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.

Filter by
Sorted by
Tagged with
60 votes
2 answers
87k views

Difference between $\Delta$, $d$ and $\delta$

I have read the thread regarding 'the difference between the operators $\delta$ and $d$', but it does not answer my question. I am confused about the notation for change in Physics. In Mathematics, $\...
Yuruk's user avatar
  • 889
35 votes
2 answers
4k views

Symbols of derivatives

What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...
Steeven's user avatar
  • 50.5k
55 votes
5 answers
15k views

Mathematically-oriented Treatment of General Relativity

Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would Prove all theorems used. Use modern "mathematical notation" as ...
48 votes
9 answers
8k views

Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
Isomorphic's user avatar
  • 1,558
18 votes
2 answers
15k views

Do derivatives of operators act on the operator itself or are they "added to the tail" of operators?

How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this? For example: say you had the ...
Mike Flynn's user avatar
  • 1,156
3 votes
2 answers
1k views

Understanding Dirac's notation

Let's say I have eigenstates $|x\rangle$ associated with measurement of position. I know that the eigenstates corresponding to their respective eigenvalues form a basis, let's call it $A$. Now let's ...
Patrick's user avatar
  • 165
9 votes
3 answers
954 views

What does the notation $\frac{1}{2} \otimes \frac{1}{2} = 1 \oplus 0$ mean exactly?

My question is very similar to the question here, however I want to ask about some specifics. I am having trouble keeping track of exactly how we are "tensor-producting" or "direct-...
Jbag1212's user avatar
  • 2,254
18 votes
4 answers
11k views

Difficulties with bra-ket notation

I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with "...
xxxxx's user avatar
  • 1,565
13 votes
3 answers
4k views

Staggered Indices ($\Lambda^\mu{}_\nu$ vs. $\Lambda_\mu{}^\nu$) on Lorentz Transformations

I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes. What are the respective meanings of $\Lambda^\mu{}_\nu$ as ...
MrLeibniz's user avatar
  • 139
41 votes
2 answers
31k views

Square bracket notation for dimensions and units: usage and conventions

One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol ...
Emilio Pisanty's user avatar
26 votes
2 answers
20k views

Uncertainty in parenthesis

In a physics text book I read the following: $$e/m=1.758820150(44) ×10^{11} \mathrm{C/kg} $$ In this expression, $(44)$ indicates the likely uncertainty in the last two digits, $50$. How should ...
Steeven's user avatar
  • 50.5k
14 votes
3 answers
4k views

What is the meaning of the double complex integral notation used in physics?

In Altland and Simons' condensed matter book, complex Gaussian integrals are introduced. Defining $z = x + i y$ and $\bar{z} = x - i y$, the complex integral over $z$ is $$\int d(\bar{z}, z) = \int_{-\...
knzhou's user avatar
  • 102k
22 votes
4 answers
7k views

Applying an operator to a wavefunction vs. a (ket) vector

I have a question regarding the effect of quantum mechanical operators. The definition that I'm familiar with says that an operator $A$ acts on a vector from a Hilbert space, $|\psi\rangle$, and the ...
Socob's user avatar
  • 489
8 votes
3 answers
2k views

Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?

The Euler-Lagrange equation for particles is given by $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$ and for fields it is $$ \partial_{\mu} \frac{\...
Hermitian_hermit's user avatar
5 votes
5 answers
4k views

What is the meaning of following expression $C=\frac{\delta Q}{dT}$ mathematically?

Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even Wikipedia) writes wrong expressions (from ...
TMS's user avatar
  • 2,051
5 votes
3 answers
4k views

Square bracket notation for anti-symmetric part of a tensor

I know that $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$ But how can write $E_{[a} F_{bc]}$ like the above? Can you provide a reference where this notational matter is discussed?
rainman's user avatar
  • 2,983
1 vote
3 answers
2k views

Convention of tensor indices

Let $g_{ij}$ be the diagonal Minkowski metric tensor diag$(g) = (1,-1,-1,-1)$, then $g^{ij}$ is defined to be $(g^{-1})^{ij}$, hence $$g_{ik}g^{kj} = g_i^{\ \ j} = \text{diag}(1,1,1,1)=\delta_i^{\ \ j}...
DKS's user avatar
  • 386
12 votes
1 answer
2k views

Difference between Cartesian product $\times$ and tensor product $\otimes$ on groups

After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1)$...
Marion's user avatar
  • 2,190
21 votes
5 answers
6k views

Inverse and Transpose of Lorentz Transformation

I've seen this question asked a few times on Stack Exchange, but I'm still quite confused why the following "contradiction" seems to arise. By definition: $(\Lambda^T)^{\mu}{}_{\nu} = \...
Shrey's user avatar
  • 706
13 votes
4 answers
3k 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 ...
AccidentalTaylorExpansion's user avatar
12 votes
4 answers
9k 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 ...
peterwright's user avatar
11 votes
1 answer
4k 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-...
LYg's user avatar
  • 1,121
10 votes
1 answer
1k views

What do term symbols with a half-integer "$L$" like $^3[3/2]_{1/2}$ mean?

Atomic term symbols are used to notate the angular momentum content of the electronic states of an atom, and are normally written down as $$^{2S+1}L_J$$ where the state has total spin $S$, spin ...
Emilio Pisanty's user avatar
13 votes
2 answers
7k views

Bra-Ket Notation

I'm having difficulty understanding the bra-ket notation used in quantum mechanics. For instance, take the notation used in the question Is there a relation between quantum theory and Fourier analysis?...
s n's user avatar
  • 133
11 votes
1 answer
1k views

How are the definitions of a coherent state equivalent?

I am trying to understand coherent states. As far as I could find there are three equivalent definitions and in general many sources start from a different one, still I fail to see their equivalence. ...
ckrk's user avatar
  • 620
6 votes
2 answers
1k views

Difference between slanted indices on a tensor

In my class, there is no distinction made between, $$ C_{ab}{}^{b} $$ and $$ C^{b}{}_{ab}. $$ All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
nate's user avatar
  • 397
6 votes
3 answers
594 views

In what subfields and how far can the naive limit $c\rightarrow\infty$ of special relativity be carried?

Even if many interesting similarities between the classical and the quantum mechanical framework have been worked out, e.g. in the subject of deformation quantization, in general, there are some ...
Nikolaj-K's user avatar
  • 8,425
5 votes
1 answer
1k views

Working with indices of tensors in special relativity

I'm trying to understand tensor notation and working with indices in special relativity. I use a book for this purpose in which $\eta_{\mu\nu}=\eta^{\mu\nu}$ is used for the metric tensor and a vector ...
MeMeansMe's user avatar
  • 703
5 votes
1 answer
1k views

Why is not ${(\Lambda^T)^\mu}_\nu = {\Lambda_\nu}^\mu$?

I am following lecture notes on SR. The author writes that the following is equivalent: $$\Lambda^T\eta\Lambda = \eta \iff \eta_{\mu \nu} {\Lambda^\mu}_\rho{\Lambda^\nu}_\sigma = \eta_{\rho \sigma}. \...
Mikkel Rev's user avatar
  • 1,326
4 votes
2 answers
3k 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, ...
Étienne Bézout's user avatar
38 votes
8 answers
42k views

Is there a symbol for "unitless"?

I'm making a table where columns are labelled with the property and the units it's measured in: Length (m) |||| Force (N) |||| Safety Factor (unitless) ||| etc... I'd like not to write "unitless" ...
Ben's user avatar
  • 981
6 votes
1 answer
10k views

Wave function and Dirac bra-ket notation

Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation? How do you get from one to the ...
Freeman's user avatar
  • 777
6 votes
2 answers
1k views

Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$

If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$. In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
SRS's user avatar
  • 26.2k
2 votes
1 answer
3k 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\...
Jpmarulandas's user avatar
2 votes
4 answers
557 views

Why do we use different differential notation for heat and work?

Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we ...
user1007028's user avatar
40 votes
3 answers
3k views

Partial derivative notation in thermodynamics

Most thermodynamics textbooks introduce a notation for partial derivatives that seems redundant to students who have already studied multivariable calculus. Moreover, the authors do not dwell on the ...
1__'s user avatar
  • 1,574
26 votes
4 answers
10k views

Why is the candela a base unit of the SI?

The candela is defined as The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540\cdot10^{12}$ hertz and that has a radiant ...
Gerard's user avatar
  • 2,730
15 votes
3 answers
3k views

Why is the $dx$ right next to the integral sign in QFT literature?

I've noticed that in QFT literature, integrals are usually written as $\int \!dx ~f(x)$ instead of $\int f(x) dx$. Why?
Craig Feinstein's user avatar
12 votes
1 answer
15k views

Principal value of $1/x$ and few questions about complex analysis in Peskin's QFT textbook

When I learn QFT, I am bothered by many problems in complex analysis. $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why $1/x$ can have a principal value because ...
346699's user avatar
  • 5,901
9 votes
2 answers
2k views

When can I use $\wedge$ instead of curl?

It seems in some circles the wedge product is used in preference to curl. I have a basic understanding of Green and Stokes' formula, I wish to use the $\wedge$ notation from now on. Can someone tell ...
Fraïssé's user avatar
  • 1,724
9 votes
2 answers
5k views

Why do we write $(v\cdot \nabla) v$ instead of $v \cdot (\nabla v)$ for $v_j \frac{\partial}{\partial x_j} v_i$ in the material derivative?

Suppose I have a steady flow and I want to find the rate of change of pressure of a bit of fluid. This depends on the velocity of the fluid and the pressure gradient, $$\frac{\mathrm{d}P}{\mathrm{d} ...
Mark Eichenlaub's user avatar
7 votes
5 answers
2k 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 ...
Mark Eichenlaub's user avatar
7 votes
1 answer
32k views

What does $\Psi^*$ mean in Schrodinger's formulation of Quantum Mechanics?

I am not a physics student. In one of my courses, some fundamental concepts of Quantum mechanics were needed, so I was going through them when I stumbled upon this. It says $$\text{probability} = \...
Jack's user avatar
  • 564
7 votes
3 answers
1k views

Vector cross product formula without a second term (Spiegel, Theoretical Mechanics)

In Spiegel's Outline Of Theoretical Mechanics (more precisely in the Moving Coordinate Systems chapter, § "Derivative Operators") I find (both in the 1968 and the 1977 edition) the following ...
Vince Vickler's user avatar
5 votes
1 answer
328 views

Atomic physics, determining levels and terms

In atomic physics I understand there a configurations, terms and levels. I think levels for instance appear because of spin-orbit interactions, so that terms are split. But I'm confused about the ...
Alexander's user avatar
  • 165
4 votes
4 answers
22k views

Index notation with Navier-Stokes equations

This is an index-notation question rather then the NS one: For incompressible flow and Newtonian fluid, the continuity equation is denoted with: $$\frac{\partial u_i}{\partial x_i} = 0, $$ which ...
miro2's user avatar
  • 43
4 votes
3 answers
4k views

Derivation of Inverse Lorentz Transformation in Index Notation

To review my special relativity I tried to work out the inverse lorentz transformation explicitly. This led to a lot of confusion; I would like to ask what the issue was with the assumptions I made in ...
doublefelix's user avatar
  • 6,872
3 votes
4 answers
542 views

Inconsistency between $d_A = d + A \wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative. $d_A A=F$ --- @...
ann marie cœur's user avatar
3 votes
1 answer
557 views

Notation for Standard Model Charges?

Does anybody know what these following numbers describing an electron $(1, 1, -1)$ represent in $SU(3) \times SU(2) \times U(1)$? Or, these numbers that describe an up quark: $(3, 1, 2/3)$? I'm ...
curiousGeorge119's user avatar
2 votes
2 answers
9k views

Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
Peter4075's user avatar
  • 3,029

1
2 3 4 5