Recent content by AwesomeTrains

I have problems deriving a formula in a paper I'm reading for a project. The paper is about putting a number of waveplates in series rotated relatively to each other to form a tuneable broadband waveplate. For the i-th waveplate the jones matrix is given by: $$ J_{i}(\delta_i, \Theta_i)=...

I ended up solving the equation numerically which gave me a pretty good fit to the measured values. The numerical solution works until approximately 500 GHz which is where the conditions from post #5 aren't holding anymore, so that is probably the explanation and then expanding to third order...

Here's my derivation: (I missed the minus sign in the original post) ## \frac{\alpha_2}{\mu_2} \tan\left(\frac{b\alpha_2}{2}\right)=-\frac{\alpha_1}{\mu_1} \tan\left(\frac{a\alpha_1}{2}\right) ## First I expand the tangents on both sides: ##...

Hello PF, I'm reading a paper for a project. In the paper they derive an equation for the effective refractive index ##n=\sqrt{\epsilon^{e} \mu^{e}}## of two stacked layers ##(n_1^2 = \epsilon_1 \mu_1, a)## and ##(n_2^2 = \epsilon_2 \mu_2, b)## where ##a,b## are the lengths and in my case...

Thanks for the help! I should be able to get it now, I was trying hard to solve for the k in the alphas :oldeyes:

Thanks for the response. Sorry for the bad quality of the picture, the font is not so clear. It's a chi and not a k, k is in the alphas. I should have written all the definitions in the first post to clarify, my bad :oldeyes: \alpha_1 = k \sqrt{n_1^2-n^2}, \alpha_2 = k \sqrt{n_2^2-n^2} and...

Hello PF, first of all I don't know where to put this post as it's not exactly a homework question but a clarification question for a project. I'm going through the derivation of the effective permeability of two stacked medias, given the polarization of an incoming EM wave but I'm stuck at the...

I cannot seem to get the differential fiddled into parts like: ##f(V)dV = g(T)dT + h(p)dp## the best I could do is something like: $$dV+V\left(\frac{dp}{p}-\frac{dT}{T}\right) = \frac{-adp}{T^2}+\frac{3adT}{T^3}$$ or $$\frac{1}{V}\left(dV+\frac{adp}{T^2}-\frac{3adT}{T^3}\right)=...

My calculation for ##F## didn't really seem to go anywhere, added it to the question just in case it could have been useful. ##dV(T,p) = \frac{\partial V}{\partial T}dT + \frac{\partial V}{\partial p}dp \implies \int \frac{dv}{v} = -\int \kappa_T dp + \int \alpha_p dT \implies## $$ln~V =...

I don't have the source but I attached a screenshot of the problem sheet as it was handed out.

Homework Statement The isobaric expansion coefficient and the isothermal compressibility are given by: $$\alpha_p = (1/V)(\partial V/\partial T)_p \quad \kappa_T = -(1/V)(\partial V / \partial p)_T$$ Suppose they have experimentally been determined to be: $$ \alpha_p = \frac{1}{T} +...

Thanks a lot for the help and patience! The last part of the question I think I can do on my own now. Have a nice weekend :)

## \rho = \sum_n c_n | n \rangle \langle n |= c_1|+\rangle_x \langle + |_x+c_2|+\rangle_y \langle + |_y\\=\frac{c_1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} \frac{1}{\sqrt{2}} \begin{pmatrix} 1^{*} & 1^{*}\\ \end{pmatrix}+ \frac{c_2}{\sqrt{2}} \begin{pmatrix} 1 \\...

Sorry about the misconception I got a bit confused myself since I've seen quite a few different notations in my lecture and on the internet. ##|+\rangle_y=\frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \\ \end{pmatrix}## and ##|+\rangle_x=\frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1...

It's because it's the eigenstate of ##\hat{S}_z = \frac{1}{2} \sigma_z =\frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix} ## if ## |+\rangle ## is represented as ## \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} ##