Indices Definition and 207 Threads

Compare this with the definition of the inverse transformation Λ-1: Λ-1Λ = I or (Λ−1)ανΛνβ = δαβ,...(1.33) where I is the 4×4 indentity matrix. The indexes of Λ−1 are superscript for the first and subscript for the second as before, and the matrix product is formed as usual by summing over...

Hi there. I am working with a numerical quadrature in some scheme to solve a set of equations. At this point I am working in two dimensions. The thing is that I have some function ##\psi_m(x,y,\Omega_m)## with ##\Omega_m=(\Omega_{x,i},\Omega_{y,j})## with ##\displaystyle...

This is probably a really stupid question , but, Does it matter whether the metric is after or before the tensor? My guess is it doesn't because tensors can be positioned in any order, the equation is unchanged. E.g ##M_{ab}B^{c}T^{m}_{nl}=T^{m}_{nl}M_{ab}B^{c}## right? However the covariant...

In the electroweak sector, we define the left-handed Weyl fields ##l## and ##\bar{e}## in the representations ##(2,-1/2)## and ##(1,+1)## of ##SU(2) \times U(1)##. Here, ##l## is an ##SU(2)## doublet: ##l = \begin{pmatrix} \nu\\ e \end{pmatrix}.## The Yukawa coupling in the electroweak sector...

The left-handed Weyl operator is defined by the ##2\times 2## matrix $$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{pmatrix},$$ where ##\bar{\sigma}^{\mu}=(1,-\vec{\sigma})## are sigma matrices.One can use the sigma...

I'm working through some intro QFT using Peskin accompanied by David Tong's notes, and have a question over notation. From Peskin I have: x^\mu=x^0+x^1+x^2+x^3=(t,\mathbf{x}) and x_\mu=g^{\mu\nu}x^\nu=x^0-x^1-x^2-x^3=(t,-\mathbf{x}) so p_\mu p^\mu=g^{\mu\nu}p^\mu p^\nu=E^2-|\mathbf{p}|^2...

Hi, I've somehow gone the past year without paying attention to the order of the indicies when one is upper and one is lower i.e. that in general ##g^{\mu}## ##_{\nu}## ##\neq g_{\nu}## ## ^{\mu}##. A have a couple of questions : 1) ##g^{u}## ##_{v} x^{v}=x^{u}## [1] ##g _{v} ## ##^{u} x^{v}...

Hi guys, I have a very basic question about the WZ model. I want to show that it is invariant under SUSY transformations. The action is \int{d^4 x} \partial^\mu \phi* \partial_\mu \phi +i\psi^† \bar{\sigma}^\mu \partial_\mu \psi The SUSY transformations are \delta\phi = \epsilon \psi ...

Homework Statement My question is regarding a single step in a solution to a given problem. The step begins at: ##\large \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{y^p} \frac{\partial x^j}{\partial y^q} - \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{\partial...

Let me see if I understand this correctly. Using the metric to raise an index converts a vector into a one form and lowering the index converts a one form into a vector. The contraction on the indices is the dot product between the two. Am I correct so far? If so, here is my question. What is...

I have learned that there is a difference between the tensors ##{T^{\mu}}_{\nu}## and ##{T_{\nu}}^{\mu}##. Does the upper index denote the rows and the lower index the columns?

I have an equation that says $$C_1\partial_{\mu}G^{\mu\nu}+C_2\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}G_{\rho\sigma}=0$$ If I want to get rid of the ##\epsilon^{\mu\nu\rho\sigma}## in the second term, I know I must multiply the equation by some other ##\epsilon## with different set...

Homework Statement I need to find the X of two substances using only the data found in lab. Which is: nmixture=1.4156 nbutan-2-one=1.3787 ntoluene=1.4970 For the mixture, I don't know what X (the mole fraction) is. That is what I must find out. Homework Equations nmixture= (Xbutan-2-one...

I've just come across an example of this on the following website; http://www.bbc.co.uk/education/guides/zqn3r82/revision/5 The example was; [40 - (2+4^2)] x 2 The site said that if there are two sets of brackets calculate the inside brackets first. I proceeded as follows; inside brackets...

Hello! New here, so please bare with me :). I am not entirely sure if this was the best forum to ask this question, so if it is not within the topic of GR, please say so. In Einstein's theory of General Relativity the metric tensor is symmetric and has the property of lowering and raising the...

