Perturbation theory Definition and 238 Threads

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...

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...

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...

Please help me try to understand this problem. It deals with the quantum-confined Stark effect in nanoparticles. For odd n, n = 1, 3, 5, ... \psi_{n}(x) = \sqrt{\frac{2}{a}} \cos (\frac{n \pi x}{a}) and for even n = 2, 4, 6, ... \psi_{n}(x) = \sqrt{\frac{2}{a}} \sin (\frac{n \pi x}{a}) and...

Homework Statement I'm given that a harmonic oscillator is in a uniform gravitational field so that the potential energy is given by: V(x)=\frac{1}{2}m\omega^2x^2 - mgx, where the second term can be treated as a perturbation. I need to show that the first order correction to the energy of a...

Hi, I'm working out the 2nd Edition of Quantum Mechanics by Bransden & Joachain and I'm a little puzzled by the sign of the last term in equation 8.30 on page 380, which reads... a_{nl}^{(2)} = \frac{1}{E_n^{(0)} - E_l^{(0)}}\sum_{k{\neq}n} \frac{H_{lk}^{'}H_{kn}^{'}}{E_n^{(0)} - E_l^{(0)}}...

-Ok..Let,s be the Hamiltonian H=H_0 +W in one dimension where W is a "weak" term so we can apply perturbation theory. -The "problem" comes when we need to calculate the eigenvalues and eigenfunction of H0 of course we set the system in an "imaginary potential well of width L" so we have the...

A quick question from a high school student, In perturbation theory, what is to be done with the found energy correction? I'm working out the solution to an a/r+br potential and using br as the perturbation. I set up the integral and normalized, but what do I do with the expression that I'm...

if we know that the divergent series in perturbation theory of quantum field theory goes in the form: \sum_{n=0}^{\infty}a(n)g^{n}\epsilon^{-n} with \epsilon\rightarrow{0} then ..how would we apply the renormalization procedure to eliminate the divergences and obtain finite...

I'm reading the Cohen-Tannoudji book and I found somthing I don't understand in stationary perturbation theory. the problem the Hamiltonian is split in the known part an the perturbation: H=H_{o}+\lambda \hat{W} H_{o}|\varphi_{p}^{i}\rangle=E_{p}^{o}|\varphi_{p}^{i}\rangle (1) and...

Hi, I'm new to this subject, so bear with me. We consider the harmonic oscillator with a pertubation: \hat{H}' = \alpha\hat{p}. (What kind of a perturbation is that anyway, it's not a disturbance in the potential, what does it correspond to physically.) Now I have to calculate the...

I have been studying Perturbation theory in my Quantum class but my professor has not really explained why physically it comes into play. The book says that perturbation theory is used to help come up with approximate solutions to the Schrodinger Equation. Is this analagous to how we use Fourier...

Can anybody explain what Griffiths means when he talks about "good eigenstates" in degenerate time-independent purturbation theory? Mathematically, I know he is just talking about the eigen-vectors of the W matrix (where Wij = <pis_i|H'|psi_j>). But what do the eigen-vectors physically...

I was studying up on an aspect of perturbation theory, and I must have strained something (there's something about Hilbert spaces that I just can't get my head around...sorry, bad joke), because I have a really bad headache now. I was wondering what a headache is, and how we get them. I know...

For the harmonic oscillator V(x) = \frac{1}{2}kx^2, the allowed energies are E_n=(n+1/2)h \omega where \omega = \sqrt{k/m} is the classical frequency. Now suppose the spring constant increases slightly: k -> (1 + \epsilon)k. Calculate the first order perturbation in the energy. This is 6.2...

Hey there, I'm working on a perturbation theory problem, and I have no clue where to start in solving an infinite series. It's an infinite square well with a delta function potential in the centre and I'm trying to find the 2nd order energy correction to Energy En. Anyway, what I've got is...

I have a problem where I should calculate the ground state eigenfunction of a particle in the box where the potential V(x)=0 when 0<x<L and infinite everywhere else with the perturbation V'(x)=\epsilon when L/3<x<2L/3. I get that the total ground state eigenfunction with the first order...

can somebody help me to find an expression for the density contrast (in fouruer space; delta_k) in a moving frame. Basically I am trying to figure out how various quantities like power spectrum P(k) etc., will look in a uniformly moving frame .

we have a particle in an infinite one-dimensional square well potential [V(x)=0 for 0<x<L and V(x) is infinite otherwise] and introduce a small potential (perturbation) in the middle of the square well potential. Then the first order energy correction for the ground state is 100 times...