Questions tagged [notation]

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Ket notation in alternate forms

I have been told that I can describe a system by its wave number states: $$|k_1\rangle|k_2\rangle,$$ and that the following is true: $$|k_1\rangle|k_2\rangle=|k_1+k_2\rangle|k_1-k_2\rangle,$$ I am ...
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27 views

Ricci Tensor component, index lowering doubt [closed]

$v^j$$v^k$$[R(e_i,e_j)]e_k$.$e_i$ gives $v^j$$v^k$$R^l$$_k$$_i$$_j$ $e_l$.$e_i$ Then $v^j$$v^k$$R^l$$_k$$_i$$_j$ $g$$_l$$_i$ Finally, $v^j$$v^k$$R$$_i$$_k$$_i$$_j$ Which upon summation on i gives $v^j$...
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Proving the Existence of Locally Inertial Coordinates

This question is regarding the proof of the existence of locally inertial coordinates, outlined in Sean Carroll's Spacetime and Geometry book (Chapter 2, page 74). In particular, I believe that extra ...
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Operators in QM acting on kets [duplicate]

I know that operators in quantum mechanics act on states (kets, assumed to be normalized), and return another state. So, an operator such as $\hat{p}$ when acted on a ket $|\psi\rangle$ will give me ...
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1answer
60 views

Chern-Simons, General relativity, and notations

Consider the Einstein-Maxwell-dilaton theory with an additional Chern-Simons term as in this paper \begin{equation} S = \int d^4 \sqrt{-g} \left[ \frac{1}{2} R - \frac{1}{2} (\partial\varphi)^2 - \...
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4answers
114 views

Calculus of variations: meaning of infinitesimal variation $\delta$ and action minimum

So I am studying classical mechanics through the MIT 8.223 notes, and encountered the derivation of the Euler Lagrange equation. There is a part I don't quite understand, which resides in the actual ...
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1answer
57 views

Squared vector field terms $\mathbf{E}^2$ and $\mathbf{H}^2$?

Consider the simple case of electromagnetic irradiation of a homogeneous isotropic dielectric, neglecting the dispersion of the refractive index. Assuming a transparent medium, the spatial density of ...
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69 views

Is notational compactness in tensors (compared to linear algebra) relevant? [migrated]

In this post you can read: A matrix is a special case of a second rank tensor with 1 index up and 1 index down. It takes vectors to vectors, (by contracting the upper index of the vector with the ...
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0answers
74 views

“$10^{\mathrm{th}}$” Index skipped in M-theory

Why is it that in M-theory, the 11-dimensional vectors are labelled with indices $0 \dots 9,11$. For example, the spatial momentum components are $p_{1}, \dots, p_{9}, p_{11}$. Why is the $10^{th}$ ...
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2answers
55 views

The $\delta$ notation in Goldstein's Classical Mechanics on the calculus of variation

In Goldstein's classical mechanics (page 36) he introduces the basics of the calculus of variation and uses it to effectively the Euler-Lagrange equations. However, there is a step in which the $\...
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190 views

Proving that the Minkowski metric tensor is invariant under Lorentz transformations

I'm studying special relativity. A general Lorentz transformation is defined by $\Lambda^T\eta\Lambda=\eta$. Now, \begin{align} \eta'^{\mu\nu} &= \Lambda^\mu_{\;\;\alpha}\Lambda^\nu_{\;\;\beta}\...
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1answer
39 views

What is the convention for tensor indices for matrices?

The Lorenz-Transformation of the EM-Tensor F is given by the equation $$ F'^{\mu \nu} = \Lambda^{\mu}_{\ \ \rho} \Lambda^\nu_{\ \ \sigma} F^{\rho \sigma}$$ Then it says that this is equivalent to the ...
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5answers
851 views

Inverse and Transpose of Lorentz Transformation

I've seen this question asked a few times on Stack Exchange, but I'm still quite confused why the following "contradiction" seems to arise. By definition: $(\Lambda^T)^{\mu}{}_{\nu} = \...
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67 views

Contravariant rank-2 tensor transformation in index notation

I'm slightly confused about the placement of upper and lower indices for the transformation of a rank-2 contravariant tensor. A contravariant rank-2 tensor transforms as $$M' = \Lambda M \Lambda^{T}$$....
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Derivation of Maxwell's equation from Faraday tensor [closed]

If $F = F_{\alpha\beta}\text dx^\alpha \otimes \text dx^\beta$ Then how do I write the covariant derivative $\nabla F$ in component form. Here $F_{\alpha\beta}$ is a component of Faraday tensor and $\...
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2answers
218 views

Tensor and Matrices

Suppose that we are dealing with the following matrix: $$A=\begin{bmatrix}a_{00} & a_{01} \\ a_{10} & a_{11}\end{bmatrix}$$ but I don't want to use matrix notation, insted I want to use tensor ...
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27 views

surface charge density $\hat{\rho}$ and surface current density $\hat{\mathbf{j}}$

I am currently studying the textbook Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Page 5, chapter 1.1.3 ...
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2answers
90 views

What does the notation $|\text{grad} \ F|$ mean?

