Questions tagged [notation]

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3
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2answers
118 views

What does this kind of notation mean?

Trying to understand quantum information. Need some help :( What does this notation $$ \langle\alpha|\hat{n}|\alpha\rangle $$ mean? Here $$|\alpha\rangle$$ is a coherent state and $$\hat{n}$$ is ...
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0answers
41 views

Gauss' Law in 2D?

At first I thought it was $$∮E.dl=Q/ϵ.$$ So i've read through some sources here and on the internet and most of them said that $$∮E.dl=2πq.$$ But I'm confused. Can anyone explain where does the $...
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3answers
429 views

Difficulty in understanding ket vectors in quantum mechanics

$\newcommand{\k}[1]{\left | #1 \right\rangle }$Dirac, in his book The Principles of Quantum Mechanics, states that: To proceed with the mathematical formulation of the superposition principle we ...
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2answers
168 views

Why are circular unit vectors often defined as $\hat{\mathbf e}_\pm = \mp (\hat{\mathbf e}_x \pm i \hat{\mathbf e}_y)/\sqrt{2}$

When dealing with circular polarizations, spherical harmonics, and generally with any vector-valued rotationally-invariant quantity, it is often a requirement to define complex-valued unit vectors of ...
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0answers
15 views

What is the syntax in scilab to derivative a function like $x/log(x)$, not a polynomial? [closed]

In scilab to do a derivative I used the syntax derivative as: Function y=f(x) y=x/log(x) endfunction disp(derivative ("f","x",4)) And my output was: Undefined variable: derivative Where was I ...
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2answers
60 views

Are sign of charges necessary to find force between two charges? [closed]

Do we need to insert signs of charges in coulomb's law to find the Forces between two charges ?
0
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1answer
45 views

Summation notation for statistical mechanics

I was taking a look to a book of statistical mechanics, many equations show something as follows: $$ Q(K,N) = \sum_{s_{1},s_{2},...,s_{N}=\pm 1}[ e^{K(...+s_1s_2+s_2s_3+s_3s_4...)} ]\tag{1}$$ then ...
1
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1answer
65 views

What are Connection Forms in General Relativity?

I'm trying to follow an article by H. Ellis (1973), where he developed the first ever metric of a traversable Wormhole (more info here). In pages 105-106 (the end of the 3rd page in the linked file ...
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1answer
37 views

Dirac delta function and Kronecker delta function

Can someone please tell me the difference between Kronecker delta function and Dirac delta function?
0
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0answers
19 views

Notation for submatrices / subspaces and their partial traces

I‘m dealing with an observable on a N-dim Hilbert space and in some sense fix the Nth coordinate and regard the system thus as composite with a 1-dim and a (N-1)-dim subspace. Now I require to perform ...
1
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1answer
57 views

Operator acting on bras

I need some help. Suppose, $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are operators and $|\psi\rangle$ is any state, so that $$ \hat{\textbf{A}}|\psi\rangle=a|\psi\rangle. $$ And I wonder if this ...
0
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1answer
11 views

Difference in notation for radial component of electric field formula

Apologies if this is a trivial question, but it is a notation confusion lingering in the back of my head. In the electric field formula, I'm having confused going back and forth between $$\frac{\vec{r}...
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2answers
52 views

Changing derivative to difference quotient

Can differential be changed to Delta or difference? In high school education, in the acceleration section of Newton's formula 2, acceleration is a change velocity (velocity difference) divided by a ...
3
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1answer
109 views

Nomenclature in QM textbook by Landau and Lifshitz

According to Landau and Lifshitz ["Quantum Mechanics non-relativistic theory", page 6], the probability of various results of any measurement is given, in general by the following expression: $$\iint \...
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1answer
28 views

Notation in 'right/left moving' modes

In Superstring Theory Vol.1 chapter 2.1 we define the general solution to the massless wave equation: $$ X^\mu(\sigma)=X^\mu_R(\sigma^-)+X^\mu_R(\sigma^+) $$ with $$ \sigma^- = \tau-\sigma $$ $$ \...
1
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2answers
68 views

$\langle x | p \rangle = \psi_p(x)$ , What this anything to do with $\psi$ (Notation in R.Shankar)

