Questions tagged [notation]

This tag is for questions on the meaning, history, and usage of various symbols and notation that are used to denote different quantities in physics. Don't forget to mention the reference, where you encountered those notations.

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How to write completeness of wavefunctions without bra ket notation?

In the quantum textbook I'm currently working from, the completeness relation is written as: $$ \sum_i |\psi_i \rangle \langle \psi_i| = \mathbb{1}. $$ But this seems to specifically require knowledge ...
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What $\mu$ represents in Gryzinski's free fall atomic model equation?

When you look into the Gryzinski's atomic free fall model equation (see wiki link here ), it has a variable $\mu$. What it is? $L = \frac{1}{2} m v^2 + \frac{Ze^2}{r}+\frac{Ze}{c}[v\cdot\frac{\vec{\mu}...
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Why is Hamilton's equations sometimes written with a gradient? [closed]

I am used to seeing Hamilton's written as: $$\frac{dq_j}{dt} = \frac{\partial H}{\partial p_j}\\ \frac{dp_j}{dt} = - \frac{\partial H}{\partial q_j}.$$ However I have also seen it written as $$\frac{...
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1 answer
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What does $\sin(a,b)$ mean in the absorber theory of radiation?

I'm doing a revision of the absorber theory of radiation by Wheeler and Feynman (that you can see here: "Interaction with the Absorber as the Mechanism of Radiation" - page 161) and I have ...
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1 answer
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Commutator between covariant derivative and a field

I have field as an element of a Lie algebra as $\Phi = \phi^at^a$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$ with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant ...
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-1 votes
1 answer
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In functional derivative the starting point confusion

how can one define the functional derivative $$\delta F= F[f+\delta f]-F[f].$$ Is it by definition or any physical reason holds for it.
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2 answers
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In an $n$ particle system, why is the Hamiltonian summed over $n$?

Suppose I am working in a system consisting of $n$ particles. Thus the phase space will be $\mathbb{R}^{6n}$, and both the momentum and position space will be $\mathbb{R}^{3n}$ each. Then, for some ...
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Question about integration limits in the special relativistic action

We can read in this article that the action of a particle in special relativity, is It seems like nitpicking maybe, but shouldn't the two coordinate time limits be changed to proper time limits, or ...
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1 answer
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Notation for contracting vectors using metric tensors

If we take a vector $A$, which has three components, my understanding is that we can write this using Einstein notation as $A_{u}$ where this is actually $A_1+A_2+A_3$. We can also write $g^{uv}A_v = ...
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7 votes
2 answers
168 views

What is the difference between the metric (tensor), $g_{\mu\nu}$, and the invariant interval, ${ds}^2$?

Here is a question from a problem sheet I found which I'm going to use to illustrate a point: The $\color{red}{\text{metric}}$ on a unit sphere is $${ds}^2={d \theta}^2+{\sin}^2\theta\, {d\phi}^2\tag{...
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Confusion about raising and lowering indices

Is it possible to take the following expression: $$U^\mu U^v\partial_\mu\partial_v$$ Where $U$ is the four-velocity, and simplify it the following way?: $$U^\mu U^v \eta_{\mu v}\partial^v\partial_v =c^...
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1 answer
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Divergence theorem in index notation

From Batchelor's book of fluid dynamics: I guess that's an easy question for anyone having more familiartiy than me in tensor calculus, anyways. First integral argument is the i-component of the ...
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3 votes
1 answer
55 views

What were the $r$ and $n$ of $\theta$s (Polchinski String theory section 8.6 page 265)?

In the Polchinski String theory section 8.6 page 265 In generic backgrounds, all the $\theta$s are distinct and the only massless vectors are the diagonal ones, $i = j$. The unbroken gauge group in ...
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0 answers
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Meaning of $\langle X(t')X(t'') \rangle$?

Context My background is not in physics so I am not very familiar with the $\langle \rangle$ notation. I am trying to understand the following in a paper that I am reading (Berglund AJ., PhysRevE., ...
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1 answer
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Problem related to classical mechanics

The actual formula for density is mass/volume. Can someone explain how this $dm = p.dx$ came?
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4 answers
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How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it

Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.
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4 votes
1 answer
95 views

What does $\delta/\delta t$-derivative represent in tensor calculus?

