In Praise of Dimensional Analysis

. You raise some good points. . One of the problems with the modern unit system (Pascals, Siemens, etc) is that the fundamental meaning of the unit is lost. For example: Pressure X Area = Force. Pascals X Square meters = Pascal Meter^2. So what? You must first convert Pascal to its fundamental definition, which is Newtons/Square Meter. Now dimensional analysis makes sense: (Newtons/square meter) X (Square meter) = Newtons, which is consistent with a unit of force. . Another example is characteristic impedance of a transmission line. We've all learned that the equation is R = SQRT(L/C), where L = inductance/unit length, C = capacitance/unit length. Resistance = (Henries / Farads)^.5? You need to get back to the fundamental relationships among voltage, current and time for inductance, and the fundamental relationships among current, voltage and time for capacitance. v = Ldi/dt. i = C dv/dt. From these fundamental equations, you can get the fundamental units of L and C: L=VoltSeconds/Ampere, C= AmpereSeconds/ Volt. Now the equation for characteristic impedance makes sense, in terms of its fundamental units: R = SQRT([(VoltSeconds/Amps)/ (AmpsSeconds/Volt)] R = SQRT(Volts^2/Amps^2) = Volts/Amps = Ohms.

...

So what is that in meters, kilograms, and seconds? Does it matter?

Jerry

(snip)

Capacitance is in cm, resistance in s/cm, inductance in s**2/cm.

Capacitance per unit length is dimensionless, inductance per unit length s**2/cm**2, so sqrt(L/C) is s/cm, just like resistance!

I used to be able to do it in MKS and CGS units about equally, and sometimes Heaviside-Lorentz units. (Similar to CGS, but without so many 4pis around.)

-- glen

That's CGS absolute, with volts in abvolts (1 abvolt = .01 microvolt) and current in abamps (1 abamp = 10 ampere) I'd rather use MKS units; so would you! :-)

Maybe you wouldn't rather. :-)

Jerry

(snip)

One professor, when explaining the unit system he expected in answers, explained that his house power line was 1/3 of a statvolt.

-- glen

I knew someone who claimed to be a professor of theoretical nuclear physics--- oh no, he really was --- who wildely objected against this.

My scientific origins are in theoretical physics but I mostly worked as a softwareengineer--- now lurking here to learn about DSP. What baffled me again and again is how engineers (in Germany) approach a problem: their question is ``where is the applicable formula''. I was trained to ask: ``what is the mathematical modell that allows me to handle this kind of problem.''

From studying mathematical modells, the physicist is probably just much more trained to handle long symbolic calculations as the average engineer and from this he has a stronger ability to analyze such formulas ``at a glance''--- w/o inserting units. To him the essential thing is the physical quality that is measured, like length or time, not the units that someone happens to use.

Measured values just tell us how often some well defined object will fit ``into'' the measured object; the sole purpose of units is to pass along what object the one who measured happened to use for reference. Thus I like to claim: all those units were completely superflous, hadn't things been hopelessly messed up, starting at beginning of the world, by merchants and engineers.

Comparing any speed to the speed of light in vacuum is perfect--- might even have some advantages, when on speed-limit signs a non-zero digit does not appeare until the 9-th (or so) position after the decimal point ...

Strongly typed languages were designed to support this style of programming. They come in two flavours: ``static typing'' and ``dynamic typing''. If you want to see powerfull representatatives of both paradigms, just look at ML and Scheme.

Static typing gives you strict compile-time type check at no run-time overhead; dynamic typing gives you strict run-time type-check .

i disagree, i guess. when i was teaching EE classes (circuits) and as a grad student in EE, i would be pretty hard on any engineering student that did it the latter way. i think Jerry's little proverb said it best: "Mathematicians routinely ignore units, but engineers do so at their peril."

the cumbersome method you ascribed to engineers, Glen, is almost as bad as ignoring the unit. (because your root algebraic equation is dimentionless. in the latter case, F = ma is actually expressed as the dimensionless F is the product of the dimensionless m and the dimensionless a, as long as you express all three in their base SI units.

r b-j

why would a theoretical nuclear physicist have a problem with that? these are one example of "Natural units" (look that up in Wikipedia) and i think these are called "electronic units" if the "h" is replaced with "hbar". but qualitatively, the are electronic units. they can be compared to Stoney units (but i think Stoney normalizes G instead of hbar), Planck units (but Planck normalizes G instead of e), or Atomic units (which normalize the mass of the electron instead of c). personally i like Planck units the best because they have no prototype particle, object, or "thing" that needs to be sorta arbitrarily chosen as a base unit. it's better, in my opinion, to normalize these parameters of free space, before introducing any special particles or objects, to define your base Natural units. i which Planck would have normalized 4*pi*G instead, that would have been more natural.

other good Wikipedia articles that are related is

Planck units Physical constant Dimensionless physical constant Fundamental unit

i had something to do with those articles, including Dimensional analysis.

r b-j

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That was one of Gary Larson's classic Far Side cartoons. "Everything is squared away. Yep, squa-a-a-a-a-ared away ..."

Mark

Nope. It's

X double dot = - (omega squared) X

-- Mark

Somewhere, often in the form of an R*C, those volt-seconds are divided by seconds to get back to volts.

Mark

Hello Robert,

That is cool you had some input on those articles. If you recall I use atomic units all of the time. There are two variations. One uses a Rydburg and the other uses a Hartree for the fundamental energy unit.

The main reason for using these is the Schrodinger equation becomes quite simple in appearance. A factor of two shows up in the energy term. So that is why there are two variations. The distance becomes scaled to Bohr radii - the mean distance to the electron in the ground state of Hydrogen. And 1 Ry is the binding energy.

