Questions tagged [notation]

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Meaning of the $\delta$ notation [duplicate]

Two examples I've seen of this so far are in statistical mechanics, when looking at the work done on a system: $$dW=-\frac{1}{\beta}\frac{\partial \ln{Z}}{\partial V}\delta V$$ and in the Noether ...
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36 views

Terminology for the symmetric vector product on the Bloch sphere

I was reading the paper "The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem" by Diederik Aerts, Massimiliano Sassoli de Bianchi. ...
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3answers
131 views

How to show $\hat{T}(a)=$exp$(-ia\hat{p}/\hbar )$ is the translation operator? [closed]

I'm trying to understand a proof my lecturer gave. We define $\hat{T}(a)=e^{-ia\hat{p}/\hbar}$ and write $\hat{x}\hat{T}(a)|x\rangle =\hat{x}(\mathbb{1}-i\frac{a}{\hbar}\hat{p}+\frac{1}{2}(-i\frac{a}{...
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2answers
125 views

Clarify definite integration of differentials in physics problems

I realized there is an issue with integration in physics problems that I had always taken for granted. As an example, the relation between work and potential energy is $dW=-dU_p$ when integrating ...
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2answers
89 views

What does it signify to have a value enclosed by angle brackets?

What is the significance of enclosing a value (say for instance, electric current $i$) by angle brackets instead of curly brackets? I am currently reading a book and there is an equation which ...
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1answer
24 views

What means $h$ which appears following cosmological measured parameters?

For example, Big Bang Nucleossynthesis says that baryonic content in the Universe is around $0.018 \leq \Omega_{b}h^{2} \leq 0.024$. I know it is something related to measurement error but I do not ...
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1answer
58 views

What is the significance of slashed lines used to represent a surface in mechanics?

Almost in all diagrams in mechanics, I noticed that a surface is represented by a line with a lot of slashes ("/") on one side, like the one shown below: I've seen this in optics where such slashed ...
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1answer
57 views

Geometrical optics problem and broader questions about the correct use of $\approx$ in physical calculations

My textbook, Fundamentals of Photonics, Third Edition, by Saleh and Teich, gives the following: This seems to be mathematically incorrect to me? Firstly, the author stated that $\phi = \psi - \...
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2answers
82 views

Taking the Hamiltonian eigenvalue problem into position space?

I'm having a hard time notationally understanding the relationship of: $$ \hat{H} \vert{\psi}\rangle = E\vert \psi\rangle $$ and $$\hat{H} \psi(x) = E\psi(x) $$ Here's my thought process: Starting ...
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1answer
61 views

Question about Einstein summation convention

I'm dealing with the following: $$\eta^{\alpha \mu} \eta_{\alpha \nu} \phi,_{\beta \mu}$$ $$\eta^{\alpha \beta} \phi,_{\alpha \beta}$$ where $\eta$ is the Minkowski metric and $\phi$ is a function ...
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1answer
70 views

Confusion about expressing an inner product using the Einstein summation convention

I think this likely comes down to the following expression, $$g’^{ab}e’_a e’_b = \delta ^a_b $$ Is this in agreement with the Einstein summation convention? Because even though the two indices are ...
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2answers
89 views

Writing Hermitian conjugate of quantum operator in integral form

I know that the Hermitian conjugate of a quantum operator $\hat{Q}$ can be represented as: $$\displaystyle{\left\langle\phi_1,\hat{Q}\phi_2\right\rangle= \left\langle\hat{Q}^\dagger \phi_1,\phi_2\...
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1answer
36 views

Completeness relation, commuting operators

I have a question about some formulars our professor wrote on the black board. Let $\hat{Q}_{1},...,\hat{Q}_{N}$ be operators, which are a CSCO. We know now that there exists a set of eigenvectors $\{...
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1answer
62 views

Difference between $g^{\alpha\beta}$ and $g^\alpha_{\space\space\beta}$

I'm working out a problem where at some point get the following product of metric tensors and momenta: $$g^{\mu\beta}g^\nu_{\space\space\alpha}(2k+\frac{q}{2})^\alpha(\frac{q}{2}-k)_\beta$$ How can I ...
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0answers
51 views

The mathematical structure of $\widehat{su(2)}_k$

Some of my colleagues work on CFT's and quantum groups and I hear them talk a lot about $\widehat{su(2)}_k$ algebras. According to them (and the general physics literature) these are what ...
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2answers
176 views

A question about the Angular Velocity Vector

First time asking here, so please forgive me if I'm doing anything wrong. I'm having quite a hard time with this equation: $$\mathbf{\omega} = \frac{1}{2} \left[ \left(\mathbf{i}\wedge\frac{d\mathbf{...
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1answer
55 views

Why can we 'act on the left' with an operator?

