Questions tagged [dimensional-analysis]
Dimensional analysis means to obtain results by analyzing the units in question, etc. DO NOT USE THIS TAG if your question is about degrees of freedom or spatial dimensions.
1,268
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Dimensional regularisation and Wick theorem [duplicate]
Consider an integral:
$$I^{ij}=\int\frac{d^d\textbf{p}}{(2\pi)^d}p^i p^j f(\textbf{p}^2).$$
How can we show that this is equal to:
$$I^{ij}=\frac{\delta^{ij}}{d}\int\frac{d^d\textbf{p}}{(2\pi)^d}\...
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Defining the loudness in terms of mass and time
The unit of loudness is $$\rm\frac{W}{m^2},$$ which is equivalent to $$\rm\frac{kg}{s^3}.$$ Since kg is the SI unit for mass and s is the SI unit for time, this led me to the following question:
Can ...
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Trouble understanding a derivation in Griffith's QM: How to nondimensionalise Schrödinger eq. to $\frac{d^2\psi}{d\xi^2}=\left(\xi^2-K\right)\psi$
I have trouble understanding a derivation in Griffith's QM: How to nondimensionalise Schrödinger equation to $$\frac{d^2\psi}{d\xi^2}=\left(\xi^2-K\right)\psi \quad .$$
The book explains it like so (...
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Is this theory renormalizable? If so, is there any proof? [closed]
Consider a kinetic lagrangian of 2 Klein-Gordon-Foch fields $\varphi$ and $\chi$ with interaction term
$$\mathcal{L}_I=g^2\bar{\chi}\chi\bar{\varphi}\varphi.$$
Is theory with such interaction ...
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Confusion regarding Stress-Energy-Momentum Tensor
The dimension of the stress-energy-momentum tensor is similar to that of pressure, according to wikipedia.
The stress-energy-momentum tensor Tμν is defined as μ momentum in a spacetime box of volume ν....
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Units of the derivative of an exponential function with respect to a quantity with units in the argument? [closed]
EDIT: this question is ill posed because it describes an equation that adds two values that I claimed have different physical dimensions, which is incorrect. But I left the question up in case someone ...
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Is there a system of measurement that consistently includes logarithmic units?
Logarithmic scales are not uncommon: magnitudes in astronomy, f-stops in photography, the Richter magnitude scale, etc.
But decibels are weird.
I am not even talking yet about the fact that we use a ...
- 1,305
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3
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What exactly does it mean for a unit to be dimensionless?
For instance, why are moles and decibels considered dimensionless, but kg and meters aren't? Or, in other words, what exactly is a "dimension" in this context? Is just about the system of ...
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Does Curved Edge exist for a smooth infinitely long right circular Cylinder? [migrated]
This question is in the continuation of this question. As it is cleared from the comments of the respective question that an infinitely long cylinder which is also a right circular, is a smooth $3$D ...
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Why is amplitude measured in meters whilst $θ$ is measured in radians? [duplicate]
In a simple pendulum, why is amplitude measured in meters whilst $θ$ is measured in radians? Shouldn't they both be measured in radians, if the trajectory follows a circular arc, how is it even ...
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Notation to give units in legend/axis label/etc
When writing e.g. an axis label for a plot or a header for a table column that contains data that is associated with a unit (e.g. a length in meters), I always used to write it down like this:
...
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Dimensional inconsistency in evaluating the canonical partition function
We know that canonical partition of an $N$-particle system is given as
$$Z=\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\text{All possible microstates}}\!\!\!\!\!\!\!\!\!\!\!\!e^{-\beta E}=\sum_E\Omega(E)e^{-\beta E}...
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Why can't I balance the units in the equation for power of a sound wave?
According to this answer (source), the time averaged power $P$ of an acoustic wave is:
$$P = \frac{1}{2} \mu v \omega^2 A^2$$
Where $\mu$ is mass density of the medium, $v$ is speed of sound in the ...
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Converting Tesla (classical magnetic field unit) to $\rm GeV^2$ (QFT magnetic field unit)
I want to do research on hadrons in a presence of an external magnetic field but I need to know how convert Tesla to $\rm GeV^2$.
Can you explain that to me?
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What's the link between Planck Mass squared and $1/8πG$?
Good morning,
In an equation of an article, we said that 1/8πG = Mass of Planck^2. But 1/8πG = 596175243.8, is much larger than the Planck Mass^2 = 1.383*10-16 kg. Is there a conversion to do? If not, ...
