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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36 views

Does appearence of $\hbar$ implies the role of quantum mechanics?

I'm currently confused with the uncertainty in statistical mechanics and in quantum mechanics. As I understand, If roll a die, then you can get an outcome from the sample space $$\mathcal{S}=\{1,2,3,4,...
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18 views

Derived units described with multiple 'per's

In a dimensional analysis problem, a rate was given textually as so many "Milliliters per Kilograms per Day". This seems ambiguous, as one could either interpret it as A:: Ml / (Kg / Day) = ...
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37 views
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34 views

Meaning of a product of two physical units [duplicate]

I am having a hard time to understand the meaning of measurements that involve a product of physical units, like $10\, \rm N\cdot m$ ($10$ Newton-meters), $3.33\, \rm J\cdot s$ ($3.33$ Joules-second),...
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Why is BMI measured in units of $\text{kg}/\text{m}^2$? [closed]

Body mass index (BMI) is a convenient rule of thumb used to broadly categorize a person as underweight, normal weight, overweight, or obese. Where $m$ is body mass and $h$ is height, it is defined as: ...
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70 views

Constraints on scalar field theories in $n$ dimensional spacetime

Let us consider the classical action in $n$-dimensional (flat Minkowski) spacetime $$ S=\int dx^0\ L=\int d^nx\ \mathcal{L}\tag{1} $$ with Lagrange function $L$ and Lagrange density $\mathcal{L}$. For ...
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24 views

Obtating the Fermi Coupling Constant from expeced muon decay

I recently performed a lab where we found the expected muon lifetime as $\tau = 2.02$ microseconds. I am now supposed to use this value to obtain the Fermi Coupling Constant $G_F$, by using the ...
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2answers
81 views

Question about Dimensional Analysis

To determine interesting relationships in the properties/variables of a phenomena, one can often use dimensional analysis. For example, say we want to determine the dependence of environmental ...
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0answers
35 views

Why is work done equal to force×displacement? [duplicate]

I have understood that work done depends on force and displacement but I have not understood that why does the product of these two gives work done and not multiplied by some constants or raised to ...
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1answer
37 views

Dimensional analysis: on the choice of fundamental dimensions

As far as I understand, the entire idea of dimensional analysis relies on the existence of fundamental dimensions that are independent. Because these fundamental dimensions are independent, each ...
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4answers
104 views

What is the difference between m/s and m/s/s?

Can I subtract, say, for example, 600 m/s - 20 m/s/s? I am confused about this topic and there is no good information on the web.
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4answers
104 views

Is it possible to divide time by distance? Is $\rm s/m$ possible? [closed]

Is it possible to divide seconds by distance/area/volume? If so, how to imagine and understand such a thing, and also what do we get by dividing them both (if possible to do so).
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Is there a theoretical interesting in another dimensionless constants?

Since my initial study in QM I have found a lot of quotations about the constant of value "1/137", the fine structure constant. Generally the author gives an introduction about it and says ...
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28 views

Normal mode energy units

In classical mechanics and solid-state physics texts, you often see the Hamiltonian $H$ of a system of harmonic oscillators written in normal mode coordinates: $$ H = \frac{1}{2}\sum_n\dot{X_n}^2 + \...
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1answer
23 views

Why can we set the external momenta to zero while calculating the 1-loop correction to non-abelian theory coupling?

Refer to the second diagram in figure 73.2 on page 440 of Srednicki's Quantum Field Theory book. On page 441 he proceeds to calculate the amplitude and argues that since the divergent part is ...
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2answers
63 views

Converting energy from units of joules into units of wavenumber

If we have an expression for energy $E$ in Joules why is it that if we wish to convert the energy into wavenumber we divide the expression by $hc$? I know that $$E = \frac{hc}{\lambda}$$ So surely ...
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1answer
73 views

What is the unit and dimension of a matrix operator in quantum mechanics?

Let us consider the higher order Born approximation that explains multiple scattering. The well-known Born expansion leads to the expression for transition "matrix" or "T-matrix" ...
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Dimensional analysis and more: Order of divergences in the $\phi^4$ theory

On page 324 of his book on qft, Peskin analyzes the superficial degree of divergence of the 1-loop self-energy correction of the $\phi^4$ scalar theory. Realizing that $D=2$, he concludes that the ...
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1answer
73 views

Why do we treat the action as dimensionless in QFT?

