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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Why do we use down and up arrows? [closed]
Using electron Quantum Superposition, electrons exist as a superposition of their different spin states (-1/2 and +1/2). If so, apart from simplicity is there any other reason we use the up and down ...
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Why is the notation/convention of the equation for an electron scattering event $e−e−→e−e−$ rather than $e−+e−→e−+e−$? [closed]
I'm trying to help out my son who is attempting year 12 physics. They have to cover Lepton and Baryon conservation, and understand 5 Feynman diagrams - Moller scattering, Bhabha scattering, electron-...
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How to go from usual vector/matrix notation to weyl spinor notation?
I've always had trouble understanding the notation with dots, also known as the notation for Weyl spinors or Van der Waerden notation. I would like to ask for help.
For example, (Becker-Becker, page ...
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Why we have $|P\rangle =\int dp \space |p\rangle p \langle p| =-i\hbar \int dx |x\rangle \frac{d}{dx} \langle x|$?
And similarly, why we have $$\langle x|P|x' \rangle =-i\hbar \frac{d}{dx} \delta(x-x')~?$$
Both equations come from the Wikipedia page Momentum operator.
My question is, if we insert the identity ...
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Occupation number doubt in QFT [closed]
Let say we have the Hamiltonian of a free scalar field in QFT:
$$\hat{H} = \mathcal{V}\int{\frac{d^3k}{8\pi^3}\hbar\omega_k\left(\hat{a}^\dagger_k\hat{a}_k+\frac{1}{2}\right)}$$
The question is what ...
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Position Eigenstate and Parity Operator
I would like to ask for help with a question that I can't solve.
I know that in quantum mechanics an abstract state $|x\rangle$ (position) is defined up to a phase $e^{i\theta}|x\rangle$.
Now we know ...
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A confusing tensor index calculation [duplicate]
We define Lorentz transformation as $\Lambda$ satisfying:$$\eta_{\alpha\beta}=\eta_{\mu\nu}\Lambda^{\mu}_{\space\space\alpha}\Lambda^{\nu}_{\space\space\beta}$$
And its inverse$$(\Lambda^{-1})^{\mu}_{\...
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A question about operators in QM and notation
I have a problem understanding a notation written by my professor. He writes:
$$\langle n | a^+ | m \rangle=\langle n | (a^+ | m \rangle) = (\langle n | a) | m \rangle$$
and he wants to demonstrate ...
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Meaning of $\mathbb{R}^{1,3}$ and $\mathbb{R}^{3,1}$ notation for Minkowski spacetime
In David Tong's lecture notes on general relativity (page 12) he denotes Minkowski spacetime as $\mathbb{R}^{1,3}$. Also, on Wikipedia, I found that both $\mathbb{R}^{1,3}$ and $\mathbb{R}^{3,1}$ are ...
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Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as:
$\chi^{(2)}:\textbf{E}\textbf{E}$
or
$\nabla\textbf{E}:\partial \...
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Odd notation $\stackrel{\leftarrow}{\nabla}$ for a gradient
I've tried working out the Heisenberg EOM for the 4-current operator. Two very beautiful articles (DOI: 10.1103/PhysRevA.84.042107, DOI: 10.1103/PhysRevA.90.012508) present this result, but I have not ...
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What are $w,x,y,z$ electronic transitions? How are they classified?
This paper's abstract mentions transitions $(x,y,z)$ and mentions another paper which talk about $w$ transitions in He-like Ti ions. The papers have diagrams which clearly show what these transitions ...
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Base change using bra-ket notation
If I have two orthonormal bases (of a vector space over $\mathbb{C}$) $A=\{|a_{1}\rangle, |a_{2}\rangle\}$ and $B=\{|b_{1}\rangle, |b_{2}\rangle\}$, the change of base matrix from $A$ to $B$ is given ...
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Explanation for the $(a,b)$ Notation in Super Conformal Field Theories [duplicate]
In the literature, there are many theories and quantities which I usually seen in the context of conformal field theory that described via the notation of $(a,b);\quad a,b\in\mathbb{Z}$
for example:
...
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One doubt about Dirac notation [closed]
I am encountering an issue with Dirac notation and I hope someone can help me. Thank you in advance.
I know that if $|n\rangle$ and $|m\rangle$ are eigenstates of the time evolution operator $U = e^{-\...