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How can the unit of Newtons measure both gravity and electromagnetism?
For context, I was thinking about forces. Force is the mass of the particle multiplied by the acceleration it undergoes. There are different kinds of forces, for example, gravity and electromagnetism. ...
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The universal value of Boltzmann constant?
So I'm quite confused about Boltzmann's constant $k_B$ being fundamental. From here:
... the Boltzmann constant. Its value is well known but even if its value were 10 times bigger or if it were ...
- 1,021
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Is amount of substance fundamentally a scalar quantity? (in the mathematical sense of scalar)
Reading the SI (and ISO) standard for units and quantities, I'm currently puzzled by something very subtle.
If I can see and understand why we talk about scalars, vectors, and tensors in the context ...
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Frequency of harmonic oscillator potential
Consider, a particle is moving in a harmonic oscillator potential :
$V=\frac{1}{2}m\omega^2x^2$.
The force on the particle will be :
$F=-m\omega^2x$.
What is the unit of $\omega$ here ? Is it $Hz$ or $...
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Derive the dimensionless Gibbs free energy equation [closed]
I'm having trouble deriving the Gibbs free energy equation:
$$ G = \sum_i n_i \left(g_i^0(T) + RT\ln P_i\right),$$
then,
$$\dfrac{G}{RT} = \sum_i n_i\left[\dfrac{g_i^0(T)}{RT} + \ln P + \ln\left(\...
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Why the Cauchy Stress Tensor & the Stress-Energy-Momentum Tensor have the same SI units?
Shouldn't adding time as a dimension changes the Stress-Energy-Momentum Tensor's units? What math operation(s) (if any) would transform the 3D Cauchy Tensor into the 4D Energy Momentum Tensor of GR?
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How does the $\beta$ factor appear in eq. (4.33) of Altland, Simons "Condensed Matter Field Theory"?
In Altland and Simons' "Condensed Matter Field Theory" book, on the partition function for the non-interacting gas section, is states:
however, immediately after that, a beta factor shows ...
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Why are we interested in the dimensional analysis/power counting in string theory?
I'm learning bosonic strings on my string theory course; here is part of my notes about the dimensional analysis on the world sheet $\Sigma$ and the spacetime manifold $\mathcal{M}$:
I learned this ...
- 1,305
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Small and large extra dimension(s) of the physical space
Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example.
Consider a two dimensional manifold like $\mathbb{R}^2$ and we are trying to add a small and ...
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When do you use $ΔU= mCΔT$ in thermodynamics? [duplicate]
I am confused between when to use $ΔU= nCvΔT$ and $ΔU= mCΔT$.
For example a question says
100g of water is heated from 30°C to 50°C, ignoring the slight expansion of water the change in its internal ...
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Why does dimensional analysis work? I need a elementary reason [duplicate]
I understand this is a very elementary question, but I haven't been able to come up with any elementary reason why it should work.
Also, why should quantities in an exponential be dimensionless?
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Looking for dimensional analysis practice [closed]
I just finished Bridgman's book on dimensional analysis, and am trying to work through the problems. Does anyone know of an answer key for Bridgman's book?
Here is the book in question:
https://www....
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Basic question about units in range formula [closed]
$$R = \frac {2v_0^2sin \theta cos \theta}{g}$$
velocity is $\frac fs$ and g is $\frac {f}{s^2}$ which would seem to be $\frac fs \times \frac {s^2}{f}$ resulting in $s$ after cancellation, yet the ...
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Why aren't the interatomic Heisenberg exchange coupling parameters reported in units of energy/units of spin squared?
The Heisenberg Hamiltonian is $H_{Heis}=-\sum_{ij} J_{ij} \langle \vec{S}_i \cdot \vec{S}_j \rangle$ (generally, save different constants depending on convention). This seems to suggest that the ...
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Is action an extensive quantity - always?
Action, the integral over time of the kinetic minus the potential energy seems to be an extensive quantity. (There is nothing serious coming up in Google on this issue. Neither on Google Scholar.)
In ...
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Units and dimensions in equations [closed]
If I said, for which value of $r$ does a circle have equal area and circumference, we could all calculate that it happens when $r=2$, but when written out, are we not saying that an area equals a ...
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How to compute the Reynolds number of a fluid without knowing the density?
I am running a simulation with LAMMPS involving a colloid suspended in a fluid. Simple shear is applied creating flow. My question is, how can I calculate the Reynolds number of the fluid given I don'...
