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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Notation in the Liouville-von Neumann equation $ i\hbar\frac{dI}{dt} = i\hbar\frac{\partial I}{\partial t} + [I, H] $ [duplicate]
The Liouville-von Neumannn equation is defined by
$$
i\hbar\frac{dI}{dt} = i\hbar\frac{\partial I}{\partial t} + [I, H]
$$
where $I$ is any operator and $H$ is the Hamiltonian. I assume that the left-...
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2
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Dirac Formalism/Notation Question
I'm reading the 3rd edition of Sakurai and Napolitano's Modern Quantum Mechanics, which (probably rightly) relegates wave mechanics to an appendix. Instead, it carefully develops Dirac's formalism, ...
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Notation of ghost fields $b$, $\tilde{b}$, $c$, and $\tilde{c}$ in Polchinski
I am terrifically confused by the notation in Polchinski's string theory book from chapter 3 to chapter 4. The ghost action of the bosonic string in conformal gauge is (3.3.24)
$$S = \frac{1}{2 \pi} \...
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1
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Spectroscopic Notation - Does $n^{2S+1} L_J$ describe a Single Electron, or the state of multiple Electrons?
I was introduced to spectroscopic notation as $n^{2S+1} L_J$ (with L = "S, P, D..").
And the meaning that n stands for the principal quantum number (energy level, as known from the hydrogen ...
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Clarification of Notation in MTW's Gravitation, Section 6.4
Three difficulties concerning notation in MTW's Gravitation Section 6.4, which introduces a coordinate system for an accelerated frame:
the time axis e0' is the observer's 4-velocity, so he is always ...
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1
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79
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What does $\partial_ν/\partial^2$ mean?
I found such notation in this article link, equations 24-25. I know that $\partial_μ$ is four-gradient, but it does not contain second-order derivatives. Only d'Alembert operator does, $\partial^μ\...
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2
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111
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Understanding this Lagrangian calculation
I was trying to understand this section of a Wikipedia article:
$$0 = \delta \int \sqrt{2T} d\tau =
\int \frac{\delta T}{\sqrt{2T}} d\tau =
\frac{1}{c} \delta \int T d\tau$$
For the life of me, ...
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1
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110
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What does the $\Delta x $ in Heisenberg uncertainty principle actually mean?
In the textbook ''Concepts of Modern Physics -Arthur Beiser'' in chapter 3 section 3.7 where the book talks about the uncertainty princple
While illustrating the physics behind the uncertainty ...
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Shankar's definition of adjoint [duplicate]
I think Shankar's definition of adjoint operator (in his QM book) differs from many other sources.
On page 26, he made the definition
$$\langle \Omega V|=\langle V|\Omega^\dagger \quad .$$
Now $\...
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Is there a way to rewrite all of circular motions in exterior algebra?
The torque is defined as the cross product between the position vector and the applied force:
$$\tau = \vec{r} \times \vec{F}. $$
The cross product only works in 2 and 3 dimensions and the ...
- 11
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2
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Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
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What does a dot over a spinor index signify?
My questions should be rather simple. I was trying to get through one of my professor’s papers, and I saw the following notation, first with regards to Dirac and Weyl spinors, but the notation ...
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Why can the time evolution operator be left or right multiplied on bra or ket? [duplicate]
In the book Modern Quantum Mechanics by Sakurai, it said that:
We see that the expansion coefficients of a state ket in terms of base kets are the same in both pictures: $$ \begin{aligned} &
c_{...
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2
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79
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Why is Lorentz Transformation defined with one super and one sub index?
I came across the Lorentz transformation in tensor form, usually written as
$$\Lambda ^\mu _{\nu}$$
I understand that the first index usually corresponds to rows and the second to columns, and while I ...
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Question on index notation
I am working my way through Carroll's text on GR and am having trouble understanding what it means when an index is inside/outside parentheses. For example, in his discussion of geodesic deviation, ...
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Inner Product of two 4-vectors
I have a question on the inner product of two 4-vectors. As per the definition the inner product of two 4-vectors is defined as,
$$\vec A.\vec B = -A^0B^0+A^1B^1+A^2B^2+A^3B^3$$
From linear algebra, ...
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1
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75
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Elementary question about Einstein notation
I have encountered, in a physics textbook, the following Lagrangian:
$$L=\dfrac{m}{2}g_{ij}(x^k)\dot{x}^i\dot{x}^j.$$
I understand that Einstein notation is being used, and therefore there is an ...
