = 1, 2 ## and has energy eigenvalues ##E_n = (n + 1)\hbar \omega \equiv (n_1 +...

i am currently self studying qm, and i am trying to plan ahead since i am relatively over with griffiths part1 (which is the theory part) and i was wondering if i should go ahead to part 2 (applications) or should i just keep this for later and attempt to stregnthen my basics in qm from another...

When we try to find the statistical correlation of some perturbation between two positions, we always calculate the quantum 2-point function. Are these two concepts really the same? Also, people say vacuum fluctuation is gaussian. For normalized fields, we always use Bunch-Davies initial...

Let's say we've a system which can be described by the Hamiltonian: $$H_0 = \dfrac{p^2}{2m} + V(x)$$ Now suppose we introduce a perturbation given by: $$H_1 = \lambda x^2$$ Our total hamiltonian: $$H = H_0 + H_1 = \dfrac{p^2}{2m} + V(x) + \lambda x^2 $$ Normally, the perturbation doesn't...