The metric tensor has the property that it can raise and lower indices, but this is on the assumption that it (the metric) is symmetric. If we were to construct a metric tensor that was non-symmetric, would it still raise and lower indices?

Say I want to find ##\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)##. Is the following alright: ##\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)##?

Suppose I have something like \left( \nabla_\mu \nabla_\beta - \nabla_\beta \nabla_\mu \right) V^\mu = R_{\nu \beta} V^\nu Can since all the terms involving ##\mu## on the left and ##\nu## on the right are contractions, can I simply do: \left( \nabla^\mu \nabla_\beta - \nabla_\beta \nabla^\mu...

Hi PF Peeps! Something came up while I was studying for my QM1 class. Basically we want to represent operators as matrices and in one case the matrix element is defined by the formula : <m'|m> = \frac{h}{2\pi}\sqrt{\frac{15}{4} - m(m+1)} \delta_{m',m+1} But the thing is we know m takes on...

Hi all, first let me post this (from "Elementary Crystallography An Introduction to the Fundamental Geometrical Features of Crystals" by Buerger): https://www.physicsforums.com/attachments/1-png.82644/ Can someone please explain me the proof of Lemma 1? I just can not see it with the "mini...

Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...

Homework Statement [/B] (a) Show Compression of FCC leads to BCC. (b) State rules for X-rays reflection in FCC. (c) What are the new Miller Indices after compression? Homework EquationsThe Attempt at a Solution Part(a)[/B] I'm quite confused as to what they mean by 'principal axes'. For an...

Homework Statement Homework Equations Relabelling of indeces, 4-vector notation The Attempt at a Solution The forth line where I've circled one of the components in red, I am unsure why you can simply let ν=μ and μ=v for the second part of the line only then relate it to the first part and...

At low speeds and assuming pressure ##P=0##, T^{\alpha \beta} = \rho U^\alpha U^\beta g_{\alpha \mu} g_{\gamma \beta} T^{\alpha \beta} = \rho g_{\alpha \mu} g_{\gamma \beta} U^\alpha U^\beta T_{\gamma \mu} = \rho U_\mu U^\beta g_{\gamma \beta} Setting ##\gamma = \mu = 0##: T_{00} = \rho...

If we know the reciprocal space basis of a BCC lattice b_1=\frac{2\pi}{a}(\vec{x}+\vec{y}),b_2=\frac{2\pi}{a}(\vec{z}+\vec{y}),b_3=\frac{2\pi}{a}(\vec{x}+\vec{z}) how do we go about finding the shortest reciprocal lattice vector and its corresponding miller index? To me all the constants in...

We've just started with this new notation in class and I'm just trying to get a little more insight into the basics so have a few very basic questions if someone here can help? 1) dxμ = (dt,dx,dy,dz) I understand that's it's just a four-vector with four components but, what would dxμ be equal...

I'm trying to do some GR self-instruction through a variety of video lectures and thought this would be a good place to seek clarification on the inevitable thorny issues. I've tried this before and didn't make it too far, but I'm trying to get back on the horse, so to speak, and give it...

Hi there. When I have dummy indices in a tensor equation with separate terms, I wanted to know if I can rename the dummies in the separate terms. I have, in particular: \displaystyle w_k=-\frac{1}{4}\epsilon_{kpq}\left [ \frac{\partial u_p}{\partial x_q}-\frac{\partial u_q}{\partial x_p}...

I would like to take the trace over spinorial indices of the following expression: (\gamma_{\mu}\gamma^{0})_{\alpha}^{\beta}=(\gamma_{\mu})_{\alpha}^{\gamma}(\gamma^{0})_{\gamma}^{\beta}. How do I go about doing this? I reckon I could expand the trace out (let's say I want to do this in 4D)...

The Fresnel equations indicate that radiation will be 100% transmitted if two mediums have the same refractive indices. If that is true, then whey is there so much reflection off of, for instance, cracks in glass? Is this because there is a microscopic vacancy where the index of refraction...

Homework Statement Question is to determine the miller indices of the plane Homework EquationsThe Attempt at a Solution I know how to determine miller indices but in this problem the y intercept lies outside the cube. Do I have to somehow shift it so that it lies inside the cube ?, or just...

Homework Statement Determine the Miller indices of the cubic crystal plane that intersects the position coordinates C (1, 1/4, 0), A (1, 1, 1/2), B (3/4, 1, 1/4), and all coordinate axes. The Attempt at a Solution This is an example problem with solution from Foundations of...