In Principal of Quantum Mechanics R.Shankar Page 137 if we project the eigenvalue equation $$P|p\rangle = p|p\rangle$$ onto the $X $ basis,we get $$\langle x|P|p\rangle = p \langle x |p \rangle$$ ...
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2answers
37 views

Finding the new force of vectors

It's been a while since the last time used vectors. I came across with the following question. Find the Net force (size and orientation) of the vectors $$ \vec{A}=15,37^{\circ},\,\vec{B}=5,162^{\...
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2answers
83 views

Dirac expression derivation

In Quantum Mechanics, 2nd Edition by Davies & Betts on page 78 it states that there is a symmetry implied by the following Hermitian operator equation: $${\displaystyle \int \phi^{*}(A \psi)d \,\...
1
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1answer
85 views

How does one write conditional expectation in bra-ket notation?

In physics, the expectation of a random variable under the bra-ket notation is $\langle u \rangle$, how do I write the conditional expectation of $u$ on another variable $v$? $\langle u | v\rangle$ ...
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0answers
42 views

This tensor equation in Gerard t' Hooft's notes on General Relativity

This question probably has a simple answer that I'm just missing, but in Gerard 't Hooft's lectures on GR he has the following: $$X^{\mu}=Y_{\mu\alpha}Z^{\alpha\beta\beta}.$$ Why is the result of ...
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4answers
194 views

The exponential of an operator

I have a problem to have an intuitive idea about these operators: $\hat{D_x}(x)=e^{-i\frac{x}{\hbar}\hat{p_{x}}}$: the spatial displacement operator, moves the wave function $\psi$ along the x ...
1
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3answers
62 views

Textbook proof error? Runge Lenz

I was reading this proof in my textbook. They say that $$\vec{r} \cdot \dot{\vec{r}} = |\vec{r}||\dot{\vec{r}}|.$$ Doesn't that mean $\vec{r}$ is parallel to $\dot{\vec{r}}$, and if so, then the line ...
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4answers
350 views

What does this notation for spin mean? $\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}=\mathbf{1}\oplus\mathbf 0$

In my quantum mechanics courses I have come across this notation many times: $$\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}=\mathbf{1}\oplus\mathbf 0$$ but I feel like I've never fully understood what ...
1
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1answer
66 views

Two basic questions about Christoffel symbols

I am trying to understand (rather than memorise) the derivation of the Christoffel symbols from the vanishing covariant derivative of the metric, the very first step is \begin{equation} \label{eq:...
1
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2answers
345 views

Confused by raising and lowering indices

I lack in understanding of some basic idea regarding 4-vectors and index raising and lowering. From what I understand that: $$ \partial^\mu = \eta^{\mu\nu}\partial_\mu$$ So then, is the following ...
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2answers
127 views

Proving the raising and lowering of the raising and lowering operator

I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$. However, in the written proof for $\hat A^{\...
3
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2answers
191 views

Meaning of tilde ($\sim$) above vector (Context: particle physics)

I have encountered a notation I am not familiar with, namely a tilde $\sim$ above a vector (i.e. a column vector), e.g. $\tilde{H}$. From the context, it is clear that it cannot mean transposition, ...
1
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2answers
105 views

Exponential form of the translation operator

In quantum mechanics, the translation operator $\hat{T}(a)$ is defined such that $\hat{T}(a) \cdot f(x) = f(x+a)$. I'm asked to find the exponential form of this operator, given by $\hat{T}(a)=e^{i\...
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3answers
616 views

Confusion with Dirac Notation

I'm trying to calculate uncertainty in momentum, and I know that $$\langle\hat P^2\rangle=\int^{\infty}_{-\infty}\hat P^2|\Psi(x)|^2\,\text dx$$ But I'm confused by what that symbol means. Does it ...
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1answer
152 views

What does lowercase-delta mean in Noether's first theorem?

Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ...
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1answer
31 views

Implied metric for the basis vectors when lowering indices

When we map a vector to its corresponding covector with the metric: $$g_{\mu\nu}x^\mu=x_\nu$$ is there a second (implied) metric being used to convert the basis vectors too? Written explicitly: $$...
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2answers
75 views

Difference between the contravariant, covariant and mixed form of the electromagnetic tensor and Minkowski metric tensor?