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Notation for vector density in Lagrangian density
Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation
$$\int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
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Symbol denoting parity eigenvalue
What is the symbol reserved for designating the parity of a parity eigenstate?
For example an eigenstate $\phi$ of the squared angular momentum operator $\hat{\mathbf{L}}^2$ is characterized by a ...
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Dirac's Bracket Notation
I have a question on Dirac's bracket notation. In particular, according to this notation, vectors and covectors are represented by $|\psi\rangle$ and $\langle\psi|$ respectively. Moreover, these two ...
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Operator's definition in Dirac picture [closed]
I have a question about the definition of quantum operators in the Dirac picture. The definition is: $$A=\sum_i \sum_j \vert i \rangle A_{ij} \langle j \vert.\tag{1}$$
By deplacing the ket vector I ...
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What does it mean when vector quantity is negative?
I’ve seen in some problems that vector quantity is negative. For example, negative Coulomb means that two charges attract… negative acceleration means that object is slowing down. However, magnitude ...
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What does $v$ actually represent in pulley problems?
Let us consider the problem of two pulleys with boxes attached to their ends. One pulley is movable, and the other is not. The problem requires us to find a relationship between velocities of these ...
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About the notation for TEM waves
I just came across this article where the term "$\mathrm{TE}_{101}$ microwave mode" is mentioned. Other than the basics of TEM waves which I learned in Griffiths, this is the first time I ...
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What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$
What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$
In John Dirk Walecka's book 'Introduction to General Relativity',...
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Multiplication of $\mathrm{U}(3)$ matrices [closed]
On this page of this paper:
I am unable to understand how they multiplied the $3\times 3$ $\mathrm{U}(3)$ matrix with $T_{3,2}$, which is a $2\times 2$ matrix, in Eqs. (26) to (28). Can anyone please ...
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What does the superscript $3$ in $d^3x$ mean in an integral?
At the risk of seeming ignorant, please explain what does the superscript $3$ in $d^3x$ mean in the integrals 5.12, 5.13, 5.14, 5.15? Why 3? Why there is no such in the 5.10 integral?
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What is $r$ in a metric signature in general relativity? If $v$ and $p$ are the time and spatial coordinates?
The Wikipedia article on metric signatures says that the signature of a metric can be written $(v,p,r)$, where $v$ is the number of positive eigenvalues, $p$ is the number of negative eigenvalues, and ...
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Would it be valid to say $y \propto 2x$ instead of $y \propto x$?
Would it be acceptable to write $y \propto 2x$ or would it be wrong to add the $2$? I just want to describe a relationship in more detail.
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Questioning the Probability Expression for Neutrino Oscillation in Griffiths' "Introduction to Elementary Particles"
In Griffiths' book, Introduction to Elementary Particles (Griffiths, D. (2020). John Wiley & Sons, p. 390), the author defines the pure electron and muon neutrino states as:
$$|ν_{e}\rangle=-\sinθ|...
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What is the meaning of the tensor product ($\otimes$) of groups? Is is the same as the cartesian product ($\times$)? [duplicate]
In the high energy physics literature and in some discussions on group theory for physics, the tensor product $\otimes$ of groups is sometimes mentioned. For example, the gauge group of the standard ...
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Index notation for the generalized form of the velocity gradient with shear and rotation
Good day guys, I was studying fluid dynamics and came across the following equations (shown in the image):
When we have both shear and and rotation, they each contribute to the change in velocity ...
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What does the notation $d𝜏'$ mean?
$\text{I was studying helmotz theorem and saw this notation, what does it mean? How is d}\tau' \, \text{ different from d}\tau \text{?}$
From :- David J. Griffiths-Introduction to Electrodynamics-...
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Understanding differentials in an equation in general relativity
I have not studied physics but I was browsing Carroll's relativity book and randomly stumbled upon the following which I would like to understand mathematically. It says
$$ds^{2} = 0 = - \left( 1 - \...
Esoog
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Complex coordinates $ds^2 = dzdz̄$ in 2d
I have a very elementary question about complex coordinates in two dimensions. When we have a 2D Euclidean space,
$$ds^2 = dx^2 +dy^2$$
and we go to complex coordinates:
$$z = x + iy$$ $$z̄ = x - iy$$
...
