Perturbation theory Definition and 267 Threads

Homework Statement Ok, so i have this online test to be completed by tomorrow and i have NO IDEA how to go about it, my notes are useless, they don't explain anything. On the up side all the questions seem to be on a very similar topic so if i could understand some key ideas then i should be...

Homework Statement 1. Considered the 2D harmonic oscillator potential, V(x,y) = m\omega^{2}x^{2}/2+m\omega^{2}y^{2}/2+ \lambda xy and showed that the energy eigenvalues could be found exactly. Now, treat this as a perturbation theory problem with perturbing Hamiltonian, H^{'}=\lambda xy...

Homework Statement Consider a quantum particle of mass m in a 3-D harnonic potential with frequency \omega and it experiences a perturbation H_{1}=az^{2} a. Determine the effect of H_{1} on the 1st exicted level of the system ( at the 1st order perturbation) b. what happen to L^{2} and...

a particle moves in one dimension in the potential V(x)=\infty \forall |x|>a, V(x)=V_0 \cos{\frac{\pi x}{2a}} \forall |x| \leq a now the unperturbed state that i use is just a standard infinite square well. anyway the solution says that perturbation theory is only valid provided that...

Hi together... When reading Sakurai's Modern Quantum Mechanics i found two problems in the chapter "Approximation Methods" in section "Time-Independent Perturbation Theory: Nondegenerate Case" First: The unperturbed Schrödinger equation reads H_0 | n^{(0)}\rangle=E_n^{(0)}...

Consider a system of a rigid rotator together with a uniform E-field directing along z-axis. So to calculate the perturbed energy and wavefunction we have to use perturbation theory. But the book said we can use non-degenerate one to calculate the result. I wonder why. It is because the original...

Dear All, I have recently read about WKB approximation and about perturbation theory. Both methods are applicable in the range of slowly varying potentials. What I have not understood is which is the range of applicability of one of the method compared with the other one. More...

Homework Statement Hi I am trying to apply degenerate perturbation theory to a three dimensional square well v= 0 for x, y,z interval 0 to a, perturbed by H' = xyz (product) from 0 to a, otherwise infinite. I need to find the correction to energy of the first excited state which I know is...

Homework Statement Why can't we use perturbation theory to calculate the effect of the spin orbit interaction in hydrogen like uranium? Homework Equations The Attempt at a Solution Is it something to do with the fact that the perturbation must be small compared to the rest of the...

Homework Statement Find the ground state energy of a particle restricted to move in one dimension subject to the potential in the attachement using perturbation theory. Homework Equations Yo = (2/a)1/2 sin(nπx/a)The Attempt at a Solution I'm not sure how to account for the potential since...

Homework Statement Material contains 10^{19}/cm^{3} Cr^{3\frac{1}{2}}. in the state \Psi(l=0, s = 3/2) with fourfold degenerate ground states. When a DC magnetic Field in x-direction is applied to the material, the spin degeneracy is lifted. At near Zero absolute temperature, only ground...

Homework Statement A particle of charge q and mass m is in a harmonic oscillator potential V0=0.5m(wx)^2. A perturbation is introduced which changes the potential to V=V0 + dV with dV=0.5sm(wx)^2 where s is small. Use perturbation theory to compute the first order shift in the ground state...

Homework Statement See attached. The problem is labeled "Peatross 1". Don't worry, it's short. I just didn't feel like retyping it.Homework Equations Included in attempt.The Attempt at a Solution I'm not sure if I am doing this correctly, but here it goes. I'll just do it for H'_{10}, since...

Hi all. I'm reading about time-independent perturbation theory for degenerate states in Griffiths' Introduction to QM. I have a question on the things he writes in chapter 6.2, page 269. What does he mean by the so-called "good" linear combinations? I hope you can shed some light on...

why in time independent degenerate perturbation we diagonalize the matrix of the perturbation part of the hamilitonian and not the original hamiltonian?

Homework Statement Assume that H0 describes a paramagnetic system that couples to a magnetic field via the Zeeman effect, i.e. V = −μB, where μ is the magnetic moment. Note that for the unperturbed paramagnetic system the probability of having an up-spin is equal to that for a down-spin. Show...

Homework Statement Given the Hamiltonian and perturbation below, what are the energy shifts for the states with l=1 Given H_{0}=(L^2)/(2I) H_{1}=E_{1}cos\vartheta Homework Equations L= r x P The Attempt at a Solution in order to find the first order correction to the energy...