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
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Identity $\nabla_{j}\partial_{i} = \Gamma ^{k}_{ij} \partial_{k}$ involving covariant derivative

I am trying to understand the identity i have came across, but i am not being able to: $$\nabla_{j}\partial_{i} = \Gamma ^{k}_{ij} \partial_{k}$$ I thought that such equality would become obviously if ...
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2 votes
1 answer
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Relativistic Euler-Lagrange equations for a four-vector (or one-form) field

I think the best way to ask my question is by considering the maxwell-Lagrangian, $$\mathcal{L}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}=-\frac{1}{2}(\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}-\partial^{\mu}...
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0 votes
1 answer
56 views

What, precisely, using mathematics, is meant by "electric field with only an $\hat{x}$ component"?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following: The Helmholtz Equation In ...
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0 votes
1 answer
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What does it mean for the $\textbf{B}$-field (Hypercharge) to be in the 0 representation within the SM?

I was reading through the wikipedia page for the mathematical formulation of the standard model and I noticed that it listed the representations of the vector bosons under the SM gauge groups as being ...
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1 answer
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In Particle Physics what does the Rest Mass notation: 95$^{+9}_{−3}$ MeV/c$^2$ mean?

On the Wikipedia page for the Strange Quark, I came across the following notation for defining its mass: 95$^{+9}_{−3}$ MeV/c$^2$ Following the reference link brings me to this page, which shows a ...
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1 answer
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Why does Griffiths Example 5.5, assume the distance from wire to be $1$?

I was under the impression the magnitude of the cross product of cursive $r$ and $dl^{'}$ would be $|\overrightarrow{r}||\overrightarrow{dl^{'}}|sin\theta$. But he simply writes it as $|\...
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1 answer
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Elliptical polarisation - unsure as to what quantity is on wikipedia page

quick question - https://en.wikipedia.org/wiki/Elliptical_polarization - what is $\alpha_{x}$ and $\alpha_{y}$? It's not defined on the page. I'm asking to answer part of a question, which requires me ...
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Reference which explains Penrose Diagramatic notation in simple way

In both Penrose's Road to reality and Spinor's and space-time, the following notation is shown: With a lot other examples for doing calculation with Tensors. Could someone give another reference ...
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2 votes
3 answers
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Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $

My course guide gives the following derivation for change in entropy w.r.t. energy, where I don't understand a step: \begin{align} E & = E_1 + E_2 \\ S & = S_1 + S_2 \\ S(E,E_1 ) & = S_1 (...
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0 votes
1 answer
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Velocities - Equation 1.46 of Goldstein 3rd edition

In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein uses the parametrization (equation 1.45') $$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$...
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Can we define $\text dW$? [duplicate]

I am currently taking applied thermodynamics at my university, and for the definition of entropy this is the formula used in the book (Thermodynamic for Engineers by Moran, Shapiro, Boettner, Bailey): ...
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1 vote
1 answer
26 views

Convective derivative N-S

This is probably an easy answer, but I've not been able to find it yet - Why in some formulations of the N-S equations (for example here https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html), is the $...
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2 votes
1 answer
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Notation and Terminology Questions from Schwartz' QFT Book

I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing. First off, on page 34 he defines a translation of a field to first order as $$...
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1 vote
2 answers
134 views

Discrete and continuous basis in Quantum Mechanics

In the context of Quantum Mechanics and Hilbert spaces, I understand that a function can be interpreted as $\psi(x) = \langle x \vert \psi \rangle$ in the position basis, and things like $$\int_a^b|\...
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0 votes
1 answer
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Quantum physics notation in statistical physics

Studying statistical physics, in particular studying paramagnets and ferromagnets, I found some quantum mechanical formulas that I don't understand how to interpret. I don't understand if they are ...
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1 vote
2 answers
80 views

Maxwell's eq-meaning of del's cross and dot product?

In maxwell's eq there is del whose cross and dot products exist. So what is del in cross vs dot product. What's the difference when it's just a partial differential operator.
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2 votes
1 answer
59 views

Does the expression "$𝑑𝑠^2$..." mean the same thing as "$\Delta 𝑠^2$... "?

I reviewed this question but sometimes I'm unsure about delta versus differential notation. Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the same thing as &...
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0 votes
0 answers
39 views

What does $\partial_{k_\mu}$ mean on the Simon Laboratory site at Stanford University? (condensed matter physics and quantum optics)

I realize this is a strange question, but I have come upon the Simon lab site a lot, and that symbol or motto of theirs is everywhere, and I am just wondering if it means anything..... http://simonlab....
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3 votes
3 answers
80 views

When or why to use the $\equiv$ symbol in place of the $=$ symbol?