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Component notation and matrix notation for gradient of vector
I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
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Coordinate-free calculations in bra-ket notation [closed]
I'm interested if bra-ket notation can be used without having to introduce a basis or coordinates in calculations and proofs.
For example, could the notation, if need be, extended with some exterior ...
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Integration over the complex plane and the completeness relation of the coherent states [duplicate]
I am studying some of the properties of coherent states using the book "Introductory Quantum Optics" by C. Gerry & L. Knight. (C. Gerry & L. Knight, Chapter 3, Section 5) And when I ...
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Understading dimensions in quark bilinears
I have encountered myself with the following definition for $\pi$-fields as quark bilinears:
$$
\pi^a = i\bar{q}\tau^a \gamma_5 q \ ,\quad\text{with }\ q = \left(\begin{array}{c}u\\d\end{array}\right) ...
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Mathematical meaning of a position eigenbra $\langle x_0 |$
Let $|x_0\rangle$ be an position eigenket. The physical picture I have for $|x_0\rangle$ is a particle located at $x_0$. Thus it should be represented by a delta function $\delta(x-x_0)$.
For $f\in L^...
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Can unit prefixes be interpreted as algebraic stand-ins?
I am currently a little confused about whether unit prefixes can be interpreted algebraically
For example:
$$4km = 4\times 10^{3} m$$
Is it incorrect to say that $k$ is simply an algebraic stand in ...
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Question about meaning of "bar"-ing in the context of Dirac fields
Following chapter 38 of Srednicki, "bar"-ing means (based on eq. 38.15)
$$\bar{A} = \beta A^\dagger\beta$$
One can show for instance that $$\bar{\gamma^\mu} = \gamma^\mu$$ My question is, ...
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Orthogonality of Lorentz transformation [duplicate]
I know this might be a general or well known topic, but please bear with me, since I have a very specific question regarding this and I want to see specifically whether there is a flaw in my reasoning ...
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What is the effect of the operator $\hat{O} = |\phi \rangle \langle \psi|$ on any state $| \psi \rangle$? [closed]
I have seen this operator in some questions,including the book Liboff,Introductory Quantum Mechanics. I am not sure how it would act on a state.
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Raising and lowering indices of Kronecker delta [duplicate]
I have some confusion about how to raise the indices of the Kronecker delta.
To raise and lower indices we use the metric tensor, let's suppose to use the metric (+---).
I should have that $$g_{\mu\nu}...
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Why do we multiply the Euler-Lagrange equations by negative one?
As I've learned classical mechanics from different sources, I've seen both
$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_k} \right) - \frac{\partial L}{\partial q_k} = 0,$$
and
$$\frac{\...
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Directional derivative $(\mathbf{A}\cdot\nabla)\mathbf{B}$ of the vector field $\mathbf{B}$
While reading Introduction to Electrodynamics by David J. Griffiths, I have encountered some issue with the notation of the directional derivative of the vector field and I was wondering if there are ...
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Einstein Summation Convention Confusion
My textbook:
The second bit confuses me. I asked a question on this site yesterday (Moment of Inertia tensor confusion) which involved the moment of inertia tensor and the term $$r_{i}r_{j}$$
The ...
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Contracting the metric tensor with its inverse yields Kronecker delta
It's probably straightforward, but I would like to see the proof of the identity:
$$g_{\mu\nu}g^{\nu\alpha}=\delta^\alpha_\mu.$$
In the book 'Spacetime and Geometry' by Carroll, this identity is the ...
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Moment of Inertia tensor confusion
This is the moment of inertia tensor about the COM.
$$I_{ij}'=\sum_{k=1}^Nm_k\left[\vert\mathbf{r}'^{(k)}\vert^2\delta_{ij}-r_i'^{(k)}r_j'^{(k)}\right]\neq I_{ij}$$
I don't understand how to compute ...
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In diagrams, how can I indicate that a body is not moving or rotating?
I often create diagrams to illustrate the scenario of an exercise for my students. These are not necessarily free-body diagrams, as creating those may be part of the task or similar or it would be ...
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Where does the negative sign disappear?
The defining equation for simple harmonic motion is such
$$a=-ω^2x$$
When we find the centripetal acceleration of an object in orbit we use the formula
$$a=ω^2r$$
As a consequence of the accleration ...
