Questions tagged [dimensional-analysis]

Dimensional analysis is the process of obtaining results by analysing the units and dimensions in questions, equations, and so on using The Principle of Homogeneity. Note: DO NOT USE THIS TAG if your question is about degrees of freedom or spatial dimensions.

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Why is the general unit for energy (in terms of energy bills) $\rm kWh$? [closed]

Why is the unit for energy (in terms of energy bills) expressed as $$\text{time} \cdot \frac{\text{energy}}{\text{time}}$$ rather than just energy? Wouldn't it be better to express it in megajoules (...
flakpm's user avatar
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-1 votes
1 answer
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How exactly are natural units used in everyday life? [closed]

I’m currently trying to understand natural units, and I have read that natural units are used frequently in everyday life. For example: “I live 100s away". But I do not understand how this is an ...
cookiecainsy's user avatar
-2 votes
0 answers
52 views

Fermi-Dirac integrals [closed]

I accounted a problem when deriving the Fermi-Dirac integrals shown below Could someone point me out why the lower limit of integtal could subsitutite from $Ec$ to $0$? 20240410 edit I have derived ...
Jack Huang's user avatar
4 votes
1 answer
285 views

White noise fluctuation amplitude

I'm trying to understand better noise processes, and have a very basic question. Suppose I have a stochastic process characterised by white noise, namely $$ \langle X(t) \rangle = \overline{X} \,;\...
user avatar
0 votes
1 answer
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Can unit prefixes be interpreted as algebraic stand-ins?

I am currently a little confused about whether unit prefixes can be interpreted algebraically For example: $$4km = 4\times 10^{3} m$$ Is it incorrect to say that $k$ is simply an algebraic stand in ...
cookiecainsy's user avatar
-4 votes
0 answers
334 views

How many base dimensions are there? [closed]

Let's imagine we have a 1-dimensional line. (This means a flat, straight line). There are electromagnets lying on it. They are like special rocks that can stick to each other and some metals. How ...
ingotangjingle's user avatar
1 vote
0 answers
51 views

Units for the Calculus of Variations [duplicate]

Just a quick question regarding the units for a quantity. I just started reading a QFT textbook, and it starts out with a little bit of Calculus of Variations. Specifically, there is a result that ...
Hobson Carion's user avatar
3 votes
1 answer
265 views

How does the Planck constant enter into the uncertainty principle?

In Stein & Shakarchi's Fourier Analysis, the Fourier transform of a Schwartz function $\psi$ is defined to be $$\hat{\psi}(\xi) = \int_{-\infty}^\infty \psi(x) e^{-2\pi i x \xi} dx$$ which gives ...
Drake's user avatar
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6 votes
2 answers
318 views

Radian unit mystery (damped oscillator)

I would be extremely grateful for any help that anyone could offer here. I am interested in solving the optical bloch equations for the excited state population Rabi oscillations with damping due to ...
HB123's user avatar
  • 61
0 votes
2 answers
58 views

Extracting the dimension of an operator from algebra

I may misinterpret the question. In the lecture note of conformal field theory, arXiv:2207.09474, it says the following where for $P^\mu=i\partial_\mu$ and $D=ix^\mu \partial_\mu$. I am confused ...
Tanmoy Pati's user avatar
3 votes
2 answers
120 views

Difference between renormalizable and super-renormalizable theories

In $\phi^n$ theory in Peskin & Schroeder the superficial degree of divergence is: $$D = d - V[\lambda] - \big(\frac{d-2}{2}\big)N \tag{10.13}$$ where $d$ is the dimension, $V$ is the number of ...
CBBAM's user avatar
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2 answers
83 views

How to think of the unit $\rm eV$?

How to get a sense of $\rm eV$? I mean when I know how much a metre is or a second is, but how to "visualize" when it is said atomic reactions are in order of $\rm eV$ and nuclear reactions ...
Questioningmind's user avatar
0 votes
3 answers
81 views

The SI-unit of the cosmological constant (vacuum energy) is $\frac{1}{m^2}$. What does that have to do with Energy?

