It is indeed a decay process of the form ##1\rightarrow 2##, in particular, a Higgs decaying to an electron pair. You're suggesting using the same procedure of going into the Higgs CoM frame on the last equation? Nonetheless, I have no idea on how to move on from there.
I've been trying for a very long time to show that the following integral:
$$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi
)^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2
f_{L}^0f_{R}^0(1+f_{H}^0). $$
can be reduced to one dimension:
$$
I_D...
I'm looking forward to have a better understanding of the polarization vector in quantum field theory in order to solve a particular problem.
In class and in several textbooks I see that ##s^\mu=(0,\vec s)## and ##|\vec s|=1##. Are polarizations vectors defined to have no temporal component in...
I want to make certain that my proof is correct:
Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it...
I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states.
How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...
I'm currently working out quantities that include the vector and axialvector currents ##j^\mu_B(x)=\bar{\psi}(x)\Gamma^\mu_{B,0}\psi(x)## where B stands for V (vector) or A (axialvector). The gamma in the middle is a product of gamma matrices and the psi's are dirac spinors. Therefore on the...
Starting from the general formula:
$$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\epsilon}{2})}{\Gamma(2-\frac{\epsilon}{2})\Gamma(n)}\frac{1}{\Delta^{n-m-2}}(\frac{4\pi M^2}{\Delta})^{\frac{\epsilon}{2}}\Gamma(n-m-2+\frac{\epsilon}{2})$$
I arrived to the following...
Consider, for example, the gluon propagator $$D^{\mu\nu}(q)=-\frac{i}{q^2+i\epsilon}[D(q^2)T^{\mu\nu}_q+\xi L^{\mu\nu}_q]$$
What exactly is the renormalized gluon dressing function ##D(q^2)## and what is its definition? My interest is in knowing if I can then write the bare version of this...
I might have not been clear, I'm sorry. I do want to use the trace identities in order to do the calculations. I just wanted to write out the indices explicitly so I show clearly that the numerator is indeed a trace.
I'm working out the quark loop diagram and I've drawn it as follows:
where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices.
For this diagram I've written...
This was my attempt at a solution and was wondering where did I go wrong: -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma...
That said, my approach was to determine the energies and 3-momenta at the center of momentum reference frame for each particle, with a fixed s, and check it corresponds to each one of the above, but I'm having some trouble proving that, for example, E_A=\frac{s+m^2_A-m^2_B}{2\sqrt{s}}. I've...
Homework Statement
After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin.
2. The attempt at a solution
I tried to apply the...
Homework Statement
Following from \hat{b}^\dagger_j\hat{b}_j(\hat{b}_j
\mid \Psi \rangle
)=(|B_-^j|^2-1)\hat{b}_j
\mid \Psi \rangle
, I want to prove that if I keep applying ##\hat{b}_j##, ## n_j##times, I'll get: (|B_-^j|^2-n_j)\hat{b}_j\hat{b}_j\hat{b}_j ...
\mid \Psi \rangle
.
Homework...
Are the first and second rows really identical? The spins for the first two terms of each mentioned row have different spin states. Otherwise yes, the determinant would be zero. However, despite the configuration being one that doesn't cancel the determinant, it is one that involves identical...
In Zettili's Quantum Mechanics, page 477, he wants to determine the energy and wave function of the ground state of three non-interacting identical spin 1/2 particles confined in a one-dimensional infinite potential well of length a. He states that one possible configuration of the ground state...
I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle:
\begin{bmatrix}
E_0 &-A \\
-A & E_0
\end{bmatrix}
Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix}
\frac{1}{\sqrt{2}}\\...
Indeed! I'm using Mathematica! Meanwhile, I recently solved my problem. I simply defined Hyfield as a function of "a" and I avoid declaring auxlist as a function of a. That did the trick!
My objective is to make a list of functions and afterwards be able to make operations with those functions.
Hyfield[list_, bits_] := Module[{i, auxList, Hy},
auxList[a_] := List[];
For[i = 1, i <= bits*2, i++,
auxList[a] =
Append[auxList[a]...
Oh, since I have 3 electrons to distribute over the the orbitals, the lowest energy state (bonding orbital) is filled with 2 electrons, each one with energy -2t (smallest energy eigenvalue from the hamiltonian) and the other electron stays in a higher energy orbital (anti-bonding orbital)...
Homework Statement
I'm considering a molecule made by three atoms, each a vertex of an equilateral triangle. Each atom has a covalent bond with its neighbours, sharing their only valence electron. I must estimate the energy gain when creating the molecule, using tight binding theory.
Homework...
Hi!
