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.
1,541
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Meaning of the transpose of a gradient
Sometimes I encounter PDE's with a term like this
$\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$
An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $...
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3
votes
1
answer
479
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Issues with operator associativity in quantum mechanics
Textbook authors often point put that operators aren't in general commutative which is a source of confusion. Another point of confusion, which I haven't seen mentioned anywhere is the issue with ...
- 1,547
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49
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$(α|0⟩ + β|1⟩)|0⟩$ in matrix/vector form
I am currently working through superdense coding with bell states and have a question regarding this value:
$$(α|0⟩ \ + \ β|1⟩) \ |0⟩$$
I understand that $α|0⟩ \ + \ β|1⟩$ can be represented in ...
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vote
1
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Notation question: what is $N_A$?
I'm reading some literature on stellar opacity, and there is some notation that I don't understand. In particular, they write that the total density $\rho$ is related to the electron number density $...
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Notation used to denote the unit $\rm mm^3$
For example, to denote the volume of a cube with the sides $1mm$ we write $1mm^3$. But I think it should be written as $1 (mm)^3$ because the prefix $m$ denote the factor $10^{-3}$ hence $$[1mm^3=10^{-...
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1
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Problem understanding the use of differentials in equation for energy stored in a capacitor
I was going through the Feynman Lecures on Physics Vol. II and arrived at section 8.1, where the equation for the energy stored in a capacitor is obtained.
We consider now the energy required to ...
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3
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214
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What is the significance of sliding a force vector?
Scenario 1:
Here $AB$ is the line of action of the force $\underline{u}$, and $AB$ passes through the center of mass of an infinitely long rigid and uniform bar. Here, $\vec{CD}=\underline{u}$. Now, ...
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1
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Is my book drawing a vector wrongly?
My book's drawing:
According to my book, $\underline{u}=\vec{AB}$. However, my book hasn't put the arrow at the end of the line segment $AB$, but rather my book has put the arrow at the middle of $AB$...
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2
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134
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When drawing a vector on paper, does it matter where I'm putting the arrow?
Let a vector $\vec{P}$ passes along the line segment $AB$. Now, from the figure, can we say that $|\vec{P}|=AD$? [$D$ is the point where I drew the arrow]
In other words, when drawing a vector on ...
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1
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390
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Vector symbols confusion: A vs |Ā| when it comes to representation of magnitude of vectors
My teacher said that $\vec{A} = \left| \vec{A} \right| \hat{A}$ , where $\left| \vec{A} \right|$ is the magnitude and  is the direction of the vector.
In this homework question, what exactly do you ...
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2
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717
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What do these subscripts mean?
I am reading an introductory article about quantum optics, I am confused what $\hat{a_j}$ means, I tried to interpret it as applying $\hat{a}$ then taking the component along $|j>$ but it's not ...
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3
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$\left\langle{\hat{O}}^\dagger\varphi\middle|\psi\right\rangle$ How do I act the operator in bra?
$$\left\langle\varphi\middle|\hat{O}\middle|\psi\right\rangle=\left\langle{\hat{O}}^\dagger\varphi\middle|\psi\right\rangle.$$
In above formula, I have confused what does mean $\left\langle{\hat{O}}^\...
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If $\underline{u}$ is a vector, does $u$ indicate its magnitude?
If $\underline{u}$ is a vector, does $u$ indicate its magnitude? $|\underline{u}|$ also indicates the magnitude, doesn't it?
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How should second differentials and differentials of one-forms be expressed?
Components with under-barred indices are expressed on a covariantly constant basis. This is equivalent saying the under-barred coordinates lie in the tangent plane. Basis vectors are written in the ...
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85
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Tensor contraction in matrix product state
From Matrix product state, the matrix product can be written as
$$
|\Psi\rangle = \sum_{\{s\}} \operatorname{Tr}\left[A_1^{(s_1)} A_2^{(s_2)} \cdots A_N^{(s_N)}\right] |s_1 s_2 \ldots s_N\rangle,
$$
...
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1
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Covariant derivative with an upper index in terms of Christoffel symbols
I have encountered expression
$$\frac{1}{2}\left(2 \dot{g}_{\mu}{}^{\lambda ; \mu}-\dot{g}_{\mu}{}^{\mu ; \lambda}\right)$$
in a GR paper.
Here we assume to be working with the de Sitter metric $g$ ...
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vote
1
answer
98
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Meaning of big $O$ notation with 2 values separated by a comma
I'm reading Classical Electrodynamics 3e by Jackson. In section 1.7 he performs a proof of the Poisson equation in the context of the electric potential. Near the end of the proof, he writes
$$
\...