I just don't get how Energy is measured in $\frac{1}{m^2}$. Wasn't it measured in Joules? (source is https://en.wikipedia.org/wiki/Cosmological_constant#Equation)
Marvas's user avatar
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6 answers
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How to interpret units of measurement like $\text{kg m/s}^2$?

Let's first take an example. I understand that if a car has $v=20 \,\text{m/s}$ this means that every second it moves $20 \,\text{m}$. But how should I interpret units that are multiplied like $\text{...
costantino corsi's user avatar
0 votes
1 answer
36 views

If the value of Coulombs Constant is high, What can we conclude about the Electric force?

I wondered about that question and regardless of the obvious answer (If $k$ increases, the Electrostatic force increases), What can we conclude from $||\vec{F}_e||$?
Mavlock's user avatar
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0 answers
83 views

Schrodinger equation with $\hbar =1$

The Schrodinger equation is given by: $$i \hbar \frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle.$$ Sometimes, physicists set $\hbar=1$. I suppose that they achieve this by changing the scaling and ...
MonteNero's user avatar
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1 answer
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What would be an experimental test of Sciama’s theory and why it has not been pursued yet?

Recently I came across a video were the origin of inertia was attributed to Sciama’s paper (1953). I have seen only a couple of questions regarding this topic on Stack Physics. Both of them are ...
Py-ser's user avatar
  • 289
1 vote
2 answers
100 views

Keplerian Frequency of Schwarzschild Black hole

The Keplerian frequency/ Orbital frequency is the inverse of orbital period and for Schwarzschild black hole it is given by $$\frac{1}{2\pi}\sqrt{\frac{M}{r^3}}.$$ its unit is Hertz. Now To express ...
zahra's user avatar
  • 21
1 vote
0 answers
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Why can Newtonian gravity provide the correct value of the Schwarzschild radius? [duplicate]

By using Newtonian gravity, we can equate the kinetic and potential energy of a test mass in order to obtain the escape velocity of an object from a large mass $M$: $$\frac{1}{2} m v^2 = \frac{GMm}{r}$...
e-diamond's user avatar
0 votes
2 answers
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What is the equation for the speed of a molecule at a specific temperature?

What is the equation for the speed of a molecule at a specific temperature? I saw two equations $v = \sqrt{\frac{3 k T}{m}}$ and $v =\sqrt{\frac{3RT}{m}}$. What is the difference?
Arjun Raj's user avatar
  • 107
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1 answer
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Charge renormalization choice in QED

In the lectures on QFT I'm following we define the renormalized QED Lagrangian as $$\mathcal{L} = \dfrac{1}{4} (F_0)_{\mu\nu} (F_0)^{\mu\nu} + \bar{\psi}_0 (i \bar{\partial} - (m_0)_e) \psi_0 - e_0 \...
Gabriel Ybarra Marcaida's user avatar
2 votes
1 answer
75 views

Why are these terms not present in the QED Lagrangian?

I am working though some questions for my QFT/ QED exam and i am having trouble with the following question: Explain why the following terms cannot be part of the Lagrangian of QED: $-g(\bar{\psi}\...
ugur's user avatar
  • 35
-3 votes
1 answer
74 views

Does the inner product of wavefunctions really have units? [closed]

Let $\psi(x)$ and $\phi(x)$ be wavefunctions. I usually see the inner product defined as $$\int dx\, \overline{\psi(x)} \, \phi(x)$$ and interpreted, I think, as "the amplitude that state $\phi$ ...
Upasker's user avatar
  • 120
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1 answer
49 views

Mass dimension of ghost Lagrangian in BRST quantization

It seems from the BRST transformation rules that the ghost fields should be dimensionless: For eg. in the Abelian case in 4D: $$A_{\mu} \to A_{\mu} + d_{\mu}c.$$ Then the ghost Lagrangian density $\...
vvs's user avatar
  • 13
2 votes
1 answer
77 views

Why are the distances in real space and Fourier space inverses of each other?