I am currently studying the textbook Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Page 5, chapter 1.1.3 ...
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1answer
42 views

Contracted indices can be interchanged?

I am working on Lorentz transformations and I get a tensor of the form $$M_{abcd}=\epsilon_{ab\mu\nu}\Lambda^\mu\hspace{0.1cm}_c\Lambda^\nu\hspace{0.1cm}_d$$ Where $\epsilon_{abμν}$ is the totally ...
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2answers
42 views

Partial derivative of the function with respect to $t$ in total derivative

In the formula description there is one extra partial derivative compared to the example solution. What's the difference here? What's the physical implication of the last partial derivative in the ...
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1answer
30 views

What do $S$, $M$, and $A$ mean in quark/color theory?

From Wikipedia [...]below, and symmetric in flavor, spin and space put together. With three flavors, the decomposition in flavor is $$ \mathbf{3} \otimes \mathbf{3} \otimes \mathbf{3} =\mathbf{10}_{S}...
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2answers
49 views

How to write units when multiple terms are involved in a derivation?

Say I am going to write down the steps of some calculations to get the final value of $s$ from an equation like this: $$ s = s_0 + \frac12 gt^2. $$ Let us say $s_0 = 20\,\mathrm{m}$, $g = 10\,\mathrm{...
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4answers
2k views

What does the notation $8.9875517923(14)$ mean? [duplicate]

The number $8.9875517923(14)$ appears in Coulomb's constant. I have read that it has something to do with the uncertainty of the accuracy of the number but answers have been unclear. Can somebody ...
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2answers
97 views

Minkowski Inner Product's Shenanigans

In the context of Special Relativity, so in flat spacetime, and with the metric tensor $g_{\mu \nu}$ chosen with the signature: $(+,-,-,-)$, lets consider the following four vectors: $$x=(x_1,x_2,x_3,...
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3answers
89 views

Expectation Value in Bra-ket notation

I've been staring at this problem for quite sometime, but I don't think I understand bra-ket notation in the form $<a | x | a>$. I understand that <a|x> is just an inner product, but I ...
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2answers
664 views

Bra ket notation rigorous way

I'm struggling to see how $\langle x|\Psi\rangle =\Psi(x)$. I have read a few previous bra ket questions in here but still not clear. Any good book for understand the bra-ket notation in more rigorous ...
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1answer
58 views

Position in generalized coordinates

In Lagrangian mechanics, when talking about a particle position expressed in generalized coordinates it is usual to find the expression: $$\mathbf{r}(q_0,...,q_k,t)\tag{1}$$ what it means this ...
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2answers
42 views

Confusion about notation for block transformation in Ising model

I'm going through Cardy's "Scaling and Renormalization in Statistical Physics", and I've run across a notational confusion. Consider a 2D Ising system with the following Hamiltonian $$\...
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3answers
127 views

How can $F_{\mu\nu}F^{\mu\nu} = 2(B^2-E^2)$ be proved? [closed]

How does $F_{\mu\nu}F^{\mu\nu} = 2(B^2-E^2)$? $$ F_{\mu\nu}=\pmatrix{ 0&E_x&E_y&E_z\\ -E_x&0&-B_z&B_y\\ -E_y&B_z&0&-B_x\\ -E_z&-B_y&B_x&0 } $$ $$ F^{\mu\...
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1answer
69 views

Need some help with Bra-Ket notation (specifically orthogonality in bra-ket notation)

I'm reading notes from a friend of mine taking a quantum mechanics class, and I see something I don't quite get. $$\left<x_i|x_j\right> = \delta_{ij}.$$ The notes say this implies orthogonality. ...
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2answers
78 views

Deriving the Covariant Derivative of the Metric Tensor

First off, I did look through some other questions: Covariant Derivative of Metric Tensor Why is the covariant derivative of the metric tensor zero? https://math.stackexchange.com/q/2174588/ But they ...
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1answer
52 views

Meaning of capital pi symbol in sum over histories integral

This question is primarily mathematical in nature. I have been reading Quantum Field Theory for the Gifted Amateur and I am reading about Feynman’s path integral approach. The definition of the “sum ...
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1answer
40 views

Energy levels and bands in solids: What does $^4\text{F}_{3/2}$ and $^4\text{I}_{11/2}$ mean?

