Perturbation Definition and 426 Threads

Homework Statement V(x) = \frac{1}{2}mw^{2}x^{2} + \lambdax^{4} Using first-order perturbation theory to calculate the energy shift of: 1. The ground state: \psi_{0}(x) = (2\pi\sigma)^{\frac{-1}{4}}\exp(\frac{-x^{2}}{4\sigma}) of the harmonic oscillator, where...

Hi everyone, I'm trying to work on Problem 5.32 from Sakurai's Modern Quantum Mechanics. In a nutshell, we need to use Perturbation Theory to find some of the energy levels of Positronium. Here's the full problem: http://img714.imageshack.us/img714/6515/sakurai532.gif Problem 3.3 isn't much...

Homework Statement The length of a pendulum is slowly doubled (l=l_0(1+epsilon*t), 0<=t<=1/epsilon). How does the amplitude q_max of the oscillations vary? Any hints? Homework Equations The Attempt at a Solution

Homework Statement Hi, i have put the question, my attempt and actual answer in the attached picture. My answer is not quite right; firstly why is the second term a minus lambda, and where does the O(lamdba^2) come from? Homework Equations The Attempt at a Solution

Hi, I am basically trying to put a wavefunction into the Time Dependant Schrodinger Eqn, as shown in my lecture notes, but i don't understand one of the steps taken... |\right \Psi (t)\rangle=\sum c_n (t) |\right u_n\rangle e^-(\frac{E_n t}{\hbar}) into i\hbar \frac{\delta}{\delta t}|\right...

The potential of an electron in the field of a nucleus is: -Ze^2/(4 Pi Epsilon0 r) r > r0 -Ze^2/(4 Pi Epsilon0 r0) r <= r0 where r0 is the fixed radius of the nucleus. What is the pertubation on the normal hydrogenic Hamiltonian? Calculate the change in energy of the 1s state to the first...

Homework Statement Calculate the 1st order probability an electron in the ground state of an infinite sqaure well (width 1) will be found in the first excited state t seconds after the pertubation H=sin(PI*x) is switched on. Homework Equations Transition frequency is omega_12 The Attempt at a...

Homework Statement 1. A particle of mass M is in a square well, subject to the potential: V(x)= V0\theta(x-a/2) for x in (0,a) and diverges elsewhere, where theta is heaviside step function. In perturbation theory, find O(V0^2) correction to the energy and O(V0)to the eigenstate. 2. A...

Hi, I am having some problems understanding this concept, I hope you can help. I studied on Hobson, Efstathiou and Lasenby, in chapter 16 on Inflationary cosmology that in cosmological perturbation theory we need to express quantities in a gauge invariant way, very clear so far. The problem...

Homework Statement Regard the nucleus of charge Ze as a sphere of radius R0 with uniform density. Assume that R0<<a0 where a0 is Boher radius/ 1. Derive an expression for the electrostatic potential V(r) between the nucleus and the electrons in the atom. If V0(r)=-Ze^2/r is the potential...

Hi all Please look at this link (Search for the phrase "The quantum state at each instant can be expressed as a linear combination of the eigenbasis"): http://en.wikipedia.org/wiki/Perturbation_theory_%28quantum_mechanics%29 If we write the wavefunction for the perturbed system as a...

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...

Is it possible to describe the measurement process in QM simply as a perturbation to the system (either a small one or a very large one) so that effectively the eigenstates of the Hamiltonian evolve into eigenstates of an additional measuring operator? So at t=0 the Hamiltonian H turns into...

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...

Is it possible to write down a statement about the asymptotic behaviour of the wave function after a small perturbation has been switched on? So I have and initial wave function \psi_0 and the Hamiltonian H=\begin{cases} H_0 & t<0\\ H_0+V+R & t\geq 0 \end{cases} where V is a small perturbation...

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)}...

Hi, I know that for degenerate states, we need to apply degenerate perturbation theory by looking at the perturbative hamiltonian in the subspace of the degenerate states. What then if the states still degenerate after we cast them the generate subspace. Is there a way to find the zero...

Am I correct in thinking string theory has an infinite number of terms so to prove finiteness to the first order means proving one (or the first) term to be finite? If so, then how can we ever prove an infinite number of terms? And what exactly does it mean to say, or prove, something is...

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...

Hi, I have been trying to model the temperature distribution along the length of a fin. With a constant 'h', the analytical solution is easy to get. But in my case, near the tip, the value of h changes significantly. Is perturbation a good technique to get a analytical solution in...

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...

Hi, there! I'm trying to understand the derivation of (linearized) pertubation equations given in my lecture. As usual the density and velocity fields are split into a homogeneous and an inhomogeneous part: \rho(t,x) = \rho_0(t) + \delta \rho(x,t) \vec{v}(t,x) = \vec{v_0}(t) +...

Homework Statement Find the perturbation approximation of the following in terms of powers of θ0. T=\sqrt{\frac{8L}{g}}\int^{\theta_0}_{0} \frac{d\theta}{\sqrt{cos\theta - cos\theta_0}} It is helpful to first perform the change of variable u = θ/θ0 in the integral Homework Equations...

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 I'm just working through a textbook and there's a line in which I'm clearly missing something. What I want to do is show that from: [tex] \bar{h^{TT}_{\mu \nu}} = A^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex] to [tex] h^{TT}_{\mu \nu} = B^{TT}_{\mu \nu} cos(\omega (t-z))...

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 Given: \mathbf{H}=V_0\begin{bmatrix} 1-\epsilon & 0 & 0\\ 0 & 1& \epsilon\\ 0 & \epsilon & 2 \end{bmatrix} \epsilon<<1 a) Find eigenvalues and eigenvectors of the unperturbed Hamiltonian (\epsilon=0) b) Solve for eigenvalues of the Perturbed Hamiltonian...

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, I'm a geologist, and I really suck in physics so I would need some help, please! This is not a homework question, I'm an academic! This is for my research... Let's say I have a "box" (a river catchment) with a mass, M, of material (sediments) in this box, which resides in the box for a...

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 "Suppose we put a delta-function in the center of the infinite square well: {H^{'}} = \alpha\delta(x-a/2) where a is a constant. Find the first order correction to the allowed energies. Explain why the energies are not peturbed for even n" Homework Equations The...

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...

Homework Statement A hydrogen atom is perturbed with the potential V(r) = \frac{\alpha}{r^{2}} (\alpha is small). Find first-order perturbation corrections to the energy levels and then exact levels of the perturbed system. Homework Equations The unperturbed hydrogen atom radial...

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...