Perturbation Definition and 426 Threads

So I'm trying to solve old qualifying exam problems, one of which is a particle on a ring with a constant electric field perturbation. The un-perturbed problem is straightforward, and we then add a constant electric field in the x-direction (the ring lies in the xy-plane) of magnitude E...

Homework Statement A two-dimensional isotropic harmonic oscillator of mass μ has an energy of 2hω. It experiments a perturbation V = xy. What are its energies and eigenkets to first order? Homework Equations The energy operator / Hamiltonian: H = -h²/2μ(Px² + Py²) + μω(x² + y²) The...

Homework Statement We have a system whose state can always be expressed as the sum of two states ##\Psi_a## and ##\Psi_b##. the system undergoes a perturbation of the form ##H'=U\delta(t)##, where ##\delta## is the delta-function in time and ##U_{aa} = U_{bb} = 0## and ## U_{ab} = U_{ba}^*##...

Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...

Hello, I am reading the book, The Quantum Theory of Fields II by Weinberg. In page 426 of this book (about soliton, domain wall stuffs), we have Eq(23.1.5) as the solution that minimizes Eq(23.1.3). The paragraph below Eq(23.1.5), the author said "The advantage of the derivation based on...

Homework Statement Part (b): Find the perturbed energy. Homework Equations The Attempt at a Solution I've solved everything, except part (b). I got an answer of 0 for part (b) for all orders, which is kind of strange, as one would expect some perturbation. \Delta E_n = \langle \psi_n...

Hi, So, I am working through section 5.2 of Sakurai's book which is "Time Independent Perturbation Theory: The Degenerate Case", and I see a few equations I'm having some trouble reconciling with probably because of notation. These are equations 5.2.3, 5.2.4, 5.2.5 and 5.2.7. First, we...

In a text a exercice says that for the Hamiltonian ##H_0 = \frac{p^2}{2m}+V(x)## the eigenfunction and eigen energy are ##\phi_n, E_n##. If we add the perturbation ## \frac{\lambda}{m}p## ¿what is the new eigenfunction? The solution is ## \frac{p^2}{2m} + \frac{\lambda}{m}p+V=...

Homework Statement Part (a): Find eigenvalues of X, show general relation of X and show X commutes with KE. Part (b): Give conditions on V1, V2 and VI for X to commute with them. Part (c): Write symmetric and antisymmetric wavefunctions. Find energies JD and JE. Part (d): How are...

Homework Statement Part (a): Explain origin of each term in Hamiltonian. What does n, l, m mean? Part (b): Identify which matrix elements are non-zero Part (c): Applying small perturbation, find non-zero matrix elements Part (d): Find combinations of n=2 states and calculate change in...

Homework Statement Part (a): Particle originally sits in well V(x) = 0 for 0 < x < a, V = ∞ elsewhere. The well suddenly doubles in length to 2a. What's the probability of the particle staying in its ground state? Part (b): What is the duration of time that the change occur, for the...

Homework Statement Homework Equations The Attempt at a Solution With a parity operator, Px = -x implies x has odd parity while Px = x implies x has even parity. Things that puzzle me 1. Why is ##[H_0,P] = 0## and ##H_1P = -PH_1##? Is it because ##H_1 \propto z## so ##Pz = -z##? Then...

Homework Statement Two identical spin-1/2 particles interact with Hamiltonian H0=ω0 S1.S2 where ω0>0. A time dependent perturbation is applied, H'=ω1 (S1z-S2z) θ(t) Exp[-t/τ], where ω1>0 and ω1<<ω0. What are the probabilities that a system starting in the ground state will be excited into each...

My study of Quantum Mechanics have brought me to perturbation theory. I'm here talking about the non-degenerate type. My questions relate to the math behind it, and the power series expansion that we do. H = H^0 + \lambda H' (Eq. 1) Question 1: So in equation 1 I think I understand...

After time t, the probability of monochromatic absorption of the ground state |1> to the energy state |n> is given by: |<n|1>|^2=4|U_{n1}|^2\frac{\sin^2((E_n-E_1-\hbar\omega)t/2\hbar)}{(E_n-E_1-\hbar\omega)^2} where U is the transition matrix. The claim is that as t goes to infinity, the...

Hello Everyone. I am very confused on the following questions and have a few confusions about the problem that I hope someone can clear up for me (explained later). Here is the question. Homework Statement The paramagnetic resonance of a paramagnetic ion in a crystal lattice is described...

So I know this might be a lot to read but I am having a very hard time understanding how to use the formulas in degenerate perturbation theory. Here is the problem I am on. Homework Statement A system of two spin-1/2 particles is described by the following Hamiltonian...