This comes from notes on perturbation theory I am learning in class. I have seen this a few times but never really understood it: how is it that we can apply an operator to the conjugate state on the ...
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3answers
111 views

What does it mean “differentiation with respect to the coordinates of particle 1 or 2”?

I was reading Introduction to Quantum Mechanics by Griffiths. In Chapter 5, Identical Particles, I came across the notation $\nabla_1$ and $\nabla_2$. Griffiths writes that it means "differentiation ...
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2answers
95 views

Covector basis derivation

On page 65 of Schutz's A first course in General Relativity, he introduces the notation $\phi_{,\alpha}=\partial\phi/\partial x^\alpha$. He then says that $x^\alpha_{\ \ ,\beta}=\delta^\alpha _{\ \ \ \...
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2answers
110 views

Why is the ångström not a metric unit? And why is the ångström spelt with the Scandinavian letters “å” and “ö”?

The website here http://unitsofmeasure.org/ucum.html tells us whether every unit is metric or not. Metric units can be multipled by a power of 10 and can be combined with a prefix. 1 ångström is ...
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1answer
30 views

How do you read the spin contributions of proton

This paper reads the number of their results. I'm attaching the how they have included all the spin contributions of quarks to proton. The contributions are from u, d, s quarks. How I can tell ...
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1answer
36 views

Spring constant of harmonic oscillator [closed]

I got a task from my lecturer to solve a differential equation for a simple harmonic oscillator: $$m{d^2\vec{r} \over dt^2}=-k^2\vec{r}.$$ So far, I have managed to find this equation only in one book....
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0answers
78 views

Is there a notation alternative to Hertz?

Hertz, as we all know, is a unit of frequency where 1 Hz equals one cycle per second. Conversely ...
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1answer
109 views
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3answers
148 views

If $\mathrm df$ is an inexact differential, how would the function $f$ look like?

I am studying thermodynamics and in the first chapter the concept of exact and inexact differentials were used to talk about the differences between internal energy, work and heat. From Blundell and ...
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1answer
159 views

Bra-ket notation in a 3 dimensional system

Recently we started to learn quantum mechanics in a three-dimensional system. However, I am very confused from the beginning. Firstly, professor told us that for a position vector $\pmb{x}={}^t(x,y,z)$...
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1answer
68 views

Question about computing Christoffel symbols

I am trying to calculate the Christoffel symbols in polar coordinates, and I am confused on one step. Given that I am here, for example: $$\Gamma_{r \theta}^{\theta}=\frac{1}{2} g^{\alpha \theta}\...
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1answer
72 views

Notation of physical dimension of complex values

I can't find a good answer on the proper way to write the physical units for complex numbers. $$ \begin{align} z &= 707 \text{ mV} + 0.707\mathrm{i} \tag{1} \\ z &= (707 + 707\mathrm{i}) \...
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1answer
91 views

Did R. Feynman know about the different notations for exact and inexact differentials? [closed]

I remember reading a long time ago, the story of a student taking R. Feynman for responsible of her (I think it was a woman, not sure though) fail at an exam of physics because what was written in her ...
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1answer
41 views

Expectation of partial time derivatives of $x$ in QM

In Ehrenfest theorem we know that $$m\frac{d\left< x\right>}{dt}=\left< p\right>+m\left<\frac{\partial x}{\partial t}\right>.$$ So how can I exactly calculate a specific $\left<\...
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4answers
2k views

How does a linear operator act on a bra?

I'm studying QM from Shankar. He introduces linear operators and says that an operator is an instruction for transforming one ket into another. But then a few lines below he says operators can also ...
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1answer
480 views

Is Schwarzschild's solution in his original paper consistent with current solutions? [duplicate]

I was reading the Schwarzschild's original paper where he derives the Schwarzschild metric for the first time(The english translated version found in arXiv : On the Gravitational Field of a Mass ...
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1answer
99 views

How to translate this equation into physicist's notation? [closed]

I asked this in math stackexchange but no one has answered there so I ask here. How to translate this equation into physicist's notation, i.e. tensors with indices? $$\left\langle R_{N}\left(u,v\...
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1answer
41 views

What does the notation $\mathcal{O}\left(\frac{1}{r^2}\right)$ mean? [duplicate]