I'm trying to follow some working by lecturer; Treating delK (previously found in first bit of question), show that the energy En of the usual hydrogenic state [nlm> is shifted by some expression given. basically we start with \[ \frac{1}{2m_{0}c^{2}} \left\langle...

Homework Statement Question: If a particle is in the ground state at time t<0, use the 1st order time dependent perturbation theory to calculate the probability that the particle will still be in the ground state at time t. Suppose we turn on the perturbation at time t=0 H(x) = ax...

Homework Statement A Hydrogen atom in its ground state (n,l,m) = (1,0,0) is placed in a weak electric fieldE(t) = 0 if t < 0 Eo *e^{\frac{-t}{\tau}} if t > 0E is in the positive z direction What is the probability that it will be found in any of the n=2 states at time t > 0 ? use...

I have been told before that virtual particles are just an artefact of perturbation theory, that if we could solve interacting fields exactly we would have no need to talk about virtual particles at all. My question then is if virtual particles are just a mathematical tool to evaluate...

I have an infinite potential well with length L. The first task was to calculate the eigenvalues and -functions for the energy of a particle in the well. The requirements were \psi(0, L) = 0 and there is no time-dependence. I've calculated: \hat{H}\psi(x) = E\psi(x) E =...

Hello, I have to learn about the classic Perturbation Theory. I'm looking for guides, textbooks etc about Perturbation Theory. I already know the basis (Poincare method), but I found it hard to find resources for more advanced material on the one hand, that will also teach it from basis on...

First of all, I'm sorry about the last topic, accidentally I switched between the previous message and this one... Sorry about the troubles. I think it's the right forum (after reading a bit), sorry if I'm wrong... My high school graduation project is about the perturbation theory, it's...

Hello, I have some trouble while trying to use the Poincare method in a free fall problem. There's some point on earth, that the vector R0 points at. from this point there is an orthonormal coordinate system, and some point of mass at (Rx, Ry, Rz). I found the expression for the sum of...

Homework Statement The Hamiltonian for a rigid diatomic molecule is H_0 = {L^2 \over {2I}} where I is the moment of inertia of the molecule. (a) What are the lowest four energy states of this system? (b) An external electric field is applied, leading to a perturbation H_1 = ED\cos\theta...

From the following attachments I understand how the roots of the equation and the perturbation coefficients were found. What I don't get is the solid line in the graph that is allegedly the plot of two of the three roots versus epsilon. Can somebody clear this up for me? Also, how would I...

"standard perturbation theory" - what exactly is meant? hi, could someone please help me out with the question in the title, in the following context: the quantization around trivial classical solutions can be done via the minkowskian path integral, while instanton solutions arise in the...

Hi , Can anybody help me to solve this question? A time varying Hamiltonian H(t) induces transitions from state |k> at time t=0 to a state |j> at time t=t' with probability P(k to j(t')). Use first order time-dependent peturbation theory to show that if P(j to k(t')) is the prababilty that...

Homework Statement We consider two spins 1/2, \vec{S_{1}} and \vec{S_{2}}, coupled by an interaction of the form H=\alpha(t)\vec{S_{1}}*\vec{S_{2}}. \alpha(t) is a function of time who approches 0 for |t|-->infinity and takes appreciable values only in the interval of [-\tau,\tau] near 0...

Hello, This is a question on perturbation theory - which I am trying to apply to the following example. Homework Statement The two-dimensional infinitely deep square well (with sides at x=0,a; y=0,a) is perturbed by the potential V(x)=\alpha(x^{2}+y^{2}). What is the first-order correction...

Hi, i am stuck at this problem , let be the divergent quantity m= clog(\epsilon) +a_{0}+a_{1}g\epsilon ^{-1}+a_{2}g\epsilon ^{-2} +a_{3}g\epsilon ^{-3}+...+ where epsilon tends to 0 and g is just some coupling constant and c ,a_n are real numbers. then i use the Borel transform of the...

I'm looking at the beginning of of Chapter 6 of the 2nd edition of Griffiths Introduction to Quantum Mechanics. He starts out by writing the hamiltonian for a system we'd like to solve as the sum of a hamiltonian with a known solution and a small perturbation: H^0 + \lambda H^\prime He...