This expression: \Gammaavc\Gammacab Can someone please show me how to multiply the two Christoffel symbol formulas for these Christoffel symbols without overloading any indices?

Hi everyone, I am doing MD simulation for zirconium (hcp). I have to input some orientation for crystal in simulation. But i have orientation in 4-index bravais miller indices. and i have to convert (plane and direction) it from 4-index to 3-index orthogonal coordinate system. Please help me...

So I have just been introduced to indices, four vectors and tensors in SR and I'm having trouble knowing exactly what I am being asked in some questions. So the first question asks to write explicitly how a covariant two tensor transforms under a lorentz boost. Now I know that it transforms...

When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write \Lambda^\mu\,_\nu or \Lambda^\mu_\nu ? i.e. if the \nu index is to the right of the \mu or they are at the same place with respect to left-right?

1. I'll post here a simplified version of my problem Say you have a matrix A, and you want all its components to be functions, for example: A11 = Sin(a_n) A12 = Cos(a_n) A21 = Sin(b_n) A22 = Cos(b_n) And I want to be able to do this in mathematica so as to have the matrix A a...

Hello, I was reading a book about general relativity and I came across these two equations $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ \mathrm{g}^{\mu\sigma}{\Gamma}^{\nu}_{\rho\sigma} -\frac{1}{2}(...

If different frequencies of light have different refractive indices for the same material and travel at different speed in the same material, isn't it inaccurate to say that the speed of light through a certain material is c/n, where n is the "standard" refractive index?

I'm currently in a modern physics class and one of our labs was an electron scattering experiment that required the use of a cathode ray tube and a target foil. We aim the electron beam through one of four quadrants on the target foil and measure the diameter of the ring diffraction pattern...

Homework Statement find the series solution to y''+x^2*y'+y=0 Homework Equations y=summation from n=0 to infinity Cn*x^n The Attempt at a Solution y=sum from 0 to inf Cnxn x^2*y'=sum from 1 to inf nC n xn+1 = sum from 2 to inf (n-1) C n-1 xn = sum from 1 to inf (n-1) C n-1 xn...

There are four miller indices (hklj) for the hexagonal lattice, the third being redudant: l=-(h+k) (1) Given the basis vectors a1,a2,a3 I can certainly see that: a3=-(a1+a2) But how does this immidiatly lead me to the relation (1) between the miller indices?

Hey guys, So I'm reading something about vector potentials and I've come across this one line which is really annyoing me. Here's how it goes \frac{d}{dt}\mathbf{A}=\frac{\partial \mathbf{A}}{\partial t}+\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial }{\partial...

Homework Statement At the He-Ne laser wavelength (L= 632.8 nm) the refractive indices of crystal quartz are n o = 1.54264 and n e = 1.55171 calculated from its Sellmeier equation. The laser is incident from the air onto the surface of crystal quartz at an angle of incidence of 45 degrees...

I have no idea how to do this or where to start. Can someone please help me? At the He-Ne laser wavelength (L= 632.8 nm) the refractive indices of crystal quartz are n o = 1.54264 and n e = 1.55171 calculated from its Sellmeier equation. The laser is incident from the air onto the surface of...

Hi, can someone confirm those or did I not get the meaning of the 4-vectors indices: \partial^{\mu}x_{\mu}=4;\partial^{\mu}x^{\mu}=2;\partial^{\mu}x_{\nu}= \delta ^{\mu}_{\nu};\partial^{\mu}x^{\nu}=g^{\mu\nu}

Homework Statement In the figure below (see image in color), light travels from material 'a', through three layers of other materials with surfaces parallel to one another, and then back into another layer of material 'a'. The refractions (but not the associated reflections) at the surfaces...

Homework Statement Give the Miller indices of the plane that is define by the two directions [111] and [2\overline{1}3]? Homework Equations h~1/a k~1/b l~1/c The Attempt at a Solution I know how to solve for miller indices once you know the the intercepts of the plane. So I found...

Homework Statement When we raise and lower indices of vectors and tensors (in representations of any groups) we always use tensors which are invariant under the corresponding transformations, e.g. we use the Minkoski metric in representations of the Lorentz group...

Hi there, I am stuck, please can someone help me to understand where I am going wrong... I have been learning about indices and i am confused.. here is an example of why.. 2t squared x 3t squared = 6t power 4 but if i give t a value of 2.. 2x2 squared = 16 3x2 squared = 36 16 x...