Clay

what i recall Clay, is that it was YOU that first told me the name for Planck Units:

i thank you profusely. because you did that, i was able to do web searches, found papers/web_sites/books by Michael Duff, Gabriele Veneziano (pioneers of string theory), Lev Okun, John Baez, and John Barrow and i've had several really neat email conversations with ALL 5 of these guys about the nature of fundamental physical constants (the only ones that count are the dimensionless ones, any notions of a "varying c" or "varying G" are not even wrong, they're meaningless, and for those who can't see that, just think of everything measured in Planck units). and then later i got into Gravito-electro-magnetism (GEM) a little and some interest in the Gravity Probe B (which*still* hasn't been able to conclusive say that frame-draggin or gravitomagnetism or gravity waves can be measured, they are behind schedule.)

none of that fun would have happened if you hadn't done that for me nearly a decade ago. thanks.

discovered a word for that, too: "Nondimensionalization." there's a wikipedia article on that also. i may have done a minor edit to that. but dunno.

isn't normalizing the Rydberg constant have similar effect as fixing the Bohr radius (with another dimensionless alpha tossed in)?

just curious.

r b-j

(snip)

It isn't ignoring units, but more like tunneling then. One must convert given quantities to the appropriate units before applying the equation. One can then use the numbers with a slide rule or calculator, and apply the appropriate units to the result.

When writing down the process, what is usually called "show your work" all units will still be shown.

I probably overemphasized the difference, but yes, the algebraic equation is dimensionless, but so are all calculators that I know of.

The problem with the way I called the physics way is that it can result in unusual or inconsistent units. One might end up with an acceleration in meters/second/hour, for example, or worse.

For the engineer way, one converts all given units to those specified, does the algebra keeping the units (except when entering them into a calculator), and then converts the result to the desired unit.

Though the division probably isn't all that strict, and it wouldn't surprise me if many EE's used the physics way.

-- glen

dunno what that means.

sure. as noted by sombuddy else, you can program computers (especially with C++ or some other OOP) to attach a unit from a known list to the numerical quantity. that way we conceptually have a dimensionful quantity in the computer.

this is what conversion factors are for. these conversion factors, like (0.3048 m/ft) are dimensionless (even though they have units inside the expression), in fact should be the dimensionless 1 so that multiplying or dividing by it changes nothing.

but the requirement is that all units chosen must be consistent. the so-called "physics" way you can actually accelerate a pound mass by (mi/hr)/s and have a meaningful equation (and an unusual unit of force) naturally pop out.

i guess i'm a partisan for that method. almost to the point of facsism.

i'm getting less tolerant in my middle age. (Jerry, does it get worse or better as we get older?)

r b-j

I can imagine circumstances in which that would quite a useful unit for acceleration.

Robert

...

Both.

Jerry

Hello Robert,

I'm glad my giving a name for Planck units put you on an interesting course. I recall reading many years ago about Paul Dirac exploring the ideas behind Planck units. And that is how I knew about them.

Studying Hydrogen has yielded some very interesting info. Balmer empirically put together a formula that fits the spectral lines of Hydrogen, but it offered no theory.

Bohr with his theory of the atom gave a theoretical basis for Balmer's formula. But spectroscopists soon discovered a slight flaw in that some of the energy levels had finely spaced details. A spectral "line" under close observation turned out to be 2 or more lines very close together.

Sommerfeld found by adding special relativity and elliptical orbits to the theory, that he was able to explain the fine splitting (fine structure) of the spectral lines. Bohr was of course ecstatic that his theory was saved. Sommerfeld introduced the notion of the fine structure constant, which in many older books is called the "Sommerfeld fine structure constant", but now many have dropped Sommerfeld's name - what a shame.

Alpha, as the constant is usually denoted, was discovered to be a dimensionless number which may be expressed as (e*e)/(2*epsilon_0 * h*c) and "e" is the charge of the electron, h is Planck's constant, c is the speed of light in a vacuum, and epsilon_0 is the permititivity of free space. We know alpha to be basically = 1/(137.0359895...). The fact that it is much smaller than one allows perturbations to simple systems to be expanded in series involving successive powers of alpha. These series may be found by using Feynman diagrams.

Even though Sommerfeld discovered alpha with his work on Hydrogen, it has turned out to be one of the most important fundamental constants with consequences far beyond atomic physics. It shows up in many places such as in the interaction of E-M fields with electrons and in nuclear and particle physics.

In terms of atomic theory, the Bohr radius may be written as (4pi*epsilon_0*h_bar^2)/(m*e^2), where m is the mass of the electron, e is its charge, epsilon_0 the permititivity of free space, and h_bar is Plank's constant h divided by 2pi.

A Hartree is simply alpha^2 * m*c^2 (27.2116 electron volts) and a Rydberg is half of a Hartree. Again "m" is the mass of the electron and c is the speed of light. Douglas Hartree did a lot of work in computational atomic physics, and he gave us "atomic units." He even built analog computers from a kid's toy called "mechano" to solve atomic physics calculations back before WW2. [Mechano (UK) is a toy very much like the Erector Set in the U.S.]

I've been working on a new algorithm to solve the Hartree-Fock equations. So far the results have been quite encouraging. I.e., more accuracy with less computation.

Clay

p.s. A book you may find interesting is "Universal Constants in Physics" by Gilles Cohen-Tannoudji, 1993 McGraw-Hill. The parts on "h" & "k" are interesting with "h" being the quantum of action and "k" being the quantum of information.