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2
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116
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Determinant of an operator in ket/bra/bracket form?
As is well known, two examples of basis invariant functions are the trace and
determinant functions. These functions can therefore be thought of as
a property of a linear operator, and not just a ...
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1
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162
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Time ordering of integral [closed]
Is $$T\int\mathrm{d}^4x\phi^4(x)$$ just notation for $$\int\mathrm{d}^4x~T\phi^4(x)$$ since after integrating we have no time dependence anymore?
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528
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Navier-Stokes equation (notation for convection term)
Incompressible Navier-Stokes in vector notation is written as
$${\partial U \over \partial t}+(U\cdot\nabla)U =-\frac{1}{\rho} \nabla P + \nu \nabla^2(U),$$
where $U$ is velocity vector field $U=(u,v)...
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1
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Generalized forces for virtual work - Why did they drop the summation?
I am going through a PDF by Subhankar Ray & J. Shamanna on virtual work here and according to the PDF, equation 29, they write gerneralized force as:
$$Q_j = -\nabla_k\tilde{V}\cdot\left(\frac{\...
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1
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Notation in Zee's quantum field theory
I am reading Anthony Zee's Quantum Field Theory in a Nutshell(1st edition). On page 123, he does an integration by integrating by parts:
$$\begin{align}\int&\frac{d\omega}{2\pi}\log\left[\frac{\...
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3
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128
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Dirac notation and spin confusion [closed]
In my book I have $$\chi = a \chi_+ + b \chi_- = \begin{bmatrix}a \\ b \end{bmatrix} \tag{1}$$
Also, $$| 1/2, 1/2\rangle = \chi_+$$
The way I see that is that $a \chi_+ =|a/2 , a/2\rangle$, but ...
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Does this particular notation for derivatives imply anything in particular? [duplicate]
In some physics textbooks (and in those of other sciences that use physics, like soil science), I've seen some derivatives written as:
$$\frac{\delta f}{\delta t} $$
Which is a bit strange. Does this ...
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1
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What does the differential $d\Sigma_{ab}$ means when integrating over a two-surface?
In the paper $[1]$, Bardeen integrated an identity between Killing vectors and the Ricci tensor. I'll reproduce the calculation and explain my question in the following.
Consider then the identity,
$$...
- 2,841
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4
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172
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Clarification on operators and completeness relation
Given the momentum operator $$\hat{p} = -i\hbar \partial_x$$ we can apply the completeness relation to get $$\hat{p} = (-i\hbar \partial_x)(\sum_{a'}\vert a' \rangle \langle a' \vert) = -i\hbar(\sum_{...
6
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2
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483
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Exchange operator with position and momentum
I need to prove that for a given exchange operator $\hat{P}_{12}$ such that,
$$\hat{P}_{12}|x_1,x_2\rangle = |x_2,x_1\rangle $$
$\hat{P}_{12}\hat{X}_1\hat{P}_{12}=x_2$ and $\hat{P}_{12}\hat {P}_1\hat{...
- 2,970
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1
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Is this explanation of the bra-ket notation correct?
I would be very grateful for feedback, particularly pointing any factual mistakes, in the below explanation of the bra-ket notation:
"Quantum mechanical expressions can be simplified using a bra-...
- 3
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0
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Error propagation formulation for intrinsic or implicit randomness
I have a question about how to formulate an equation with implicit uncertainties for error propagation, i.e. a quantity $y$ that is determined by input values $x$ and $t$ but also has its own ...
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What does this means $| + ; \hat{x}\rangle$? [closed]
I know that the $+$ means spin up and that $\hat{x}$ is the position operator but I don't undestand the meaning of this two sign together in the same bra (like this $| + ; \hat{x}\rangle$ )?
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What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
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Distributing operators inside of the bra and kets confusion
I'm reading Griffiths and he has this section where he states that $|\hat{Q}f\rangle$ is mathematical nonsense and that really we should write $\hat{Q}|f\rangle$, where the latter makes more sense to ...
- 111
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58
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About $\mathfrak{sl}(2,\mathbb{C})$ and its generators
I am taking a class of SUSY at my university and the teacher was explaining $\mathfrak{sl}(2,\mathbb{C})$ has generators $\sigma_{ab}$, for $a<b$, on the $\rho_0(N)$ representation, with:
\begin{...
