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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1answer
45 views

Is this differential equation (for damped & driven physical pendulum) physically valid?

Following is the equation of motion for a physical pendulum which is damped and driven by a force of frequency $f$: $$\frac{d^2 \theta}{dt^2} + b \frac{d\theta}{dt} + sin(\theta) = Tsin(2\pi ft)$$ ...
-1
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1answer
36 views

What are dimensions of co-ordinates which are used to define an electric field?

A possible electrostatic field is: $ E_x = 6xy$ $ E_y = 3x^2-3y^2$ $ E_z = 0$ Suppose we are using SI system. So unit for components of field is volts/meter. Then what are dimensions of $x$ and $...
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0answers
27 views

Decay rate revisited

According to Peskin&Schroeder (pp. 107), we have $$ d\Gamma=\frac{1}{2m_A}\left(\prod_f\frac{d^3p_f}{(2\pi)^3}\frac{1}{2E_f}\right)|\mathcal{M}(m_A)\rightarrow {p_f}|^2(2\pi)^4\delta^{(4)}(p_A\...
0
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1answer
49 views

Why does $k^z/E$ have dimensions of inverse velocity rather than velocity?

I'm studying quantum field theory and I want to prove the cross section. In Peskin's book, equation 4.77 says that: $$ \frac{1}{\left | \frac{k_{A}^{z}}{E_{A}}-\frac{k_{B}^{z}}{E_{B}}\right |}=\frac{...
4
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4answers
588 views

Are quantum operators dimensionless?

I'm slightly confused as to whether quantum (hermitian) operators, which we get by promoting observables to operators, are dimensionless or not? Clearly the Hamiltonian of the system, say of the ...
4
votes
2answers
583 views

Check dimensions of the integral of a function

I and a colleague are arguing about the dimensions of: $$\int_0^x f(x) dx $$ in this particular case $[f(x)]=m^2/s^3$ and $[x]=m$. Does it follow that $[\int_0^x f(x) dx]=m^2/s^3$ or $[\int_0^x f(x)...
4
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2answers
75 views

Non-dimensionalizing incompressible Navier-Stokes

I have a question about non-dimensionalization of the incompressible Navier-Stokes (NS) equations. My understanding is that the purpose of non-dimensionalization is to "collapse" solutions onto one ...
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0answers
25 views

units of measurement in Bloch equations

I am trying to understand the Bloch equations for understanding MRI and the equation describes the change of magnetization during NMR excitation and relaxation as: $$ \frac{dM}{dT} = \gamma M \times ...
0
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3answers
65 views

Is there an official list of independent units of measurements?

When I say 'independent units', I mean those which cannot be broken down anymore, and simultaneously forms the basis for any more, complex measurements. For example, height, length, and width can all ...
1
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1answer
51 views

Are there physical law that are not unit-free?

One of the prerequisites of the Buckingham π theorem is that the physical law in question should be unit-free. I couldn't find an example of a physical law that is not unit-free. Is there such thing? ...
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4answers
65 views

Is the concept of work only defined in mechanics?

I'm studying energy and work, so far it looks like work only makes sense in kinematics (objects that move), but energy makes sense in many other ways (electric, thermodynamic, mechanic). Is work a ...
2
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1answer
45 views

How to identify the sign of a derived nondimensional parameter and its physical meaning?

I think that the nondimensional group is ordinarily defined to be positive value in a physical problem. But in some particular case, we probably need to decide the sign of a derived dimensionless ...
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2answers
100 views

What is the dimensional formula of angular velocity?

I have problem to determine the dimensional formula of angular velocity. My friend said that the dimensional formula of angular velocity is $T^{-1}$. It's come from rad/s, rad is dimensionless, the ...
0
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0answers
66 views

What does it mean to take a derivative with respect to $\hbar$?