I am currently studying Diode Lasers and Photonic Integrated Circuits, second edition, by Coldren, Corzine, and Mashanovitch. In chapter 1.2 ENERGY LEVELS AND BANDS IN SOLIDS, the authors say the ...
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1answer
51 views

I don't know why i can't use this equation (in the pic) in flatspace, I'm learning general relativity (metric tensor)

I don't know why this equation cannot be used in flatspace, Who can help me?
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2answers
133 views

Equivalence of Hermitian operator and Hermitian matrix in Quantum Mechanics

I learned that a Hermitian matrix $A$ is defined as a matrix that satisfies $$A^\dagger=(A^*)^\intercal=A,$$ i.e. its Hermitian conjugate $A^\dagger$ is the same as the original matrix $A$. I also ...
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3answers
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Critique on tensor notation

I am studying tensor algebra for an introductory course on General Relativity and I have stumbled upon an ambiguity in tensor notation that I truly dislike. But I am not sure if I am understanding the ...
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1answer
44 views

How can we write creation and annihilation operators in first quantized notation and second quantized notation?

We have learned creation operator $\hat{a}^\dagger_i$ adds a particle in $i^{th}$ state, and annihilation operator $\hat{a}_j$ remove a particle from $j^{th}$ state. They can be interpretated in such ...
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3answers
143 views

Wave function as a ket vector in a Hilbert space

There's something I don't understand: I've learned that quantum wave functions can be described as a "ket vector" in an abstract vector space called Hilbert space. The position wave function,...
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1answer
98 views

Dirac notation: what if we display tensor products vertically?

Ok hear me out. One thing I have always liked about Dirac notation is that it visually displays where expressions expect inputs/outputs. For example $\langle\psi|$ expects an input to the right to ...
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0answers
33 views

Why don't we use vector sign for quantities in 1D motion? [duplicate]

My school textbook says that we don't need to use unit vectors (i ,j,k) to represent the direction of vectors in 1D motion as + and - sign indicate direction. But that is creating a lot of confusion ...
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2answers
249 views

How do slanted indices work in special relativity? [duplicate]

What is the difference between $T^{\mu}{}_{\nu}$ and $T_{\nu}{}^{\mu}$ where $T$ is a tensor?
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1answer
26 views

Understanding concepts and notation in basic acoustic equations [closed]

I am reading a book describing the physics of acoustic sound waves. I stumbled across the equations: $$\text{grad }P = -\rho_0\frac{\partial V}{\partial t}$$ $$\rho_0\text{ div }V = \frac{\partial \...
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2answers
52 views

Index notation and total differential

I understand that the gradient $\partial_i$ is covariant. Let f be a function of 3 variables So I can write the total differential as $$ df=\partial_1fdx^1+\partial_2fdx^2+\partial_3fdx^3 = \...
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3answers
99 views

Problem with matrices in Dirac notation

Let $|q\rangle$ be the eigenvectors of the position operator, let $|\psi\rangle$ be a state and let $\hat{p}$ be the momentum operator. In my book it's stated that i can interprete the quantity: $$\...
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3answers
56 views

Showing the equivalence between the chain rule's Leibniz and Lagrange Notations

This may seem more math related but this question crossed my mind as I was reading the derivation of the Euler-Lagrange Equation. In math, we were introduced to the Lagrange notation of the derivative ...
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1answer
70 views

Question about some symbols in quantum mechanics

I'm tring to understand some symbols in physics formula for example in Ehrenfest’s theorem $$\frac{\partial}{\partial t}\langle \hat Q \rangle = \left\langle \frac{\partial \hat Q}{\partial t} \right\...
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1answer
49 views

What is gravitational constant in this context? [closed]

I've been reading an article and it gives me the following formula: $\vec{v}_B(t) = \vec{v}_0 - \mu_sgt \hat u_0$ It governs the velocity of a ball. In its explanation, it says: The ...
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1answer
41 views

Tensor gymnastics

I am working on a "tensor gymnastics" exercise, and have arrived at the following line to simplify: $\delta_{ik} y^{i} X_{ij}$ where $\delta_{ik}$ is the Kroenecker delta. Does this simplify to: $y^...
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1answer
72 views

What does the comma mean in this commutation rule between quantum operators?

The Theorem about quantum operators commutation relation says: Consider pairs $(U, V )$ of unitary representations on a Hilbert space $H$, satisfying the commutation rule: $$U(x) V(y)=\exp (i \...
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1answer
65 views

Notation for vector time derivatives [closed]

So I am self-studying mechanics using Marion and, as many books, it uses the notation of the dot over the function to express a time derivative, as in $$x = x(t)$$ $$\dot{x}= \frac{dx}{dt}(t) $$ The ...

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