In literature, I read the following: A typical relationship*, often appearing in the literature, is: $$|-\nabla(\bar p+\rho g z)|\equiv \rho g J=q(\mu w+\rho Bq^m)$$ The nomenclature does not define ...
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0 answers
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What does this curly bracket notation mean?

I came across this $$\langle H_A^{i\{1,\,k\}}\rangle=f^{i\{1,\,k\}},\,\langle H_{A\{1,\,k\}i}\rangle=f_{A\{1,\,k\}i},\,i,\,k=2,\,3$$ curly bracket notation to denote VEVs (vacuum expectation values) ...
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1 vote
2 answers
67 views

What is $M_N$ in the Goldberger-Treiman relation?

$$g_{\pi NN} F_\pi = G_A M_N .$$ Does it stand for the magnetic moment of the neutron? One place I came across it was on Wikipedia, on their QCD Vacuum page, in the section about experimental evidence,...
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0 votes
1 answer
54 views

Terminology of $SU(3)_F$

From a terminological point of view, what is the relatioship between the flavor symmetry group $SU(3)_F$ of strong interaction and the group $SU(3)$ (without subscript $F$) of 3x3 unitary matrices ...
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1 vote
1 answer
42 views

Matrix index notation and Einstein summation

Can I ask why these two expressions are not equal? $$\begin{align}A_{ij}V^j&\ne V^kA_{ki}\\A_{ij}B^{ij}&\ne A^i{}_jB^j{}_i\end{align}$$
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0 answers
51 views

Srednicki eq. (1.27): $\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}$

Srednicki, QFT, p. 8 writes $$\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}\tag{1.27}.$$ What does exactly $ab$ here denote? Assume I have a matrix X [0 1] [2 3] and does a ...
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0 votes
2 answers
73 views

Position and Momentum operators in Heisenberg and Schrödinger picture

The position operator $\hat{x}$ has eigenstates $$\hat{x}|x\rangle=x|x \rangle.$$ Usually in the Schrödinger picture the operators are time independent and the states carry the time dependence. ...
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0 votes
1 answer
37 views

How does the net vertical force equation for a non-relativistic string derived?

In the following image from "A first course in string theory", we get the net vertical force of a string, dFv. While I understand the first equation, I don't understand how the second ...
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5 votes
2 answers
348 views

Different formal definitions of Lorentz Transformations

The formal definition for Lorentz Transformation is a matrix $\Lambda$ such that $$\Lambda^\mu_{\ \ \alpha}\Lambda^\nu_{\ \ \beta}\eta_{\mu\nu}=\eta_{\alpha\beta.}$$ In some books I have found a ...
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1 vote
1 answer
49 views

What does Leggett mean by quantum states like $|\psi\rangle=(a|\psi_1\rangle+b|\psi_2\rangle)^N$?

In his article (p. 1986) Legett uses the notation $|\psi\rangle=(a|\psi_1\rangle+b|\psi_2\rangle)^N$ to classify "macroscopic quantum phenomena". Does the "$^N$" mean "$\...
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-4 votes
1 answer
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What do the symbols $\mu$ and $\nu$ mean in General Relativity?

I'm not an expert on general relativity, and below the tensors in the Einstein Field Equations, there are two confusing symbols: $\mu$ and $\nu$ below them. What do they mean? Any equations are ...
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1 vote
1 answer
81 views

Wavefunction in polar coordinates and its bra ket notation

The wavefunction of $|\psi\rangle$ is given by the bra ket $ \psi (x,y,z)= \langle \vec{r}| \psi\rangle$ . I can convert the wavefunction from Cartesian to polar and have the wavefunction as $ \psi (...
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Is $\bar{\psi} \psi$ its own complex (*)? transpose ($T$)? hermitian conjugate ($\dagger$)?

Related to this earlier A common standard model Lagrangian mistake? Here I am treating Dirac equation of Dirac field as QFT. You may want to consider the quantized version or the classical version. ...
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2 votes
4 answers
187 views

Why do we use different differential notation for heat and work?

Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we ...
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1 vote
1 answer
114 views

How to make notation like $Y_{l m_{l}}(\theta, \phi)\chi_{m_s}$ more rigorous as a tensor product?

Sometimes in quantum mechanics we come across notation like $Y_{l m_{l}}(\theta, \phi)\chi_{sm_s}$ where $Y_{lm_l}$ is a spherical harmonic representing the spatial part of some particle wavefunction ...
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