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2
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64
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Elementary question about the Euler-Lagrange equation
I need to calculate the Euler-Lagrange equation for a given Lagrangian density $\mathcal{L}$, that depends on a field $\phi=\phi(t,x)$. The statement of the problem provides me with the following ...
- 424
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How relative measurements are represented
This might be an easy question. But I need a clear answer.
When we represent a measurement relative to another we usually write it as, for example
acceleration of a, relative to b: aa,b
potential in ...
2
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2
answers
57
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Why is the mass of small elements taken as $∆m$ in center of mass of a continuous body?
A continuous body has continuous distribution of mass. Doesn't $\Delta m$ mean $m_f - m_i$? But, is the mass Changing? If yes, how is the mass varying? Why is the mass of the small elements in a body ...
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Is there a common notation for sets of quantum states, Hamiltonians, and gates having both physical and logical counterparts?
I'm a control theorist writing an article for a physics journal. The paper features a subsystem code $\mathcal C$ comprising a subset $S_{\mathcal C}(\mathcal H)$ of physical quantum states (density ...
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429
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Notion of Co- and Contravariance in Dirac-Notation
$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$
A (...
- 175
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Unpacking integrals in QFT and their notation
In a standard QFT course integrals are often written $$\int d^4pf(p)=\int dp^0\int dp^1\int dp^2 \int dp^3f(p^0,p^1,p^2,p^3).$$ This is just the standard notation for packing in a lot of math into a ...
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Notation: What's the group $G_1\backslash G_2$ compare to $G_2/G_3$?
Quote Clifford Johnson D-brane page 108
This group includes the T-dualities on all of the $d$
circles, linear redefinitions of the axes, and discrete shifts ofthe $B$-field. The full space of torus ...
- 3,044
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What does $g^{a}{}_{b}$ mean in general relativity? [duplicate]
I'm having a hard time getting a grasp of what $g^{a}{}_{b}$ truly means in GR. I understand that somehow it is the metric tensor in a sense, but I don't know its properties. I have some wild guesses ...
1
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2
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Help to understand $\mu\nu$ notation and differential forms
I'm hoping someone might be willing to help me obtain an understanding of the differential forms notation ubiquitous in QFT and GR. My problem is I'm fairly new to tensor algebra/calculus but I don't ...
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Notation of trace in QCD
I have been reading about QCD, and I have found a notation about the trace at Schwinger Dyson equation like $Tr_D[exp]$ and I don not know whats does means. Thank you.
Update reference: equation 27
...
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1
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195
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Bra-Ket Notation for creation and annihilation operators
I'm reading from Landau's book about second quantization and I confused about the bra-ket notation for the creation and annhilation operators.
From the book, annhilation oparator defined as
$$
a_i|N_i\...
1
vote
1
answer
76
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Unclear on spacetime vectors
I am self-learning QFT as a hobby, which requires me to first learn special relativity. I have come across the notion of a spacetime vector and I am unclear on a few aspects regarding it.
What I do ...
- 2,110
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4
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86
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Angular velocity: why is integral written with $t$ and with $\tau$?
In a physics exercise book I am using, the integral to get the angular velocity $\omega$ from some angular acceleration $\alpha (t)$ is written like this:
$$\omega(t) = \int_0^t \alpha(\tau) d \tau = \...
- 131
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0
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129
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Thermodynamics Chain Rule And Independent Variables
I was reading my textbook and I came up across the entropy $S(T,V,N)$ where temperature $T$, volume $V$, and number of particles $N$ are the independent variables. According to the chain rule the ...
- 73
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0
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107
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Advanced index notation
I am currently profficient with basic Einstein index notation, however there are many advanced things with Lorentz indices (such as derivatives, differences between which index is written first on a ...
-4
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1
answer
117
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How do I talk to engineers about vector components? [closed]
I'd like to make what I regard as a very simple Pythagorean point using notation that engineers will be familiar with. I am not an engineer. Here we go:
Let there be some vectors (in 2-space) that ...
- 1
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1
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74
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How does an operator act when taking an expectation value? [closed]
So I am reading this book where they say this:
My understanding of how operators work is that :
$$\langle n|\hat{a}|\alpha \rangle = \langle n|\hat{a}\alpha \rangle = \langle n\hat{a}^\dagger |\alpha ...
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