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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28 views

In Srednicki's quantum field theory, page 71. Why is $Z_1 = W_1$?

Srednicki defines here: https://arxiv.org/abs/hep-th/0409035 on p.71 a "Z1", which is $exp(iW_1)$ where $iW_1$ is the sum of all connected diagrams with sourceless ...
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1answer
24 views

Need help understanding product of Kronecker delta function with an outer product

I need help solving the the below expression which is from the book Modern Quantum Mechanics by J. J. Sakurai and Jim Napolitano. $$A=\sum_{a''}\sum_{a'}|a''\rangle a'\delta_{a'a''}\langle a'| =~?$$ I ...
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2answers
103 views

Do physicists use a different way of dealing with Function notation?

I've started studying physics concepts, in Mathematics we tend to define $f$ to be some function that we apply to a variable $x$, where as in Physics and Mechanics work I sometimes find that people ...
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158 views

Uncertainty notation: I am unsure of how the parentheses notation works

If I have a value of $5.868709...×10^{−7}$, and an uncertainty of $7.88431...×10^{−12}$, is it correct to write this as $5.86871(8)×10^{−7}$ or $5.8687(8)×10^{−7}$? A problem I have with the first is ...
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25 views

Most common convention for distinguishing dimensionless variables from dimensional? [closed]

When normalizing variables (i.e. transforming them from dimensional into dimensionless units), it's useful to distinguish them from their dimensional counterparts. I've seen many conventions for this (...
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1answer
111 views

Question about bra-ket notation for inner products in Quantum Mechanics

I'm trying to understand the notation used to denote inner products in my introductory quantum physics textbook (Introduction to Quantum Mechanics, Griffiths). In section 6.1 where Griffiths ...
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1answer
27 views

A notational confusion in a Bell-type inequality

In the tripartite Bell-type inequality know as Svetlichny inequality, given in this (freely available) article. The quantity $M_{ijk} = Tr [\rho(\sigma_i \otimes \sigma_j \otimes \sigma_k)]$, $i,j,k\...
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52 views

Euler-Lagrange equation in vector form

If we have a Lagrangian which is a function of vectors $r$ and $r'$ where $r$ is the position with an $x$, $y$ and $z$ component and $r'$ is the time derivative of $r$ are we able to express the euler-...
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24 views

Dirac spinor and derivative

I have question about four-derivative on spinors. $$ \bar{\psi}\partial_\mu $$ Does the derivative act on the spinor psi bar? $$ (\bar{\psi}\partial_\mu)\psi $$ $$ \bar{\psi}\partial_\mu\psi $$ Is it ...
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56 views

What does $\partial\phi$ mean for a scalar field $\phi(\vec{x},t)$?

If I have a scalar field $\phi \equiv \phi(\vec{x},t)$, and the Lagrangian density is $$\mathcal{L} = \frac{1}{2}(\partial\phi)^2 + ...$$ what does $\partial\phi$ mean or in other words how do I ...
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3answers
139 views

Quantum Harmonic Oscillator Wavefunction: $\langle x| \hat{p} |0\rangle = \hat{p}\langle x|0\rangle$?

I wanted to rigorously understand the derivation of the ground state (of the quantum harmonic oscillator) using the position representation $\Psi_0(x)=\langle x|0\rangle$ and $a|0\rangle=0$. There I ...
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49 views

Difference between $\left \langle x \right \rangle$ and $\left \langle \hat{x} \right \rangle$

I am having a difficult time understanding the meaning of the expected value of the operator and the difference between that and the expected value of the physical quantity. If I take for example the ...
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1answer
51 views

Spin hamiltonian matrix representation

To preface, I'm an applied mathematician trying to parse the meaning of physics notation I've come across in a paper. My goal is to understand the setting in terms of matrices and vectors so that I ...
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69 views

Double Einstein summation simplification

Suppose that we are given the EFEs for the SEM tensor $$T_{\mu\nu}=\frac{1}{8 \pi}G_{\mu\nu},$$ where the Einstein tensor $G_{\mu\nu}$ is defined to be $$G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}.$$ ...
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96 views

Question about the notation of Lie algebra valued differential forms

In mathematical differential geometry, one generally writes the structure equation as: $$\Omega = d\omega + \omega \land \omega.$$ However in the following Physics Forums post, user romsofia writes ...
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Double Sums in Einstein Notation

Suppose that you are given a metric $g_{\mu\nu}$ s.t. the metric is defined as $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}=-e^{2\phi}dt^2+\left(1-\frac{b}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2(\theta)d\phi^2)....
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70 views

What is the best way to notate probability in QM?

