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.

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Are there any significant integer constants that are not unitless? [closed]

When it comes to meaningful integer constants, the only ones I can come to think of (except zero) are unitless, for example 1 (the multiplicative identity), 2 (the base of the binary system), 10 (the ...
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EFT's $\hbar$ counting at loop level

In the Saclay Lectures on EFT, the author claims: Note that $\hbar$ counting still works at the loop-level. To see this, one should take into account that, when $\hbar$ is retrieved in the action, ...
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EFT matching: using tree-level to perform 1-loop-level

I'm reading the Saclay Lectures on EFT, and I don't understand how it uses the tree-level matching to compute the 1-loop-level matching. To simplify, in this post I'll put its $C_6,\lambda_1=0$ since ...
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The dimension analysis for the langevin force [duplicate]

In wikipedia(https://en.wikipedia.org/wiki/Langevin_equation), the langevin force formula is given as $$\langle\eta_i(t)\,\eta_j(t')\rangle = {2\gamma\,k_B\,T\,\delta_{i,j}\,\delta(t-t')}.$$ However, $...
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Is there a relationship between the beta function and the dimensions on which a theory is built? [closed]

The beta function tells us the relationship between the coupling parameter and the energy scale on which we study a system. What happens to this function when a QFT is formalized in, say, 3 dimensions?...
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What are cubic gallons? [closed]

I was working on my physics homework and thge answer came out to gallons^3, and the textbook said I had the right answer, so I'm trying to figre out what exactly a cubic gallon is? does anyone have a ...
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Need clarity on dimensional analysis

I was reading about dimensions where in my book it said Note that in this type of calculation the magnitudes are not considered. It is equality of the type of quantity that enters. Thus, change in ...
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2 answers
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Is the Planck force a truly "Planck unit"?

The Planck force appears to be defined as the ratio of the Planck energy to the Planck distance, $ F_P = E_P/l_P $ that can be rewritten as $$ F_P = \frac{ E_P }{ l_P} = \frac{ c^4 }{ G }. $$ Isn't it ...
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How would the following image look like, if we didn't use $ct$ for time?

I just wonder how spacetime would look like if we didn't use $ct$ for $t$ and we just used $t$ instead? I guess the $t$-axis would just scale. Would that mean that, the hyperbolas would be very hard ...
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Is Planck length constant in String Theory? Does it have a renormalization flow?

Is Planck length constant? Planck length $l_p$ is dependent on Newton constant $G_N$ which is related to coupling constant of interaction of gravitons, but from field theory point of view, we know ...
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Logarithmically divergent Feynman diagrams in $\phi^4$ theory

I am going through the lecture notes for my class and I can't seem to follow the logic. Maybe this is considered a homework problem, but I could not find anything that directly answers my question on ...
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On the range of validity of General Relativity and Quantum Field Theory in terms of energy and impact parameter (from Rovelli & Vidotto's book)

In Fig. 1.1 on page 5 in Rovelli & Vidotto's 2015 book Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (PDF), there is this graph giving a general ...
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The definition of Time and second

From the book "University of Physics 15th edition", in chapter 1, they talk about the fundamental units. They stated that the definition of unit of time is based on an atomic clock, where 1s ...
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Why is the part of a sphere's area directly proportional to the square of its radius?

Solid angle in the book is explained in this way: "...Let $dA$ be a small area element of the surface of the sphere. If the points situated on the boundary of this area be joined to $O$ (the ...
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Where is the $\mu_0$ in this Jackson equation?

In Jackson's Electrodynamics, he derives a spherical harmonic expansion for the electromagnetic field in the radiation zone: \begin{align} \mathbf{B} &\rightarrow\frac{\mathrm{e}^{\mathrm{i( k ...
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Question about using dimensional analysis on an analytically derived equation

Say we have the following analytical relationship for a force: Say we want to non-dimensionalise this equation. I would do this by writing: $$F_L = f(\mu, V, R_1, R_2) $$ ... and going from there. ...
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Computing the anomalous dimension of $\phi^2$ via Peskin and Schroeder

I am trying to understand the example that Peskin and Schroeder present at section 12.4 where they calculate the AD of $\phi^2$. Specifically they give a renormalization condition in 12.113 which does ...
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How to show that in 2D CFT the marginal operator must have $(h,\bar h)=(1,1)$?