Homework Statement Problem in question is problem 5.6 in Dodelson's Modern Cosmology (https://www.amazon.com/dp/0122191412/?tag=pfamazon01-20) Take the Newtonian limit of Einstein's equations. Combine the time-time equation (5.27) with the time-space equations of exercise 5 to obtain the...

Homework Statement Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state Homework Equations ψn(x) = \frac{α}{√π*2n*n!}1/2 *...

http://farside.ph.utexas.edu/teaching/qm/lectures/node53.html So I was reading this and I don't understand how he goes from 658 to 661 using the completeness relation. In 661 if you use the completeness relaton can you get rid of the I n,l''>s by doing the outer product and ignoring the...

In the theory of degenerate perturbation in Sakurai’s textbook, Modern Quantum Mechanics Chapter 5, the perturbed Hamiltonian is H|l\rangle=(H_0 +\lambda V) |l\rangle =E|l\rangle which is written as 0=(E-H_0-\lambda V) |l\rangle (the formula (5.2.2)). By projecting P_1 from the left (P_1=1-P_0...

Greetings, Does anyone know of some good sources that explain classical perturbation theory, preferably using the Lagrangian formalism? The sources that I have seen more-or-less say, "write L=L_{0}+λδL, where L_{0} is an unperturbed, soluble Lagrangian, δL is the perturbation, and λ is a small...

Homework Statement Consider a particle confined in a cubical box with the sides of length L each. Obtain the general solution to the eigenvalues and the corresponding eigenfunctions. Compute the degeneracy of the first excited state. A perturbation is applied having the form H' = V from 0...

Why do we often treat the electromagnetic radiation effects on Hamiltonian of a matter as a perturbation? In the other words, why the effects of radiation is so little that is treated as a perturbation?

Homework Statement I need to find the corrected ground state for perturbed harmonic oscillator (1D) with perturbation of the form V(x) = λ sin(κx), κ>0. My problem is I have no idea how to handle a potential that has its operator as an inner function. Homework Equations The perturbed...

I have found an expression for the estimated energy contribution a term |I> will bring to a wavefunction |K> \Delta E = \frac{|\langle I|\hat{H}| K\rangle|^2}{(E_K - \langle I |\hat{H}| I\rangle)} Is there a simple way to extract the coefficient that will be associated with |I>? Even a link...

Hello all, I have boiled a very long physics problem down to the point that I need to solve the coupled equations \frac{\partial^2 x}{\partial u^2} + xf(u) + yg(u) = 0 \frac{\partial^2 y}{\partial u^2} + yf(u) - xg(u) = 0 We may assume that |f| ,|g| << 1. and that both f and g are...

Homework Statement Consider a molecule with an electric dipole moment d. The Hamiltonian of a molecule in the external electric field E is: \hat{H} = \frac{\hat{L^2}}{2I} - dE \cos{\theta}, where the polar angle \theta characterises the orientation of the molecule. (We have chosen the field...

Hi there. I'm dealing with this problem, which says: At time ##t=0## a constant and uniform electric field ##\vec E## oriented in the ##\vec x## direction is applied over a charged particle with charge ##+q##. This same particle is under the influence of an harmonic potential...

Hi there. I have to find the energy corrections through the perturbation method, and then give the exact result for the hamiltonian: ##H= \begin{pmatrix} E_A & \epsilon & \epsilon & \epsilon \\ \epsilon & E_B & 0 & 0 \\ \epsilon & 0 & E_B & 0\\ \epsilon & 0 & 0 & E_B \\ \end{pmatrix} ## So...

Homework Statement Assume that there is a deviation from Coulomb’s law at very small distances, the Coulomb potential energy between an electron and proton is given by V_{mod}(r)=\begin{cases} -\frac{e^{2}}{4\pi\varepsilon_{0}}\frac{b}{r^{2}} & 0<r\leq b\\...

Homework Statement I required to make a perturbation expansion in ε of the function: Homework Equations A(X,y,z)=A(x-εsin(wy),y,z). X=x-εsin(wy) The Attempt at a Solution Solution: A(X,y,z)=A0(X,z)+ε[A1(X,z)+∂/∂XA0(X,z)]sin(wy)+o(ε^2) I get the terms A0(X,z) and ∂/∂XA0(X,z)sin(w,y) with the...

In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode y'' + P(x)y = 0 using perturbation theory. However he points out that the domain must be finite in order for this to work, I'm wondering how you would phrase a question...

Suppose we know the perturbation series E = E(\epsilon) = E_0 + \epsilon E_1 + \epsilon^2 E_2 + \ldots converges, where E_0 is a discrete eigenvalue of H_0 and we are considering a Hamiltonian H = H_0 + \epsilon H_1. Does this mean that we know E - E_0 = O(\epsilon) as...

