I'm kind of confused on how to evaluate the principal value as it's a topic I've never seen in complex analysis and all the literature I've read so far only deals with the formal definition, not providing an example on how to calculate it properly. Therefore, I think just understanding at least...
Homework Statement
Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length ##l_1## and energy ##E_1##) or short axis (of length ##l_2## and energy ##E_2##). Suppose that the chain is subject to tension...
Homework Statement
A long straight cylinder with radius ##a## and length ##L## has an uniform magnetization ##M## along its axis.
(a) Show that when its flat extreme is placed on a flat surface with infinite permeability (i.e. a ferromagnet), it adheres with a force equal to:
$$F=8\pi a^2 L...
Homework Statement
A conductor sphere of radius R without charge is floating half-submerged in a liquid with dielectric constant ##\epsilon_{liquid}=\epsilon## and density ##\rho_l##. The upper air can be considered to have a dielectric constant ##\epsilon_{air}=1##. Now an infinitesimal...
Homework Statement
Let x, y and z satisfy the state function ##f(x, y, z) = 0## and let ##w## be a function of only two of these variables. Show the following identities:
$$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial...
Homework Statement
We have two semi-infinite coplanar planes defined by z=0, one corresponding to x<0 set at potential zero, and one corresponding to x> set to potential ##V_0##.
a) Find the Green function for the potential in this region
b) Find the potential ##\Phi(r)## for all points in...
Homework Statement
Find the electric potential of an infinitely long cylinder shell of radius ##R## whose walls are grounded, when in its interior a line charge, parallel to the cylinder, is placed at ##r=a## (with ##a<R##) and that has a lineal charge density ##\lambda##.
Homework Equations...
Hello. I'm currently beginning my 2nd semester of a dual Msc/PhD in Physics in the top university of my home country, but lately I've been struggling with self-doubt and uncertainty about the job prospects and whether research is actually for me.
To give you some background. Since I was in high...
This is one of the problems I'm currently working on but understanding how to deduce the Feynman rules for this case would give me a better idea on how to do it for more general cases besides ##\phi^4## theory (which is the example commonly covered in books like Peskin and Greiner).
1. Homework...
Homework Statement
Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates.
Homework Equations
$$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...
This one may seem a bit long but essentially the problem reduces to some matrix calculations. You may skip the background if you're familiar with Lorentz representations.
1. Homework Statement
A Lorentz transformation can be represented by the matrix...
Homework Statement
A particle with spin s=1/2 moves under the influence of a magnetic field given by:
$$\vec{A}=B(-y,0,0)$$
Find the eigenvalues of the corresponding Pauli hamiltonian. Repeat the same process for:
$$\vec{A}=\frac{B}{2}(-y,x,0)$$
Explain your result by relating the...
Homework Statement
Show that the coherent state ##|c\rangle=exp(\int \frac{d^3p}{(2\pi)^3}c(\vec{p})a^{\dagger}_{\vec{p}})|0\rangle## is an eigenstate of the anhiquilation operator ##a_{\vec{p}}##. Express it in terms of the states of type ##|\vec{p}_1...\vec{p}_N\rangle##
Homework Equations...
Homework Statement
For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form
$$L(p)=exp(i\phi J^{(21)})exp(i\theta...
Homework Statement
The SO(3) representation can be represented as ##3\times 3## matrices with the following form:
$$J_1=\frac{1}{\sqrt{2}}\left(\matrix{0&1&0\\1&0&1\\ 0&1&0}\right) \ \ ; \ \ J_2=\frac{1}{\sqrt{2}}\left(\matrix{0&-i&0\\i&0&-i\\ 0&i&0}\right) \ \ ; \ \...
Homework Statement
The degenerate parametric amplifier is described by the Hamiltonian:
$$H=\hbar \omega a^\dagger a-i\hbar \chi /2 (e^{(2i\omega t)}a^2-e^{(-2i\omega t)}(a^\dagger)^2)$$
Where ##a## and ##a^\dagger## as just the operators of creation and anhiquilation and ##\chi## is just a...
I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it.
1. Homework Statement
Given an infinite base of orthonormal states in the Hilbert space...
Hello. I've been struggling for a day with the following problem on Quantum coherent states, so I was wondering if you could tell me if I'm going in the right direction (I've read the books of Sakurai and Weinberg but can't seem to find an answer)
1. Homework Statement
*Suppose a Schrödinger...
Lately AdS/CFT seems to have been a very promising tool to simplify calculations in HEP (ex. quark-gluon plasma) and offer some insights into quantum gravity. I was considering doing a Master or PhD thesis in this field, but I'm wondering if the prospects are more reasonable than just string...
Hello there. I just finished my Bsc. in Engineering Physics+M.eng and I'm about to enter my Msc in Physics (no direct PhD in my country), so I'm choosing which electives to take on my first semester (as my professors suggested me to skip most of the basics).
While I entered with the idea to...