Intuitively, I'd say that adding a 4-divergence to the Lagrangian should not affect the eqs of motion since the integral of that 4-divergence (of a vector that vanishes at ∞) can be rewritten as a surface term equal to zero, but...
In some theories, the addition of a term that is equal to zero...
If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term.
SO how should I approach this? And how the Δ is relevant in this calculation?
I'm getting confused by the perturbation theory aspect of problem 2.2 in this book. We have to show that the energy eigenvalues are given by
$$E_n = \left(n + \frac{1}{2}\right) \hbar \omega + \frac{3\lambda}{4} \left(\frac{\hbar}{m\omega}\right)^2 (2n^2 + 2n + 1)$$
For the Hamiltonian...
Homework Statement
Homework Equations
VD= -1/(8m2c2) [pi,[pi,Vc(r)]]
VC(r) = -Ze2/r
Energy shift Δ = <nlm|VD|nlm>
The Attempt at a Solution
I can't figure out how to evaluate the expectation values that result from the Δ equation. When I do out the commutator, I get p2V-2pVp+Vp2. This...
I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric.
Something is unclear when we try to find fake vs. real perturbation of the metric.
We use an arbitrary small vector field to have the effect of a chart transition map or...
I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$
The solution up to first order is
$$ \psi(x) =...
Homework Statement
A point mass m hangs at one end of a vertically hung hooke-like spring of force constant k. The other end of the spring is oscillated up and down according to ##z=a\cos(w_1t)##. By treating a as a small quantity, obtain a first-order solution to the motion of m in time...
Hi everyone,
I am doing a time dependent perturbation theory, in a case when the electron is prepared in a state of the continuous part of the energy spectrum. Existence of the discrete part and the degeneracy of the continuous part is irrelevant at the moment and will not be considered...
Homework Statement
Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential
$$ \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...
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...
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
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...
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...
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
"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 =...
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...
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...
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...
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...