In a post I was reading upon, I saw the following: $$\text{probability} = \int_a^b\Psi^*\Psi\,\mathrm{d}x\quad\biggl(= \int_a^b\Psi^2\,\mathrm{d}x,\text{ if }\Psi\text{ is a real function}\biggr)$$ Is ...
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94 views

Einstein summation convention

Suppose that $M$ is a differentiable manifold which represents spacetime. Located at ever point $p$ in spacetime there exists a tanget space at that point. For any tanget space there also exists a ...
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44 views

What does "DC" stand for? (white light source)

I found this website where I can enter the power spectrum of a (white) light source to calculate its color rendering index (CRI). For some of my spectra, no CRI is calculated and the desciption says, ...
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1answer
38 views

What does a star on a spherical harmonic mean?

So I was studying multipole expansion and the book I am using introduced spherical harmonics. While I could understand the concept of the functions themselves, the book suddenly started putting a “ * “...
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85 views

Definite integral over a vector field

This article on Wikipedia showed that if the force field is conservative, then the work done on a mass between $t_1$ and $t_2$ is $$\int_{\vec{x}(t_1)}^{\vec{x}(t_2)} \vec{F} \cdot d\vec{x} $$ where $\...
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1answer
70 views

Spinor index, dirac field equation

Sometimes I read anticommute $$\{\psi(x),\psi^\dagger(y)\}=\delta^{(3)}(x-y)$$ Sometimes, $$\{\psi_a(x),\psi_b^\dagger(y)\}=\delta^{(3)}(x-y)\delta_{ab}$$ Are they the same, second one just emphasis ...
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53 views

Why can't we stick with a consistent notation for both QM and GR? [closed]

In QM, there's stuff like $\langle a | H| b \rangle$. In GR, the same stuff is $H_{ab} v^a w ^b$. In QM, there's $|a\rangle \langle a|$. In GR, the same stuff is $v^a v_b$. In QM, linear operators ...
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1answer
53 views

Pauli matrices: lower index vs upper index

I have read some identities about the Pauli matrices in 4-vector notation and I am a little confused. as $$\sigma ^\mu=(I,\sigma ^i);\qquad \overline{\sigma}^\mu=(I,-\sigma ^i).$$ But what exactly is $...
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35 views

A little clarification on Cartesian tensor notation

Goldstein pg 192, 2 ed In a Cartesian three-dimensional space, a tensor $\mathrm{T}$ of the $N$th rank may be defined for our purposes as a quantity having $3^{N}$ components $T_{i j k}$.. (with $N$ ...
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34 views

What do $<111>$, $[010]$, $(111)$, and $\{100\}$ represents for a cubic crystal lattice?

I want to know the difference between different notations in crystal lattice. I know what Miller indices are but what does these four different notations really mean.
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2answers
160 views

Question about indices and matrix

This is essentially a trivial question, which can be answer probably immediately, but i have this doubt anyway. If, say, $$\Lambda^{a}_{b} = \begin{pmatrix} f & -fc\\ -fc & f \end{pmatrix}$$ $...
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3answers
62 views

Vector component and component's magnitude notation

For example, $\vec{A}_{x}$ can signify $-5\hat{\imath}$. That for me is clear. However, $A_x$ seems to be a signed variable (-5) in some cases and an unsigned one (5) in others. This has caused a ...
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1answer
51 views

Space-time metric in tensor form

In space time metric in tensor form: The distance is given by $$ds^2=c^2dt-dx^2-dy^2-dz^2$$ Which in tensor form is: $$ds^2=\sum_{\alpha \beta}g_{\alpha \beta}dx^\alpha dx^\beta$$ Using Einstein ...
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1answer
59 views

Abuse of notation in GR? $f(x)$ vs. $(f \circ \psi^{-1})(x)$

I see some GR books write $f(x)$ (or even $f(x^\mu)$) when talking about a function on a manifold. But with $f: M \to \mathbb R$ and coordinates defined by a chart $\psi: M \to \mathbb R^n$, shouldn't ...
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2answers
104 views

Squared Summation of Terms using Einstein's summation convention

In working with QFT and Maxwell's equations, terms such as:$$\left(\partial_\mu\,A^\mu\right)^{2}$$ often appear. Since I am new to this, I am not sure of the expansion. That is, is it 4 terms ...
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8answers
1k views

Vector notation

I'm working with vectors and resolving them into their components on vector form, like this: $$\vec{F} = F_1\hat{\imath}+F_2\hat{\jmath}.$$ I'm wondering whether we need to put vector notation over ...
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1answer
64 views

What are the $V$ and $L$ in free electron theory?