When determining whether the couplings in a QFT Lagrangian are relevant/irrelevant/marginal, we set $\hbar = c = 1$ and use the fact that the action is dimensionless to find the dimensions of the ...
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2answers
32 views

Why is it called specific gravity, when it defines the ratio of densities?

I recently learned about the term "specific gravity" and was surprised to see nothing in the definition that relates to gravity, or indeed, any sort of force. I'm sure there's a historical ...
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79 views

Implications of the Buckingham-Pi theorem

According to the pi-theorem any dimensionally homogenous equation $$f(x_1, x_2, x_3, \ldots) = x_N$$ can be expressed as another function of dimensionless variable products $$g(\pi_1, \pi_2, \ldots) = ...
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1answer
42 views

Power of electron density in Bloch's exchange term

Exchange term introduced by Felix Bloch is given by $$ E_x = c_2 \int n(\vec{r})^{4 / 3} d\vec{r} $$ In the example 1.6.7 of the attached reference, the power of $n(\vec{r})$ is derived as following $$...
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4answers
171 views

Multiplication allows for different units: why can't we multiply apples by apples?

In school, we are taught that addition must use the same units (we can't add 3 apples + 4 bananas). On the other hand, multiplication (and division) is allowed between quantities of different units, ...
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1answer
54 views

Dimensional analysis on absolute magnitude of a star

Absolute magnitude $M$ of a star is $$M = m-5\log \Big(\frac{d}{10}\Big)$$ where $d$ is measured in $parsec$ but this raises couple of problems. $M$ and $m$ are dimensionless quantities but $d$ is in $...
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2answers
82 views

Where does the unit kelvin come from?

The zeroth law of thermodynamics implies, that there exists a function of state variables $\theta(U,X_i)$ such that when two systems are in contact with each other, their thetas are equal$$ \theta_1(...
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50 views

Renormalization theory in system with two coupling constants

Suppose a system has 2 coupling constants, $t$ (temperature) and $h$ (applied field). Let $K_{1}$ and $K_{2}$ be coupling constants, such that $$\left[ \begin{array}{c} K_1 \\ K_2 \end{array} \right] =...
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1answer
36 views

Are the units for the magnetic moment correct for this paper?

Can the units of $$\frac{e\,c}{2\pi\,f},$$ where $c$ is the speed of light, $e$ is the elementary charge and $f$ is the frequency, be correct for the magnetic moment? I see that the units of magnetic ...
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1answer
52 views

Are dimensions of physical quantities and 1D/2D/3D/4D… spacetime dimensions the same?

Are the [L]/[M]/[T]... dimensions of length/mass/time... somehow related to 1/2/3/4D spacetime dimensions? If not; why are they called dimensions and what do they actually mean?
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1answer
78 views

Where does Coulomb's constant, $k_e$, come from?

I'm familiar with Coulomb's law, $\vec{F} = k_e\frac{q_1q_2}{r^2}$, and know how to apply it (having used it extensively in practice, on exams, etc). I find that the $\frac{q_1q_2}{r^2}$ "part&...
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1answer
55 views

Confusion about dimensions of a functional, its functional derivative and its variation

Let's take a functional $F[\phi]$ as defined in this answer $$ F[\phi] = \int d^4x \, \phi\, \partial^2 \phi $$ whose dimensions are, if the coordinates have dimensions of a length, as it's customary, ...
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3answers
159 views

Misunderstanding the dimension of QCD

From my point of view, the definition of the tension tensor is contradictory. $$F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu} +ig[A_{\mu},A_{\nu}]$$ $$[A_{\mu}]=\frac{1}{cm \times g};[F_{\...
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2answers
55 views

Transformation of Dimensions

I'm reading Applied Dimensional Analysis and Modeling by Thomas Szirtes and struggle with the transformation of dimensions. In particular, given $1\,\textrm{m} = x\,\textrm{cm}$, its easy to see that $...
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1answer
39 views

Units of angular frequency in a simple harmonic oscillator [closed]

The equation of a simple harmonic motion can be $x=A \cos(\omega t)$. $\omega$ therefore has units of $radians/sec$. I was solving some problems when I found a statement on my notes $x=\left(1+\...
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2answers
43 views