Problem 6.32 of Griffiths Introduction to Quantum Mechanics, 2ed is In part (b), we take a derivative with respect to $\hbar$. Since $\hbar$ is a constant, what does it mean to take a derivative ...
1
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1answer
30 views

Differences in notation of momentum 4 vector

I have noticed three ways to write the 4 momentum vectors: $P = (E/c, \vec{p})$ $P = (E, \vec{p})$ $P = (E, c\vec{p})$ I know how to derive equation 1, and as far as I know, one can use the ...
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0answers
42 views

Photon Propagator in QFT

Please forgive my temporary split-brain, but I'm a little thrown off by something when considering units at the moment. In QED (depending on the guage), the photon propogator is written as $<0|\...
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1answer
79 views

When can the constant of proportionality in an eq be set equal to 1 and when not? [duplicate]

In $F=kma$, $k=1$ but in $F=kx$, $k$ is not equal to 1?So what are the conditions for the constant of proportionality to be set 1?
0
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1answer
62 views

Issues of normalization & differential final state momenta in analysis of normalized differential quantum-field-theoretic probability of scattering

The normalized differential quantum-field-theoretic probability $dP$ of scattering is given by $$dP=\frac{|\langle f |S|i\rangle|^{2}}{\langle f|f\rangle\langle i|i\rangle}d\Pi,$$ where $|i\rangle$ ...
0
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1answer
54 views

The true dimension of Dirac field

In natural units with $\hbar=1$ and $c=1$, as we know, the energy dimension of the Dirac field $\psi(x)$ in QED is $\frac{3}{2}$. But in cgs units, what is the true dimension of the Dirac field $\psi(...
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0answers
38 views

Finding the exponent of $\lambda$ in Wien's displacement law

I am reading this paper on a short history of the $T^4$ radiation law. In particular, on page 5, By assuming that the wavelength of radiation emitted by a molecule was a function only of its ...
0
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2answers
60 views

Proper units for physical quantities when $\hbar$=$1$

How to deal with the units of quantities if $\hbar=\tfrac{h}{2\pi}=1$? For example, the energy $E=\hbar\omega$: If I have chosen $\hbar=1$, how do I use the units to properly differentiate between ...
3
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1answer
101 views

Is meters per second equivalent to seconds per meter?

I know this question is probably ridiculous, but bear with me for a moment. This thought emerged while I was converting between nm and wave numbers ($\rm cm^{-1}$). In order to prove this conversion, ...
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2answers
87 views

Use Runge Kutta method to solve schrodinger equation

The schrodinger equation in spherical coordinates after seperation of variables as a solution of hydrogen atom is given by $$ \frac{-\hbar^{2}}{2 \ m} \left[ \frac{1}{r^{2}} \frac{d}{dr} \left(r^{2} \...
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0answers
22 views

Arguing on dimensions of logarithms and exponentials [duplicate]

Suppose you have some physical quantity $x$ of dimension $l$. We all know that the dimension of $x^2$, for example, will be $l^2$, and that of $\dfrac{1}{x}$ is $l^{-1}$. However, what will be the ...
117
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13answers
10k views

Are units of angle really dimensionless?

I know mathematically the answer to this question is yes, and it's very obvious to see that the dimensions of a ratio cancel out, leaving behind a mathematically dimensionless quantity. However, I've ...
28
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12answers
6k views

Do all equations have identical units on the left- and right-hand sides?

Do all equations have $$\text{left hand side unit} = \text{right hand side unit}$$ for example, $$\text{velocity (m/s)} = \text{distance (m) / time (s)},$$ or is there an equation that has different ...
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1answer
148 views

Difference between theoretical equations and empirical equations

Some equations are theoretical in the sense that they are derived from an underlying theory. Other equations are empirical in the sense that they were selected only because they fit experimental data ...
5
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6answers
147 views

Why do constants have dimensions?

I am just a beginner in dimensional analysis, and I see that $G$, the universal gravitational constant, is independent of everything. Speed, for example, depends on distance and time, but $G$ does not ...
3
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1answer
96 views

The dimension of the energy-momentum tensor and the Einstein-Hilbert action

I have been thinking recently what will happen if one uses the energy momentum tensor of the Dirac field as a source in the Einstein Field equations. It is well known that in this case $$ T_{\mu\nu}=...
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3answers
50 views

Is “power to mass ratio” in fact independent to mass?

I was trying to find power to mass ratio of sun (work done per second at each unit mass at average), but I found the unit is quite straight: W=kg m^2 s^-3 Then W/...
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0answers
50 views

What kind of unit is $m^2s^{-4}$ in terms of gyro/accel?

Background While working on something in the field of avionics, I have discovered the following unit as part of an inertial-physics equation... $$m^2s^{-4}$$ I am trying to figure out the formal ...
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9answers
2k views

How to interpret the units of the dot or cross product of two vectors?