I just came across a paragraph in a set of physics notes where they implicitly claim that imposing a cut-off $k<\Lambda$ to the modes in Fourier space is equivalent to smoothing the field in real ...
Wild Feather's user avatar
0 votes
0 answers
30 views

Dimensionless non-relativistic energy of fast particles

In Plasma Astrophysics, Part 1, Equation (4.7) the dimensionless non-relativistic energy of fast particles ($m$) in a plasma with electrons ($m_e$) and protons ($m_p$) is defined as, $$ x = \frac{m v^...
Refrigerator's user avatar
2 votes
4 answers
120 views

How do units work out in logarithms? [duplicate]

In a problem I am doing, it boiled down to an integral that resulted in $$\ln(x+1\text{ m})\Big|_{x=0\text{ m}}^{x=3\text{ m}}$$ this is basically just $${\color{red}{\ln(4\text{ m}) - \ln(1\text{ m})}...
Captain Chicky's user avatar
0 votes
1 answer
61 views

Relation between Coulomb's law and Fine-structure constant

There should be some relationship between them due to the nature of FSC, but I could not find anything about it.
AlexGenesis's user avatar
1 vote
3 answers
199 views

Are we allowed to cancel the units of a derivative?

Since the volume of a sphere $v(r)=\frac{4}{3} \pi r^{3} \left[m^{3}\right]$, its derivative relative to the radius is: $$ \frac{dv}{dr} =4\pi r^{2} \left[\frac{m^{3}}{m}\right] $$ Which is also a ...
Stanislav Bashkyrtsev's user avatar
3 votes
4 answers
250 views

Can there be two different physical units (or dimensions) for a same physical quantity?

I was going through this book "Nonlinear optics" by Robert W. Boyd for my postgraduate subject in Nonlinear optics and I came across the different orders of nonlinear susceptibilities. From ...
soulfourier's user avatar
4 votes
3 answers
145 views

When we apply these concepts to physics, where do we put the UNITS in vector spaces and manifolds? Do units have a clear mathematical meaning?

We know that the space of all displacements is a vector space. The vector space is defined as a mathematical object $(V,k,+,\cdot)$ such that it satisfies the 8 properties, where $k$ is a field. We ...
ZhenRanZR's user avatar
0 votes
1 answer
117 views

How to convert quantities between SI units and a natural unit system?

Let's say I'm working in a natural unit system defined by a set of physical constants set to dimensionless numbers. How can I convert quantities between that natural unit system and a more ...
MattHusz's user avatar
  • 229
2 votes
1 answer
86 views

How is dimensionality of $S$ preserved term by term in a perturbative expansion?

In a schematic notation, the scattering matrix element $$\langle p_{out}|S|p_{in}\rangle := 1 + i (2 \pi)^4 \delta^4(p_{in} -p_{out}) M$$ between an incoming state with momentum $|p_{in}\rangle$ and ...
Albert's user avatar
  • 307
0 votes
5 answers
107 views

Dimensional constants

I'm having trouble understanding the rules for dimensional and dimensionless constants. In dimensional analysis, you can only add or subtract quantities with the same dimension. For example, if $f=12t+...
user124910's user avatar
2 votes
1 answer
57 views

Right notation for SI units/dimension in polynomial function?

I integrated a jerk-function three times (acceleration, velocity, position) to get the resulting function $s(t)$ for the position. I am not sure how to use the SI units or dimension in the function. ...
user avatar
0 votes
3 answers
135 views

Why does the equation of a wave contain the term $\omega t$ instead of $vt$ in the wave equation $y=A\sin (kx-\omega t)$?

Why does the equation of a wave contain the term $\omega t$ instead of $vt$ in the wave equation $$y=A\sin (kx-\omega t).$$ I think of the constant $k$ which for higher values increases the frequency $...
Jeffy James's user avatar
0 votes
1 answer
49 views

Non-dimensionalization of the Navier-Stokes equations

In 2D simulations using Large Eddy Simulation (LES) methodology, Favre averaging is commonly applied to the variables involved in the Navier-Stokes equations, resulting in: \begin{align}\label{aq} ...
Somestudent01's user avatar
1 vote
2 answers
101 views

Why are domain sizes of all lengths when the correlation length is infinite at $T_c$?