Exponential function and natural units

The argument of the exponential function has to be dimensionless. By switching to natural units, velocity (for example) becomes dimensionless. Surely, I cannot take the exponential of a velocity now ...
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3answers
76 views

What is the physical meaning behind a meter (long) x a meter (long)? [closed]

I have searched the internet for an answer to this question, but nothing satisfactory comes up. I have been told that a meter x a meter always equals to a square meter, but I have decided this cannot ...
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26 views

Unit of measurement of torque/moment of force [duplicate]

During my mechanics class I studied that moment of forces is measured in $N \cdot m$ and I also studied that kinetic energy is measured in Joules $J$. I understand that one is a scalar quantity and ...
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3answers
80 views

Force per velocity per time squared signifies what? [closed]

Anyone know what force per velocity per time squared signifies? It has to do with magnetic fields. I was doing dimensional analysis and it popped up. (Charge is assigned units of force and the rest ...
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3answers
80 views

Difference between Dimension and Direction [closed]

What is the difference between Dimension and Direction? Because when you compare the 3 Axis of 3 Dimensional Space, to the 3 Axis in Euclidean Space, or Euclidean Geometry there doesn't seem to be any....
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1answer
108 views

Is there an analogue of Hawking radiation for other massive bodies?

Without resorting to formal GR, it is possible to derive the structure of the Hawking temperature formula through simple dimensional analysis after assuming an ansatz of $$T \sim R^a\hbar^bk_B^cc^d,\...
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2answers
79 views

What is the unit of Klein Gordon field?

Normally I don't care about units in the derivations on relativity or QM. Just set $\hbar = c = 1$. But learning about the energy momentum tensor for the Klein Gordon equation, I couldn't make $T^{00}$...
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1answer
55 views

Measuring a speed in Kelvin $K$

Since the Kelvins are a temperature unit and temperature is basically molecule/atoms excitation (so they have a "speed of excitation"), could it be possible to measure a speed (let's say 10 ...
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3answers
123 views

Why in 3+1D QFT a scalar field has (mass) dimension $1$ but in 3+1D QM the wavefunction has dimension $3/2$?

In this question we assume as usual that $\hbar = c =1$, so the word "dimension" means the dimension in mass. From the fact that an action has the same unit as $\hbar$, or dimensionless ...
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27 views

How is the expression for Reynold's number valid in steady flow?

Reynold's number is defined as the ratio of inertia force to viscous force. A common expression for this inertia force is (density)x(Area of cross section of tube)x(velocity of fluid)x(velocity of ...
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1answer
50 views

What is $[\mathrm{J}]$ supposed to denote in this case?

I am currently studying Laser Systems Engineering by Keith Kasunic. Chapter 1.2 Laser Engineering says the following: If the energy of the incident electromagnetic field more-or-less matches that of ...
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1answer
34 views

Conceptual question on current vs current density Ohm's law [closed]

The current density $j = \sigma E$ has units number/(seconds * area). Sigma is the conductance $\sigma = 1/R$, $R$ is the resistance. $E$ is the electric field. We want to make the analogy to $i$, the ...
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1answer
39 views

Dimensionless physical constants that used to be calculated experimentally but have a known closed form now

What are some examples of dimensionless physical constants that historically could only be calculated to greater and greater precision through ever finer and more precise experiments until one day, a ...
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4answers
48 views

Clarification in definitions of physical quantities

What does "per unit" mean? For example is defined as mass per unit volume. Mathematically we can write this as:$$ρ= \frac{dm}{dV}$$ The same goes for velocity and power. I can't understand ...
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0answers
32 views

Dimension of matrix elements in scattering cross section

This question is probably going to be somewhat trivial, but I am a little confused about the dimension of the matrix element that appears in the formula for the cross section of a scattering process. ...
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4answers
118 views

Is $E=mc^2$ strictly just a conversion formula? [duplicate]

$E=mc^2$ Does not this imply that there is no energy without mass, or is it strictly a conversion formula?
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1answer
57 views

Planck constant equation question

I’ve had a lot of contradicting answers around this equation and I want to know if it’s hypothetically correct. I know Planck constant isn’t a force but I want to know if this equation is correctly ...

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