Suppose I have two vectors $a=\left(1,2,3\right)$ and $b=\left(4,5,6\right)$, both in meters. If I take their dot product with the algebraic definition, I get this: $$a \cdot b = 1\mathrm m \cdot 4\...
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1answer
67 views

A simple explanation of Kepler's Third Law

Is there a simple way to explain how Kepler's third law follows from the inverse square law that of gravity (and laws of motion) For example for Kepler's second law we can say it's because Gravity ...
3
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1answer
70 views

Dependence on UV cut off of some $\phi^4$ diagrams

Consider the one loop corrections to the propagator and the vertex in $\phi^4$-theory:                    &...
0
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1answer
31 views

What is “unity constant form” and why is it useful?

I read the following in a tutorial: The standard transfer function of a first order system is: $$G(s) = \frac{k}{s + a}$$ Arranging this into unity constant form gives: $$G(s) = \...
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2answers
159 views

Dimensions of physical quantities in quantum mechanics

In most introductory quantum mechanics classes, we are introduced to the Dirac notation, concept of the 'state' of the system being represented as an abstract vector in the Hilbert space associated ...
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0answers
28 views

Initial value in rescaling differential equation

I've re scaled the simple harmonic oscillator differential equation as below: original equation: $d^2x/dt^2+\omega^2x=0$ re scaling factor: $\omega t\to t'$ re scaled (dimensionless) equation: $d^2x/...
2
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3answers
112 views

How can geometrized units have more than one constant equal to 1?

I can understand how you could manipulate units to make a certain constant equal to $1$, like $c$ or $G$, et cetera. But how can you make it so two constants (in this case $c$ and $G$) are equal to $1$...
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0answers
56 views

What's the unit of fine-structure constant? [closed]

What's the unit of fine-structure constant? I mean in SI units.
5
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0answers
81 views

Dimensional inconsistency in first law of black hole thermodynamics

The first law of black hole mechanics (let's simplify by considering a uncharged and non-rotating black hole) can be written as $$\delta M = T \delta S$$ If I use the definition of Hawking ...
3
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1answer
26 views

Dimensional interpretation of inverse gradient length $\frac{d}{dx} \ln(Y)$

Preliminary definition: inverse gradient length Let me first explain what I mean by that term. The inverse gradient length of some quantity $Y$ (often thermodynamic temperature $T$) $L_Y^{-1}$ is ...
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1answer
47 views

Proving that Planck's Law is dimensionally homogeneous [closed]

I would like to know whether it is possible to show that Planck's Law is dimensionally homogeneous, as well as the steps taken to prove it. $$B_\lambda(\lambda, T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ ...
3
votes
2answers
87 views

Functions and Length Scales

Regretfully I have to start with an apology as I fear I might be unable to express the question rigorously. Often reading physics papers the concept of "length scale" is used, in statements such as "...
3
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0answers
33 views

Pressure inside a typical white dwarf

Does any one know the order of magnitude of pressure inside a typical white dwarf (better with reference)? Thanks! I think it should be $m_e^4c^5/h^3$ (may be multiplied by $\pi$), which is $10^{22} \...
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1answer
97 views

What defines a physical property? [closed]

The physical world around us has all sorts of properties, shape, color etc. If you move on to more complex systems, there are even more like some emotional properties etc. Why do we deem only ...
3
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1answer
151 views

Newton's Second Law of Motion

Newton originally wrote his second law as: "The rate of change of momentum of a body is directly proportional to the resultant force applied to the body, and is in the same direction as the force." ...
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0answers
52 views

Why is one Telsa equal one weber per square meter instead of one weber per cube meter?

Lines of magnetic flux exist in three-dimensions, so how can they be measured per area unit?
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6answers
271 views

Dimension of an angle [closed]

Usually angles are described as dimensionless, justifying this by saying that they can be viewed as length divided by length. As a student of mathematics I'm asking myself wether this is a convention ...
1
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1answer
48 views

With radian as a unit, should action and angular momentum have the different units?

If one accepts radian as a fundamental unit, does it make sense that action and angular momentum have units differing in radian to the power of one? The same question applies for energy and torque. ...
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0answers
32 views

Dimensional analysis of explosive energy, no temporal element

I have a question about dimensional analysis, using the calculation of the explosive energy of the Bomb by Taylor in the 1940s as example. I am neither physicist nor mathematician so will have ...