I'm trying to show how the chemical potential depends on the temperature and I'm advised to use the Sommerfeld expansion. I'm using it on the density of charge n=\int^{+\infty}_{-\infty} \rho(\epsilon)n_Fd\epsilon , which gives n=\int^{\mu}_{0} \rho(\epsilon)d\epsilon...
I'm reading Gasiorowicz's Quantum Physics and at the beggining of chapter 2, SG introduces the concept of "wave packet" and gaussian functions associated to them. The first attached image is the 28th page of the book's 1st edition I suppose, and my question is about the paragraph inside the red...
Hi!
For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...
I found at least one mistake, when solving the quadratic equation, dumb me!
The final equation I get, solving the quadratic equation for \omega^2 is :
\omega^2=\frac{(4\omega_0^2+2\tilde{\omega_0}^2)\pm \sqrt{(4\omega_0^2+2\tilde{\omega_0}^2)^2-4(3\omega_0^4+4\omega_0^2 \tilde{\omega_0^2}})}{2}...
The relevant information for this problem is that the masses are the same m_A=m_B. The \omega_0 and \tilde{\omega_0} are the natural frequencies associated with each spring. The long springs have constant \tilde{k}, so \tilde{\omega_0}=\sqrt{\frac{\tilde{k}}{m}}. The short springs have constant...
Homework Statement
I have to determine the frequencies of the normal modes of oscillation for the system I've uploaded.
Homework Equations
[/B]
I determined the following differential equations for the coupled system:
\ddot{x_A}+2(\omega_0^2+\tilde{\omega_0}^2)x_A-\omega_0^2x_B = 0...
Homework Statement
Hi! I'm trying to solve a simple problem of mechanics, but I'm getting the wrong results and I suppose I don't yet grasp the concept of instantaneous axis of rotation very well.
So, a cone (see attached picture) is rolling without slipping on a plane. Vp is point P linear...
I am to study how fast an iterative method for nonlinear system of equations converges to a certain root and found out that I can evaluate my rate of convergence by using the following formula: ##r^{(k)}=\frac{||x^{(k+1)}-x^{(k)}||_V}{||x^{(k)}-x^{(k-1)}||_V}##. My question is which vectorial...
Homework Statement
I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
Pheww, I'm reassured to know you got the same parameters as me! I actually used Mathematica and had to code the algorithms since that was one of the requirements of the problem given to me. It's the first time I've heard about Maple. Is it a Mathematica-like software? Is it widely used? Once...
I've been meaning to post my results here, but I forgot to do so in the last days. So, I did minimize the sum of squared errors using the partial derivatives and found the parameters using a variety of methods (Newton's, fixed-point and Broyden's) to solve the system, coded by myself in Wolfram...
I see! I was too obsessed in trying to solve the problem through the examples I mentioned earlier and kept avoiding the derivatives of the sum of squared errors. I will try to work on it now and maybe later post the results I've arrived to. Thank you very much for the help! :)
I see. So I have to build my set of normal equations and solve it using, for example, Newton's Method (which is actually the method I'm being asked to use in later problems). My problem is coming up with a set of linearly independent functions that I can use to build my set of normal equations...
Thank you for your suggestion, but I'm trying to find a way of doing it analytically, by building a set of non linear equations and solving it. I'm aware that I must use non-linear Least Squares Method in this case, but my problem is the parameter "a" that stops me from using the logarithm to...
Homework Statement
Hi! I've been interpolating a data set using Legendre polynomials and Newton's method and now I've been asked to, given a data set, approximate the function using the Least Squares Method with the following fitting function: ##g(r)=a+be^{cr}##, where ##a##, ##b## and ##c##...
Hi! I've been studying the time-independent Schrödinger equation and the infinite square well and was faced with this problem from Griffith's "Introduction to Quantum Mechanics". Rewriting the equation this way $$\frac{d^2\psi}{dx^2}=\frac{2m}{\hbar^2}[V(x)-E]\psi$$, I have to show that E must...
Homework Statement
A rigid, square-shaped, structure with negligible mass contains 4 disks in rotation as you can see in the figure. Each disk has mass m, moment of inertia I about its rotation axis and angular velocity ws. Also, the plane of the structure coincides with the horizontal plane...
Oh! Indeed! It is that simple. And so, the sum of all contributions to the torque along that axis equals the cross product R x F, in this case is zero, since the angle between R and F is zero.
Thank you!.
Hi!
At the moment I'm studying rigid body motion, more specifically, the gyrocompass. As you can see in the attached picture (Introduction to Mechanics -Kleppner-Kolenkow-Chap.7), the gyrocompass rotates about the z axis and the spin angular momentum is reoriented towards the z axis, creating a...