A related post might be What are marginal fields in CFT? where Qmechanic♦ pointed to Ginsparg secion 8.6. However, I heard about two argument. Claim 1:In a $D$ dimension CFT, the marginal operator ...
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Gravitational wave radiation power from dimensional analysis

Let us try to find a formula for the power emitted through gravitational waves (GW) from a binary system in quasi circular orbit. The relevant quantities are the Newton's constant $G_N$, speed of ...
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Convert BLDC motor thrust in grams ($\rm g$) to Newton ($\rm N$)

I am making a quadcopter with takeoff weight equal to 2 Kg. Using thrust to weight ratio of 3:1 the required thrust per motor is 1500g. (No experimental thrust has been obtained for thrust.) Now I am ...
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Units in Actions (Functional Field Integrals)

When one rewrites the partition function of a grand-canonical ensemble (quantum version) as functional field integral $$ Z = \operatorname{Tr}_{ \mathscr{F}} \mathrm{e}^{ - \beta \left( H - \mu N \...
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Angular momentum dimensions

In Cohen-Tannoudji's book vol.1 page 648 the following is said: $|\psi \rangle$ an arbitrary state. $\langle\psi|\vec J^2|\psi \rangle=\sum_{i=1}^3\langle\psi|J_i^2|\psi \rangle=\sum_{i=1}^3 ||J_i|\...
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What does it mean that dimensionless physical constants cannot be calculated but only measured?

I have read a passage in Wikipedia about the List of unsolved problems in physics and dimensionless physical constants: Dimensionless physical constants: At the present time, the values of various ...
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Dimensions of perturbative parameter in $\varphi^3$ theory?

In QFT, $\lambda\varphi^4$ is one of the most studied interactions for the scalar field. The parameter $\lambda$ is adimensional, which makes the perturbative treatment straightforward. In the case of ...
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How convert 1 Debye to atomic units

I would like to convert 1 Debye to atomic units. I have that $$ 1~\mathrm{Debye} = 10^{-18}\mathrm{StatC}\cdot \mathrm{cm}\\ = 10^{-20}\mathrm{StatC}\\ = 10^{-20} \mathrm{cm}^{3/2}\mathrm{g}^{1/2}\...
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3 answers
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Notation for rule of thumb, without breaking dimensional homogeneity?

I'd like to know how to write rules of thumb in a concise way, without breaking dimensional homogeneity. For example, if a runner has an average speed of ~10 km / h, an approximation of the covered ...
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Why cant this situation is possible when we are considering a dimensional analysis approach in any problem?

Suppose a unknown quantity (whose dimension we know) depends on known three quantities like (Length of object L) , (Energy of Object E), (Density of Object D), when we try to get the relation among ...
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2 answers
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How does the constant proportionality balance an equation?

The Newtonian universal law of gravitation, Every object in the universe attracts every other object with a force that is proportional to the product of their masses and inversely proportional to the ...
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Canonical dimensions in quantum field theory

Is there anything wrong if i construct a qft model with fields having canonical dimensions other than 3/2 (fermions) and 1 (scalars)? Is there any rigid constraints in qft against it? A detailed ...
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Transduction coefficient of an electromechanical system

I have been recently doing a physics experiment which requires me to determine the transduction coefficient of a speaker system, undergoing electromechanical resonance, in an AC circuit. The quantity ...
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Proca field in the ultra-relativistic limit

I was recently asked by a professor of mine to show that in the ultra-relativistic regime a Proca field has mass dimension of 2 instead of 1 (in $d=4$). As a hint, I was advised to look at the Proca ...
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1 vote
0 answers
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Hawking radiation temperature from dimensional grounds in Zee

In Zee's Einstein Gravity in a Nutshell he gives an argument for the temperature of Hawking radiation of dimensional grounds only (Introduction, p 15). It goes as follows. Both the black hole mass $M$ ...
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1 answer
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I'm studying analytical mechanics and it states that it always true that generalized coordinates times generalized forces have the dimension of energy

Since the terms $q$ "generalized coordinates" are not necessarily ‘lengths’, the quantities $Q$ "generalized forces" also do not necessarily have the dimension of a ‘force’. ...
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2 votes
2 answers
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Unit of a log normal probability density function

How do I find the unit of a log-normal probability density function?
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1 answer
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Ratio of thermal energy in a parcel of water between two time periods (metric vs imperial units)

I am having a problem figuring out something that should be quite basic with determining the ratio of the thermal energy content of a parcel of water (let's say $1 kg$) in between two time periods and ...
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2 answers
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Why is the Fermi Constant sometimes listed in units of 'joule metre$^3$'? How is that related to its normal units of GeV$^{-2}$ or J$^{-2}$?