Given that the correlation length diverges at the critical point, why are domains of finite size? What is the relationship for a ferromagnet between correlation length and domain size?
Roger's user avatar
  • 31
0 votes
1 answer
55 views

Insights from non-dimensionalization

I have this partial differential equation which models the diffusion of a contaminant substance in a 1-dimensional river. The terms represent the following: The first term represents the diffusion ...
Niklas's user avatar
  • 3
2 votes
6 answers
968 views

Does a squared unit require squaring the value?

I would like to know if an acceleration number would remain squared in $$ v=v_{o}+at $$ Such as 1.35 m/s^2, for example, would end as $$ v=v_{o}+(1.35^2)t $$ or simply as $$ v=v_{o}+(1.35)t $$ Thank ...
jvno's user avatar
  • 39
1 vote
0 answers
65 views

Asking for correct ways to do "power counting" for gauge theories

I am looking into Weinberg "Quantum Theory of Fields" Volume 1 Ch. 12-1. There, he discusses the general rules of power counting. He defines the UV asymptotics of the propagator $\Delta_f$ ...
Keith's user avatar
  • 1,431
0 votes
0 answers
56 views

How to derive the dimension of conformal Killing vector fields on the Riemann sphere? Is it metric independent?

Context In all string textbooks and lecture notes, they derive the CKV on the sphere by considering the flat plane first, i.e. $(\mathbb{C},\delta_{\mu\nu})$. Then, write it in complex variables $$z = ...
Steven Chang's user avatar
2 votes
1 answer
183 views

What is the dimension of $2^{4 \text{ Newton}}$?

If you have a number with a dimensional quantity in its exponent, what is the dimensionality of this number then? For example when you have $e^{(4J)}$ or $2^{(4N)}$, with $J$ and $N$ respectively ...
stancallewier's user avatar
0 votes
0 answers
50 views

How can Torque and Work have the same dimensions (both can be expressed in N.m because Joules = m.N = N.m) but they represent diffrent quantitie?

So torque and work are both distance times force but how can they be so different conceptually? I understand the torque is an outer product and is a vector while work is a dot product and is a scalar ...
Mahran Yousef's user avatar
3 votes
0 answers
74 views

Mismatch in the mass dimensions of the dilaton field

In chapter 7 of David Tongs' string theory lectures, the low-energy effective action of string theory is presented, and given by eq.(7.16): $$S=\frac{1}{2\kappa^2_0}\int d^{26}X\sqrt{-G}e^{-2\Phi}\...
Daniel Vainshtein's user avatar
4 votes
1 answer
128 views

Buckingham's Pi Theorem: Why does $G(\pi)=0$ imply $\pi$ must be a constant?

I'm learning Buckingham's Pi Theorem and there are some conclusions that I just can't wrap my head around. What I Understand Given a unit-free equation F($v_1$, ..., $v_n$)=0 where vi are physical ...
William's user avatar
  • 143
2 votes
0 answers
48 views

Can a relevant operator's OPE with itself only include the identity and irrelevant operators?

I am interested in correlation functions in critical spin chains, and I'm trying to understand the consequences of conformal field theory for these correlation functions. I should warn that I do not ...
user196574's user avatar
  • 2,040
0 votes
1 answer
159 views

Massless tadpole integrals in dimensional regularization

I'm trying to prove the following: \begin{equation} \int_0^\infty x^a dx = 0, \hspace{2pt} \forall a\in \mathbb{R}. \end{equation} This should work in dimensional regularization. I found a lot of ...
Alex's user avatar
  • 111
0 votes
1 answer
63 views

What is the dimension of wavefunction in spherical polar coordinates?

I understand that the dimension of the wavefunction in Cartesian coordinates is $L^{-3/2}$ since the norm squared of the wavefunction is the probability density, where probability is a dimensionless ...
Dainsleif's user avatar
1 vote
0 answers
46 views

Is there something special about the 7 fundamental quantities or are they just convention? [closed]

I mean could we do as good with any fewer (or more) fundamental quantities, or is seven really a magical number when it comes to them?
NotAScientist's user avatar

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