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Einstein coefficient unit
Einstein-A coefficient of diatomic molecule can be written as (W. A. Brown, 1970):
[unit]
A= 1/sec (transition probability, Einstein coefficient)
h-bar= Joule*Second (planck constant)
Re=Coulomb*...
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A Problem in Scaling
Question: - If the size of the nucleus (in the range of $10^{-15} m$ to $10^{-11} m $) is scaled up to the tip of a sharp pin, what roughly is the size of an atom? (Assume the tip of the pin to be in ...
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$ℏ$ in the canonical commutation relation
I am wondering what the physical meaning of the introduction of a "new" constant $\hbar$ in the CCR $[\hat{x},\hat{p}]=i\hbar$ is if you compare it to the classical Poisson-bracket $\{x,p\}=...
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Unit of $v$ in $v=rω$ [duplicate]
unit of velocity is m/sec but in $v=rω$
unit of $r$ is metre and unit of $ω$ is radian/sec so unit of $v$ should be radian*metre/sec but it is m/sec how
I know this is dimensionally correct but I dont ...
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The Ricci scalar in FRW, where am I getting wrong?
I'm trying to derive Ricci scalar with FRW metric, but additional $c^2$ makes me confused.
The book by D. Baumann says
\begin{align}
R &= g^{\mu\nu}R_{\mu\nu} \\
&= -R_{00}+\frac{1}{a^2}R_{...
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Normalized units versus dimensionless units
In a molecular dynamics code, suppose, the distances are expressed in units of a characteristic length of the simulated system, $R_0$. In some papers it is written as, " distances are normalized ...
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Dimensional Analysis Does Not Check Out [closed]
I'm looking over the lecture notes found here, and if you scroll down to the end of page 2, the writers make the statement $ A(\omega _d) = \dfrac{f_0}{\sqrt{\omega_0^2 - \omega_d^2 + \omega_d^2\Gamma^...
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How do you go from watts to lumens?
let's say you have a light source and using a solar panel or photoelectric diode you can absorb all the emitted light, which would produce some amount of power outputted by the solar panel. How would ...
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At What Scales Turbulence Matters?
So from what I understand(like from this site: https://www.britannica.com/science/transport-phenomenon) is that turbulence is when the flow is almost unpredictable at different parts but on average ...
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Electric field and potential of a point charge in a (strictly) 2D 'world'
I was trying to figure out how the electric potential and electric field are different in a 3D system versus in a 2D system (I take such a 2D 'world' to be the $xy$-plane, i.e. $z=0$, in a Cartesian ...
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Why can we only use radians in the arc length formula? [closed]
Why can't we use degrees in the arc length formula ($S = R\theta$)? Why only radians?
I don't understand because radians and degrees are both dimensionless.
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Question about unit for intensity
The picture is from one physics textbook about treating sun as a blackbody and compare the electromagnetic radiation spectrum. Note the unit used for Intensity in the picture shown, it's Watts / (...
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Are there any similarities between the Action $S$ and Angular momentum $L$, since they have they same units? [duplicate]
Action is defined as:
$$ S = \int_{t_1}^{t_2} L dt,$$
And has units of joule-second. Angular momentum has the same units, but has a completely different application and interpretation. Are there any ...
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How to reinvent measurement units?
Imagine you're in deserted island. You will eventually need to know how much there is of something or how long is some thing.
Is there a way to get all main measurement units (kg, m, °C, m$^3$, etc.) ...
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What is the logic behind Planck units?
I was curious to know the logic behind “Planck Units”, I read this question but did not understand it. Do you have a better (simpler) explanation for setting $c = G = \hbar = 1$?
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Why use units of $\rm 1/Hz$ instead of $\rm s$?
The Wikipedia page for Planck's constant frequently includes the constant in units $\text{J/Hz}$ or $\text{J} \times \text{Hz}^{-1}$. Is there a reason these units are used instead of $\text{J} \times ...
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Showing that loop corrections are quantum effects in Quantum Field Theory
Say I have a theory in four dimensions with the Lagrangian density
\begin{equation}
\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu\phi - \frac{1}{2} m^2 \phi^2.
\end{equation}
This has the ...
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How to rigorously put back dimensions in equations involving natural units?
I was watching the first lecture of Special Relativity by Leonerd Susskind (link:Youtube) whereby setting the speed of light to 1, i.e. $c = 1 \dfrac{[l]}{[s]}$, where $[l] = 3 \cdot 10^8 \dfrac{[m]}{[...