What is the difference between the contravariant, covariant and mixed form of the electromagnetic tensor and Minkowski metric tensor? I know the difference in indices (superscript and subscript). ...
0
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2answers
58 views

Tensor ordering in index lowering operation

If we take two vectors and want to contract them with the metric tensor to find some frame invariant quantity: $$A^a B^b g_{ab}=\vec A\cdot \vec B$$ is there a convention on where the metric tensor ...
2
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3answers
80 views

Why does index contraction have to be done between upper and lower indices?

If I had to give a guess based on limited understanding, I would expect it to be something to do with the resulting object no longer obeying tensor transformation properties. However, if that is the ...
0
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1answer
41 views

Mathematical convention when using spatial indices: numerical $(1,2,3)$ versus Cartesian $(x,y,z)$ [closed]

When writing a document I find that I am switching back and forth between indicial notation for spatial coordinates. I would like to get your thoughts on the following examples accompanied with ...
0
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1answer
22 views

Clarification on the notation of a paper about hybrid

Here is a screenshot from this paper by J. P. Foster and F. Weinhold. This paper focuses on a model of hybridization. It therefore considers movement of electrons in three dimensions. The author does ...
6
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2answers
2k views

Existence of adjoint of an antilinear operator, time reversal

The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a well-...
1
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1answer
54 views

Euler-Lagrange equations in QFT and metric signs

I'm having a probably dumb problem with the Euler-Lagrange equations and the dot-product in Minkowski spacetime. I know that some objects are defined naturally with lower-indexes, e.g. $\partial_{\mu}$...
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3answers
78 views

What's the point of this killing vector notation?

Reading Sean Carroll's spacetime and geometry he says If $x^{\sigma_*}$ is the coordinate which ${\mu\nu}$ is independent of, let us consider the vector $\partial_{\sigma_*}$ which we label as $$...
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2answers
134 views

Covariant and contravariant coordinates - index notation

I am learning about electrodynamics and have recently been introduced to the four vector. I also come fresh to the idea of covariant four vectors and contravariant four vectors. My question concerns ...
2
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0answers
40 views

How do the Weyl spinors differ from dotted and undotted spinors? [duplicate]

As asked in the title, how do the Weyl spinors $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ differ from dotted and undotted spinors?
0
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1answer
49 views

4-velocity lowering index question

The 4-velocity in contravariant form is given by $$V^\mu=\frac{dx^\mu}{d\tau}$$ for some general co-ordinates $x^\mu$ and proper time $\tau$. Is the 4-velocity in covariant form given by $$V_\nu=V^\...
1
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0answers
23 views

Notation of basis functions for irreducible representations

In character tables for symmetry groups, there are typically basis functions for each irreducible representation given. There are basis functions given like $xy$, $S_x$ or $R$. Could someone explain ...
3
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3answers
92 views

Relation between metric tensors

$g^{ab}$ is the metric tensor (Minkowski) Im trying to understand if this is true: $g^{ab}g_{ab}=?=g^{aa}g_{aa}$ Have a nice day.
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1answer
60 views

Understanding bra-ket notation as in quantum computer in terms of pure algebra

Since I'm not a Physics student, but a Mathematics student, the only physics course I went through is elementary physics (일반물리학, textbook by Jearl Walker, David Halliday, and Robert Resnick). Not even ...
1
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3answers
378 views

Notation of plane waves

Consider a monochromatic plane wave (I am using bold to represent vectors) $$ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, $$ $$ \mathbf{B}(\mathbf{...
2
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2answers
1k views

Showing Four Acceleration and Four Velocity are Perpendicular

I want to show that in general, $\vec{a}\cdot\vec{u}=0$, where $\vec{u}$ is the four-velocity and $\vec{a}$ is the four-acceleration. The four acceleration is defined as $\vec{a}=\nabla_{\vec{u}}\vec{...
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0answers
27 views

Question on notation convention regarding partial derivatives

H.Risken's book "The Fokker-Planck Equation" contains the following formula for the general 1D Fokker-Planck equation: $\frac{\partial W}{\partial t}=\left[-\frac{\partial}{\partial x}D^{(1)}(x)+\...
3
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3answers
147 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 ...
5
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5answers
522 views

Why isn't there a minus sign in Ohm's law, $V = IR$?

Suppose current runs through a resistor from left to right, and we define the left-to-right direction as positive. Then from left to right, the potential decreases. So $V,$ the voltage across the ...

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