Normally, the Fermi Constant is valued as $1.1663787\times10^{-5}$ GeV$^{-2}$ or its equivalent in Joules. But on Rampfesthudson and Oxford Reference, it says, $1.435\times10^{-36}$ joule metre$^3$ I ...
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2 votes
1 answer
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Interpretation and units of propagators

Quantum field theory is usually expressed in natural units in which $\hbar=c=1$. This simplifies equations and one can always get back to other units by inserting $\hbar$ and $c$ in appropriate places....
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4 answers
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Validity of dimensional analysis in theoretical physics

My textbook mentions the following lines about the validity of dimensional analysis. ..... if an equation fails this consistency test, it is proved wrong but if it passes it is not proved right. Thus ...
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How to show that the Action has units Energy·time?

The Lagrangian, which has units of Energy, is defined as that which when summed over time gives the Action, the action being more fundamental. But how does summing over units of Energy across time ...
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Metric elements dimension [duplicate]

When we have a metric, what is the dimension of its elements for example when we have $$ds^2=f(r)dt^2-g(r)dr^2-r^2(d\theta^2-\sin^2{\theta} \ d\phi^2)$$ what is the dimension of $dr^2$ and $g(r)$?
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2 votes
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Second functional derivative and its units

Say I have a functional $I[\phi,g]$ with $\phi(p)$ and $g(p)$ functions from $\mathbb{R} \to \mathbb{R}$. Also say that this functional obeys the property: $$\frac{\delta I}{\delta g(p)} = -(g(p))^{-1}...
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Scaling of "non-reduced" parameters in RG theory

I'm studying quantum phase transitions using the Renormalization Group (RG) method. In Continentino's book "Quantum Scaling in Many-Body Systems: An Approach to Quantum Phase Transitions" ...
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3 votes
5 answers
698 views

Do squared units grow in size or shrink in size if the value can be represented by a different unit?

I was doing a very simple equation, $\frac{1}{2}kx^{2}$, when I realized that, if I represented the distance in centimeters, $x^2$ would grow in size (because x is 10 centimeters), but if I ...
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1 answer
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Conversion factors derivation

Unit conversion chart Queries regarding MKS to CGS system for the following formulas : Force $=$ Newton (MKS), Dynes (CGS) $$\mathrm {1\ N = 10^5 \ dynes}$$ Work $=$ Joule (MKS), Ergs (CGS) $$\...
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3 answers
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Meaning and differences between adding & multiplying two different functions in Physics

We all know that acc. to Newtonian mechanics , $F = ma$ and acc. to Lagrangian-Hamiltonian mechanics , $H = T + V$. I want to ask what makes the Hamiltonian, $H = T + V$ and not $H = T × V$? Similarly,...
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4 answers
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Why is 1 newton defined as 1 $\rm kg · m/s^2$?

From my limited understanding, one newton is defined as the amount of force that gives a mass of 1 kilogram an acceleration of 1 meter per second squared. What I don't understand is why it corresponds ...
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1 vote
1 answer
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Non-dimensionalizing laser system of diffeqs, Strogatz Nonlinear Dynamics and chaos 3.3.1 D

The system of equations in question is $$ \dot{n} = GnN - kn$$ $$\dot{N} = GnN - fN + p$$ Where ${N(t)}$ is the number of excited atoms, ${n(t)}$ is the number of photons, ${G}$ is the gain ...
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1 answer
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Does it make sense to add eigenstate bras, such as $\langle x|+\langle p|$?

For a given state vector $|\mathcal{S}(t)\rangle$ in a Hilbert space, it's known that we can express it in different bases. For instance, we can express it in the position basis as $\langle x|\mathcal{...
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1 vote
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Units in vector spaces

I am confusing myself about where physical quantities become mathematical objects. Where does one end and the other begin? E.g. displacement $\vec{s}$. A displacement is a physical quantity that can ...
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1 answer
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Scale invariance in curved spacetime?

Question What does it mean for the metric to be scale invariant in curved spacetime (in the sense when I say a property is scale invariant in thermodynamics)? I'm confused as to how to define this. It ...
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