The number of valence electrons in solid based on free electron theory, $N$ is given by $$N = \frac{Vk_F^3}{3\pi^2} \tag{1}$$ The $V$ in the formula, is it the volume of the whole, bulk crystal or ...
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0answers
21 views

Notation for molecular vibrational levels and population inversion

How do I interpret the notations (e.g. (0 0$^0$ 1)) for molecular vibrational levels like in the following diagram for CO$_2$? Also, since one can excite multiple quanta of the same vibrational mode, ...
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2answers
92 views

Ambiguity in Notation for Operators in Quantum Mechanics

Let's say I am trying to find the commutator of operators $\mathbf{A}$ and $\mathbf{B}$, and I get $$[\mathbf{A},\mathbf{B}]=\nabla^2 f(x,y,z).\tag{0}$$ There seems to be some ambiguity here. In ...
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29 views

Unruh modes inconsistent notation

The Unruh modes as defined in Sean Carroll's "Spacetime and Geometry" are: \begin{equation} h_k^{(1)} = \frac{1}{\sqrt{2\sinh(\pi \omega/a)}}\big(e^{\pi \omega/2a} g_k^{(1)} + e^{-\pi \omega/...
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77 views

Why are physists using intergrals without specifying limits? [closed]

Take an integral like this as an example: $$\int U(\vec{r})e^{i\vec{q}\cdot\vec{r}/\hbar}d^3\vec{r},$$ which has somethng to do with scattering. The integration is over all space so correctly it ...
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1answer
34 views

Writing the columns or Rows of a Matrix in Suffix Notation

I have the following matrix: $$M=\begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{bmatrix} = \begin{bmatrix} \vec{a}\\ \vec{b}\\ \vec{c} \end{bmatrix}$$ ...
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23 views

Square shoulder potential: does it matter if you use $<$ or $\leq$?

This may be a bit of a nitpicky question but when you consider the hard sphere or the square shoulder potential, which are both piecewise functions, does it matter at all if you use $<$ or $\leq$? ...
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1answer
108 views

What does a Umlaut (double dot) above an angle mean?

I'm reading a paper on double pendulums and there is an equation of motion that contains a double dot (Umlaut) above an angle. What does this mean / is this a standard notation in equations of motion?...
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55 views

Why is cancellation of differnetial not allowed here?

This is about cancelation of differentials .I am learning basics of tesnor from "Mathematical Methods " by Boas. There I encountered this epression which author says are equal. $$ \frac{\...
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36 views

What do chi ($χ$), psi ($ψ$), and capital gamma ($Γ$) stand for in these equations about molecular geometry (combination bands and overtones)?

Special case If one of the symmetries is doubly degenerate in the excited state a recursion formula is required to determine the symmetry of the $v$th wave function, given by, $$\chi _{v}(R) = \frac{1}...
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18 views

Molecular dipole moment transition in bra-ket notation

Below is equation for dipole moment transition: $$\langle \Psi_k| p |\Psi_i \rangle = \langle \Phi \chi_{N,k}|p_e+p_N|\Phi \chi_{N,k} \rangle = \langle \chi_{N,k}|\langle\Phi_k|p_e|\Phi_i\rangle |\...
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3answers
189 views

Covariant and contravariant for a beginner

I saw that people were representing matrices in two ways. $$\sum_{j=1}^n a_{ij}$$ It is representing a column matrix (vector actually) if we assume $i=1$. $$\begin{bmatrix}a_{11} & a_{12} & ...
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0answers
61 views

Matrix multiplication in Feynman's paper on quantum mechanical computers

In a paper by Feynman, on how to embed a Turning machine in the ground state of a many-body quantum system, a product of two matrices is defined (equation 1). This product does not appear to have the ...
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1answer
112 views

How to express a $u$-substitution in Dirac notation?

Suppose I have a set of functions $g=g(x)$, that form a basis in the Hilbert Space. I can define states $|g\rangle$ associated with this basis. Suppose I have some integral in the $x$ basis. I can ...
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10 views

What are Crystal Electric Field Parameters?

What does the following notation mean? $$A_{20}\left<r^2\right> = -100 \mathrm{K}$$ Where $A_{20}$ is termed a Crystal Electric Field Parameter. I often see this notation when reading papers on ...
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1answer
133 views

Different adjoints in particle physics

I am currently reading Quantum Chromodynamics on the Lattice by C. Gattringer C.B. Lang and I am confused about an expression in the book. The expression is $$\langle \text{tr}[S(\textbf{m}, \textbf{n}...
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53 views

Beta decay equations in shorthand

I'm not sure what the correct notation for $\beta^{+}$ decay is when using the shorthand notation. For the reaction $$a+X\rightarrow b+Y$$ it would be $ X(a,b)Y$. However, for $\beta^{+}$ decay, $$ X\...
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394 views

How to multiply matrices (the physicists way) for a total beginner?

N.B For this question we are only working in in flat cartesian space, not curved space-time. Question: Consider $A$, $B$, $C$ and $D$ to be $n \times n$ matrices. Write down the following matrix ...

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