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-2
votes
1
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Bras and kets question
If a state $|\psi\rangle$ can be written as a linear combination of the orthonormal states $|\phi_{n}\rangle$ as:
$|\psi\rangle=\sum_{n=1}^{\infty}c_{n}|\phi_{n}\rangle$
then is it valid to write:
$\...
1
vote
0
answers
125
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The Euclidean vector momentum operator in spherical coordinates
I am having some trouble with the notion that the different components of a vector operator can be hermitian in one coordinate system but non-hermitian in another. I have seen e.g. Bra-ket notation in ...
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2
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Dirac notation and matrix
If I have an operator $A$ whose elements can be written as $<\phi_{m}|A|\phi_{n}>$ does this mean that I can write the operator $A$ as:
$$A=|\phi_{m}><\phi_{m}|A|\phi_{n}><\phi_{n}|$$...
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0
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Understanding an integral from Quantum Field Theory
In Eberhard Zeidler's Quantum Field Theory I: Basics in Mathematics and Physics, under section 11.6.4 - Integration tricks, the following integral which is then evaluated using a Schwinger ...
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2
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80
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Dirac notation notation equality
I'm going trough my quantum mechanics notes but I don't understand why:
$$H|\phi_{m}\rangle\langle\phi_{n}|-|\phi_{m}\rangle\langle\phi_{n}|H=a_{m}\langle\phi_{n}|-a_{n}|\phi_{m}\rangle.$$
What is ...
2
votes
2
answers
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Is $\Delta$ notation commonly used for the difference in a quantity between two objects? [closed]
In a standard Atwood machine, the acceleration is
$$a = g\dfrac{m_1 - m_2}{m_1+m_2}.$$
Would writing this as
$$a = g \dfrac{\Delta m}{M}$$
where $M$ is the total mass be an abuse of notation to most ...
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1
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Question on Bracket Notation
Is it valid to say that that if $A$ is an operator then:
$$\langle f|A|g\rangle=\langle f|g\rangle A~?$$
0
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1
answer
121
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Dirac Notation and Delta Kronecker
I have one question regarding the Kronecker delta and the Dirac notation. Is it possible to write
$\vert\phi_{m}\rangle\delta_{nq}\langle\phi_{p}\vert=\delta_{nq}\vert\phi_{m}\rangle\langle\phi_{p}\...
1
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1
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141
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How is the projection operator derived?
I am having a difficult time understanding how do we go from
$$ C_{n}=\langle n | \psi\rangle $$
and
$$ \psi=\sum C_{n}|\psi\rangle$$
to
$$ \psi=\left(\sum |n\rangle\ \langle n| \right) |\psi \rangle.$...
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0
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Adding terms to the Lagrangian using quantum numbers (symmetries)
The SM is build on $SU(3)_C\times SU(2)_L\times U(1)_Y$.
Left quarks behave like $Q_l\sim(3,2,1/6)$ (colour triplet, $SU(2)$ doublet and $1/6$ hypercharge)
Or the SM Higgs $\Phi\sim(1,2,1/2)$
Since ...
0
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1
answer
127
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What is the Cauchy principal value in Sokhotski-Plemelj formula?
I met the Sokhotski-Plemelj formula in a paper:
in which $P$ is the Cauchy principal value. But the principal value in wiki is this form:
It is a limit of an integration. But the $P$ in the first ...
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1
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73
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Question on Dirac Notation
I understand that the ket $|x>$ can be written in matrix form as:
\begin{equation}
|x>=\begin{bmatrix}
x_{1} \\
x_{2} \\
\vdots \\
x_{n}
\end{...
1
vote
1
answer
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Why is the subscript $f$ used to refer to properties relating to that of saturated liquid?
In the formula for calculating Vapour Quality, the properties relating to the saturated liquid have a subscript $f$. The properties relating to that of saturated gas have a subscript $g$. I understand ...
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Please help me understand how these vectors work
Say I have a vector $R$ in general relativity where $$R = sin(\theta)\partial_{\theta} + cos(\theta)cos(\phi)\partial_{\phi} $$
What does the index of $R_{i}$ mean? I'm currently thinking of this as ...
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votes
1
answer
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The need for placeholder in tensor notation
In many books of GR, e.g., Sean Carroll, Wald, empty-space placeholders are added on the tensor component, e.g., in Wald's book
$$
T = \sum^n_{\mu_1, \cdots, \nu_l =1} T^{\mu_1 \cdots \mu_k}_{\,\,\,\,...
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2
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Justifications for the index upper lower labels in tensor component transformations
A (1,0)-type tensor may be written as
$$
V = V^{\mu} e_{\mu}
$$
The component transforms as
$$
V^{\nu} = A^{\nu}{}_{\mu^\prime} V^{\mu^\prime}
$$
(the basis can transform similarly)
My question is, ...
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1
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How to apply potential operator $V(\hat{x})$?