Sharks use an organ known has the ampullae of Lorenzini to sense electric fields, they can sense fields as weak as 5*10^-9 V/m. Which got me thinking, they should in principle be able to detect vacuum perturbation in the field, interesting?

Homework Statement Show that for a general potential energy V(r) that the form of the spin-orbit Hamiltonian becomes \hat{H}_{S-O}=\frac{1}{2m_{e}^{2}c^{2}|\hat{\textbf{r}}|}\frac{d\hat{V}}{dr} \hat{\textbf{L}}\cdot\hat{\textbf{S}} Suggestion: Start with \textbf{B} = -( \textbf{v} /c)...

Hi, I have an equation of the form (-i \lambda \frac{d}{dr}\sigma_z+\Delta(r)\sigma_x) g =(\epsilon + \frac{\mu \hbar^2}{2mr^2}) g where \sigma refers to the Pauli matrices, g is a two component complex vector and the term on the right hand side of the equation is small compared to the other...

Example from Schaum's Quantum Mechanics. Picture of the example is attached. What I don't understand is part (c). What are those wavefunctions ##\mid \psi^{(0)}_{1,2} \rangle## and ##\mid \psi^{(0)}_{2,1} \rangle##? How do I find these wavefunctions, if the unperturbed wavefunction is...

Hamiltonian is in the form ##H = H_0 + \lambda W##, where ##\lambda \ll 1## and ##W## is the perturbation. Assume the eigenstates ##\mid \psi(\lambda) \rangle## and engenenergies ##E(\lambda)## can be expanded in a power series of ##\lambda##. $$\mid \phi(\lambda) \rangle = \mid 0 \rangle +...

Hi all, I have a question about the concept of complete set when I apply the perturbation theory in two situations -Finite Hilbert Space and Infinite Hilbert Space. Consider a Hamiltonian H=H0+H', where H0 is the unperturbed Hamiltonian and H' is the perturbed Hamiltonian. Let ψ_n be the...

Homework Statement Given the equation \ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta} Determine a first-order uniform expansion for small but finite theta. Homework Equations Other than the equation above, none so far as I am aware. The Attempt at a...

Hi guys, this is my first time posting, I'm studying physics at uni, in my third year and things are getting a bit tough, so basically my question is in relation to solving problem 1, (i included a picture...) I missed the class and don't really know what I'm doing. Any help would be appreciated.

Homework Statement Suppose we put a delta function bump in the center of the infinite square well: H' = \alpha \delta(x -a/2), where \alpha is constant. a) Find the first order correction to the allowed energies. b) Find the first three non-zero terms in the expansion of the correction...

Homework Statement the problem and a possible solution(obtained from a book) is attached as a pdf to the post.However Iam unable to understand it.Please download the attachment. Homework Equations equation no (2) in the pdf.Is there any use of space translation operator in here.The Attempt at...

Homework Statement We have A \in R^{mxm} \text{ and } b \in R^{m} \text{ and } b \neq 0 \text{. Show that } Ax = b \text{ and } A(x+ \delta x) = b+ \delta b The Attempt at a Solution I did the first part just by the definition of A being non singular. The second part is tripping...

I'm reading through this pdf (http://www.pa.msu.edu/~mmoore/TIPT.pdf) on simple quantum perturbation theory and I'm quite confused with equations 32 through 34. They have E_{n}^{(2)} = <n^{(0)}|V|n^{(1)}> = - \sum_{m \neq 0}{\frac{|V_{mn}|^{2}}{E_{mn}}} but I would have done E_{n}^{(2)} =...

The L4 position is stable in the Earth Moon and I perturbing a satellite by km in the x direction to see the trajectory over the course of the year. However, the satellite isn't moving. Can anyone see if there is something wrong? I gave the satellite no initial velocity. In[2587]:=...

Hi, I am recently reading Weinberg's Cosmology, and getting subtle on Ch5, small fluctuation. One of the subtle point is on P.225-226 (same as (F.11) -> (F.13) and (F.14) in appendix F). The equations of motion (5.1.24)-(5.1.26) are decomposed into many parts. For example, (5.1.24) is...

What does it mean for a matrix to be diagonal, especially in Quantum Mechanics, where we get to Perturbation theory (Degeneracy). I don't get it. Please if you can explain in 'simple' language.

Homework Statement On page 251 of Griffiths's quantum book, when deriving a result in first-order perturbation theory, the author makes the claim that <\psi^0|H^0\psi^1> = <H^0\psi^0|\psi^1> where H^0 is the unperturbed Hamiltonian and \psi^0 and \psi^1 are the unperturbed wavefunction and its...