I was reading a text about quantum scattering, and I faced a notation I don't understand. The equation is the following: $$ \nabla \psi_{\text{scattered}} = \frac{i k f(\theta) e^{ikr}}{r} \mathbf{\...
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1answer
84 views

Can anybody identify these two equations for me? [closed]

I am trying to identify these two equations: $$S=\int L\ \gamma \ d\tau \quad \tag{1}$$ $$\quad \frac{\partial L}{\partial \phi}-\partial_\mu\left[\frac{\partial L}{\partial(\partial_\mu \phi)}\...
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1answer
93 views

Spacing in indices, and relation to matrices, in special relativity notation

I have some general confusion regarding notation on tensors in special relativity, and how indices correspond to the matrix representation of second-rank tensors. When one has a second-rank tensor $...
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2answers
65 views

Commutation of position four-vector with spacetime derivatives

I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134 $$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
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2answers
244 views

Particle notation: What does the asterisk stand for?

For instance, what's the difference between $\Sigma^0$ particles and $\Sigma^{0*}$ particles? I know they have the same quark configuration $sdu$, so how exactly are they different? Some context: I'm ...
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2answers
105 views

How do you write $A A^T$ in Einstein notation?

In index notation it makes sense as $$\sum_j {A_{ij} A_{jk}^T} = \sum_j {A_{ij} A_{kj}}.\tag{1}$$ But this doesn't make sense for Einstein notation where in $$A^\mu_\sigma (A^\sigma_\nu)^T = A^\...
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1answer
119 views

Gauss's law for magnetism : double integral

Gauss's law for magnetism is stated as followed with the beautiful closed surface double integral (by wikipidia): $$ \mathop{\vcenter{ \huge\unicode{x222F}\, }}_{S} \mathbf{B} \cdot \text d\...
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1answer
114 views

How to read bra-ket notation? [closed]

Good afternoon, I am trying to understand the basics of some quantum mechanics theorems (e.g. Uncertainity principle). I'm looking for the correct way to read this expression while I'm speaking. For ...
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2answers
71 views

Is it reasonable and common to interpret $dt$ as a time point (a point in time)? [duplicate]

I heard some one talked about the instantaneous and average velocities. He was using $\Delta t$ to denote a time frame, $dt$ denote a time point. average velocities $\bar{v} = \dfrac{\Delta s}{\...
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1answer
65 views

How can uncertainty be represented as a chance [closed]

How can uncertainty be represented as a chance when we write in Heisenberg's principle? Why do we use Delta for uncertainty and mathematically use it the same way as if it was a change in position or ...
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2answers
154 views

Position operator acting on wavefunctions. Why is $\left<x\right|\hat{X}\left|{\psi}\right> =\hat{X} \left<x|{\psi}\right> $?

Gennaro Auletta in his book makes the following argument to show that multiplicative operator acting on his eigenvectors acts in a multiplicative way on eigenfunctions as well. Here's the argument. ...
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2answers
57 views

What is the value of the inner product $_x\langle+|-\rangle$? [closed]

I know that $\left|_x\langle+|-\rangle\right|^2 = 1/2$, so is it just as simple as taking the square root of $1/2$? Thanks.
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1answer
221 views

The spinor metric, basic spinor calculations and spinor indices

I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24. Background Scharf introduces the index-raising symbol (spinor metric)...
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3answers
103 views

What does $\Delta$ stand for? [duplicate]

Newton’s first law states that $\Delta v=0$ unless acted on by an external force, $F_{\mathrm{net}}\neq0$. Can someone explain to me what the $\Delta v$ symbol means?
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1answer
316 views

Einstein solid degree of freedom

I was studying from Schroeder's thermal physics book. When it talks about Einstein solids it says that they have 2 degrees of freedom thus $U=NkT$ However, I thought when we talk about Einstein ...
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0answers
47 views

Difference between position of indexes in tensor notation (SR) [duplicate]

I am learning SR, and don't understand the difference between the following notations of a Lorentz transformation $\Lambda$ $$\Lambda_{\mu\nu} , \Lambda_{\mu}\ ^\nu , \Lambda^{\mu\nu}$$ I know that ...
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3answers
151 views

What is the point of sub and superscripts in metric tensor

I am new to QFT and have to catch up with some things that I did not learn from my SR courses. I am learning about the tensor notation, in particular the metric tensor, $$g_{\nu\mu}=g_{\mu\nu}=\begin{...

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