I want some clarification on the potential operator $V(\hat{x})$. Can you please help me
Is the action of $V(\hat{x})$ defined by its action on the position kets as $\hat{V}(x)|x\rangle=V(x)|x\...
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Subscript $Q$ in $U(1)_Q$
In most quantum field theory books, you read something like
QED is a QFT from Abelian $U(1)_Q$ that describes the electromagnetic interaction ...
But what does the subscript $Q$ stands for in the $U(...
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1
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160
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How to compute divergence of a metric tensor?
I am reading a paper where the author defines the divergence to be
$$\left(\delta_{g} \dot{g}\right)_{\mu}:=-\dot{g}_{\mu \kappa;}{}^{\kappa}$$
where $g$ looks like the De Sitter metric,
$$g=(3 / \...
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votes
1
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Mapping from tensor notation to matrix notation, left right or upper lower to row column?
In the matrix notation $M_{ab}$, the left index goes through the rows ($a = 1,2,3\dots m$ means there are $m$ rows). The right index go through the columns, ($b = 1,2,3\dots n$) means there are $n$ ...
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4
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How do I distinguish $\rm m$ for "meter" and $m$ for "mass"?
For example, an object in height 10 meters has 60 Joules of potential energy.
PE = mgh
60 J = m⋅(9.8 m/s²)(10m)
60 J = m⋅(98 m²/s²)
m ≈ 0.6
How can I distinguish it?
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Doubt on dotted indices notation for Weyl spinors
I'm starting Bailin and Love "Supersymmetric gauge field theory and string theory" and trying to get used to dotted indices.
Let's consider a Dirac spinor written in terms of left and right ...
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2
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Derivative of Delta fuction? [closed]
In Shankar's Quantum Mechanics book p-64 the last equation reads:
$$ \delta'(x'-x) = -\delta'(x-x'); $$
I am confused because if I think of it using the gaussian approximation then since:
$$ g(x' -x) ...
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1
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How does the symmetry of indices of tensors work?
Very quick and short clarification:
Knowing $R_{abcd} = - R_{bacd}$, is this symmetry true for for a,b components (i.e $R_{acbd}=-R_{bcad}$) or for the first two 'slots' of the tensor (i.e $R_{acbd}=-...
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Reordering Terms in Abstract Index Notation — General Relativity
In General Relativity by Wald, the author makes a claim that I am trying to understand. The crux of my question comes down to understanding why $\eqref{eq1}$ is true, but I have included the context ...
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0
answers
110
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Marginalisation of a joint probability distribution in bra-ket notation
Given a wave function $\Psi(\vec r_1, \vec r_2)$, where $\vec r_1$ and $\vec r_2$ are the positions of particle 1 and 2, respectively, the probability of finding particle 1 at position $\vec r$ (...
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3
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Is it possible to solve cross products using Einstein notation?
I'm considering a case where I have an equation of the form $\mathbf{x}\times\mathbf{b}+\mathbf{c}=0$; I wish to solve for $\mathbf{x}$ given that $\mathbf{b}\perp \mathbf{c}$. It was in the context ...
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2
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Grounding in Circuits [closed]
The question is to find the current through the 8 Ω resistor. My textbook says that it is 3A as there is 0 potential at points "a" and "b" and thus it must drop by 24 V across the ...
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1
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Nuclear physics notation
This is probably a very basic question about notation. If we have the following notation for Boron nuclei:
$$^{12}_5 B(1^+)\;\text{or}\; ^{10}_3B(3^+)$$
What does the number in the parentheses mean? I ...
- 75
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vote
2
answers
240
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What does Griffith mean by adding a prime on integration variables?
In the book "Introduction to Electrodynamics" by Griffith, the author mentions electric potential as a point function writes the equation for electric potential as
Then in a side note he ...
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1
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What does $p$ mean in the energy-momentum relation?
Suppose I have several particles each with some momentum vector $\mathbf{\vec p}_i$, what is $p$ in the relation $$E^2 = E_0^2 + (pc)^2$$
Is it the magnitude of the vector sum of all $\mathbf{\vec p}...
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1
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78
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What does it mean to "plug" a gradient?
Researching Cotton tensor I found in Wikipedia (https://en.wikipedia.org/wiki/Cotton_tensor):
$$\tilde{C}=C+\text{grad }\omega\lrcorner W $$ with a comment "where the gradient is plugged into the ...
1
vote
1
answer
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Action of permutation operator on other operators
I'm watching MIT 8.06 Quantum physics, lecture $23.2$ See for example [1] Particularly See $5:41$. It is shown that
$$P_{21}B(1)P^\dagger_{21}|u_i\rangle_1\otimes |u_j\rangle_2=|u_i\rangle_1\otimes |...
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