Homework Statement I'm trying to calculate the energy shift given an electron in a 1D harmonic potential has a wavefunction \Psi_{0}(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}exp\left(\frac{-m\omega x^{2}}{2\hbar}\right) The shift in E_{0} = \frac{\hbar\omega}{2} = 2eV due to...

I'm trying to bridge the gap between several expressions describing the insertion of a constant perturbation: a_{f}(t) = \frac{1}{i\hbar} V_{fi} \int^{t}_{0} e^{i(E_{f}-E_{i})t'/\hbar}dt' = \frac{1}{i\hbar}V_{fi}\frac{e^{i(E_{f}-E_{i})t/\hbar} - 1}{i(E_{f}-E_{i})/\hbar} so...

[SOLVED] time-dependent perturbation theory Homework Statement My book uses time-dependent perturbation theory to derive the following expression for the transition of \psi_{100} to \psi_{210} in the hydrogen atom in a uniform magnetic field with magnitude \mathcal{E} \frac{131072}{59049}...

Homework Statement In each of my QM books, they always say something like "we can write the perturbed energies and wavefunctions as" E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots |n\rangle = |n^{(0)}\rangle + \lambda |n^{(1)}\rangle + \lambda^2 |n^{(2)}\rangle + \cdots...

I've been reading a paper at the following link: www.cims.nyu.edu/~eve2/reg_pert.pdf I have several questions: In the first example they use the method to approximate the roots for x^2 - 1 = "epsilon" x I was under the impression - wrongly perhaps - that f(x) had to have...

while I`m reading the griffiths` textbook.. got my curiosity from "Typically, the diagonal matrix elements of H` vanish" i.e. <a|H`|a>=0 in general.. If V(x) does not have an angular dependence.. the selection rule implies <a|H`|a>=0 (since Δl=0)..yes.. but what if it does...

Homework Statement An electron is inside a magnetic field oriented in the z-direction. No measurement of the electron has been made. A magnetic field in the x-direction is now switched on. Calculate the first-order change in the energy levels as a result of this perturbation. The Attempt...

Homework Statement Determine approximately the ground state energy of a helium like atom using first order perturbation theory in the electron-electron interaction. Ignore the spins of the electrons and the Pauli principle. Homework Equations given that \intd\tau1\intd\tau2...

Homework Statement Initially, you have a one dimensional square well potential with infinitely high potential fixed at x = 0 and x = a. In the lowest energy state, the wave function is proportional to sin (kx). If the potential is altered slightly by introducing a small bulge(symmetric about...

H=H0 + lambda * W lambda << 1 must hold and the matrix elements of W are comparable in magnitude to those of H0. More precisely, the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0. I don't understand what is the meaning of " the matrix...

H=H0 + lambda * W lambda << 1 must hold and the matrix elements of W are comparable in magnitude to those of H0. More precisely, the matrix elements of W are of the same magnitude as the difference between the eigenvalues of H0. I don't understand what is the meaning of " the matrix...

for a Hamiltonian H=H_0 + \epsilon V(x) my question is (for small epsilon) can WKB and perturbative approach give very different solutions ?? to energies eigenvalues and so on the index '0' means that is the Hamiltonian of a free particle. problem arises perhaps in calculation of...

This's a question from Griffiths, about degenerate pertrubation theory: For \alpha=0, \beta=1 for instance, eq. 6.23 doesn't tell anything at all! What does it mean "determined up to normalization"?. Equations 6.21 and 6.23 involve 3 unknowns (\alpha, \beta, E^1), and Griffiths solved them...

We discussed this problem in class to some extent, and I'd just like to post it here so that I can continue the discussion on the conceptual physics of it as well as the algebra. I believe a lot can be learned from this problem. "When an atom is placed in a uniform external electric field...

For a particle in a two-dimensional box. The particle is subject to perturbation V=Cxy. What are the eigenenergies and eigenfunctions of the unperturbed system and what is the first-order energy correction?

Note that the post is long but only because I wanted to make the content cristal clear. The same post could easily have been 10 lines long. Homework Statement A spinless particle of charge q is in a spherically symetric potentiel V(r). The energy levels depend on l but not on m_l. The system...

Let's suppose we have a theory with Lagrangian: \mathcal L_{0} + gV(\phi) where the L0 is a quadratic Lagrangian in the fields then we could calculate 'exactly' the functional integral: \int\mathcal D[ \phi ]exp(iS_{0}[\phi]/\hbar+gV(\phi)) where J(x) is a source then we could...