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In Praise of Dimensional Analysis

J

Jerry Avins

Jan 1, 1970
0
robert bristow-johnson wrote:

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

Both.

Jerry
 
C

Clay

Jan 1, 1970
0
what i recall Clay, is that it was YOU that first told me the name for
Planck Units:http://groups.google.com/group/sci.physics/msg/e1d9352d0a6b64b3
and 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

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.
 
R

robert bristow-johnson

Jan 1, 1970
0
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.

i remember studying this in 3rd semester physics. from the NIST site,
they were saying that the very first occurance of alpha that
Sommerfeld had as that it came out to be the ratio of the speed of the
electron in the lowest shell of the Bohr atom to the speed of light in
vacuo.
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...).

check NIST, there are new 2006 CODATA:

1/alpha = 137.03599956 +/- something

2002 CODATA had it at 137.03599911.

this mathematician, James Gilson (i had an email convrsation with him,
too) says he has some theory that calculates alpha to be

alpha = cos(pi/137)/137 * tan(pi/(29*137))/(pi/(29*137)) .

it comes out almost within one stadard uncertainty to the latest
accepted value. it might be just numerology. i dunno.

it turns out that sqrt(alpha) is the ratio of the elementary charge to
the Planck charge and that's how i like to look at it. i like to
think that alpha takes on th value it does because of the amount of
charge, measured in Natural units, that nature has bestowed upon the
electron, proton, and positron. (what are the other charged
particles?)

because i think that it would be more natural to normalized 4*pi*G and
epsilon_0 (instead of what Planck did normalizing G and
4*pi*epsilon_0), then the elementary charged measured in these more
natural Planck units would be

sqrt(4*pi*alpha) = 0.30282212

and THAT is the number i think that theoretical physicists should be
putting up on their walls. that dimensionless number is the charge of
the electron measured in the most natural units that are defined soley
normalizing the parameters of free space, without any use of a
prototype object, particle, or "thing". and alpha results from that.
at least this is my armchair physics opinion.
It shows up in many places such as in the interaction of E-M fields
with electrons and in nuclear and particle physics.

sure, given a geometry or constellation of charges, all made up from
some given integer number of fundamental charged particles, the force
between any pair of charges, measured in natural units, is
proportional to e^2 which is proportional to alpha. increase alpha by
5% and the EM force has also increased by 5% (relative to the other
fundamental forces).
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.

sounds like a real physics text since it is McGraw-Hill. i have the
Barrow book for "light" reading.

r b-j
 
S

Steve Underwood

Jan 1, 1970
0
robert said:
i'm getting less tolerant in my middle age. (Jerry, does it get worse
or better as we get older?)

It sounds like you failed at life. Success at life is to observe it, and
become deeply cynical. Its hard not to tolerate anything when you are
sufficiently cynical. :)

Steve
 
G

glen herrmannsfeldt

Jan 1, 1970
0
dunno what that means.

A large calculation can be made up of many small steps.
If one computes intermediate results on a calculator,
one can attach the appropriate units onto the intermediate
result from the calculator. The numbers go through the
calculator, the units go around and are attached appropriately
onto the result. Usually the steps will be small enough not
to lose track of the units.



(snip)
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.

I used to know of a computer based physics teaching system
using PLATO: http://en.wikipedia.org/wiki/PLATO
For physics problems the user was expected to enter an answer
with units. As I understand it, each unit was given a numerical
value, and the resulting expression was evaluated and expected
to be close to the correct answer. As an example, m (for meter)
might be 123.45, cm would then be 1.2345, in (inch) 3.13563, etc.
Once for a problem expecting velocity units entered erg**0.5 g**-0.5
(I forget the actual exponential operator), and the answer was
judged correct.
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.

It is ChE that uses pound for mass, and adds another constant into the
equations to make them consistent. I used to work in a lab with ChE
people, with many experiments related to absorption or emission spectra.
I once did an emission spectrum in BTU/pound mole, a unit that only
ChE would use. (A pound mole, similar to the more common gram mole,
is the amount of some substance such that its mass in pounds equals
its molecular weight in Daltons.)

-- glen
 
J

Jerry Avins

Jan 1, 1970
0
glen said:
Robert said:
I suppose so, but my first thought is to convert to meters/minute**2

Miles/hour/second is a pretty common unit. Of course, an hour-second
equals a square minute. :)

Jerry
 
J

Jerry Avins

Jan 1, 1970
0
Phil said:
glen said:
Nah, furlong-seconds per cubic fortnight.

A furlong per fortnight is a velocity. That leaves second per square
fortnight (a frequency) if we factor it out. Frequency of what?

Microlightyear per century is another interesting speed.

Jerry
 
J

Jonathan Kirwan

Jan 1, 1970
0
Hello Jerry et al,
I can assure you that my students are forced to show units all of the
way through their physics calculations. I also show the utility of
expressing things as ratios, so the units cancel out. Such as given a
pendulum clock that runs normally on Earth, how much faster/slower
will it be on the Moon? (g_moon approx 1/6 that of the Earth's)

A venerable approach. Galileo used it, in fact. Algebra, at the
time, he felt was not yet rigorously founded while ratios had been for
quite some time.

Jon
 
J

Jerry Avins

Jan 1, 1970
0
glen said:
Jerry Avins wrote:

(snip)


One of my least favorite units is the one power companies like
to use to describe usage: Kilowatt hours/day. Kilowatt seems
like a fine power unit, without an extra factor of 24.

I think you mean that a KW-hour is a unit of energy. It is also
(indirectly) a unit of revenue. Hence the usage.

Jerry
 
J

Jerry Avins

Jan 1, 1970
0
Jerry said:
I think you mean that a KW-hour is a unit of energy.

No, you meant that KW-hr/day is a unit of power. My mistake.

...

Jerry
 
G

glen herrmannsfeldt

Jan 1, 1970
0
Jonathan Kirwan wrote:
(snip)
A venerable approach. Galileo used it, in fact. Algebra, at the
time, he felt was not yet rigorously founded while ratios had been for
quite some time.

I had known that Galileo's first experiments with rolling balls
were an attempt to slow down the fall of gravity, and allow him
to understand the effect. I hadn't known why he decided to do it
that way, without the algebra to show what the result would mean.

-- glen
 
G

glen herrmannsfeldt

Jan 1, 1970
0
Jerry Avins wrote:

(snip)
Miles/hour/second is a pretty common unit. Of course, an hour-second
equals a square minute. :)

One of my least favorite units is the one power companies like
to use to describe usage: Kilowatt hours/day. Kilowatt seems
like a fine power unit, without an extra factor of 24.

-- glen
 
T

Tim Wescott

Jan 1, 1970
0
glen said:
Jerry Avins wrote:

(snip)


One of my least favorite units is the one power companies like
to use to describe usage: Kilowatt hours/day. Kilowatt seems
like a fine power unit, without an extra factor of 24.

-- glen
Unfortunate for technical people, but good for 'normal' customers.

Everyone knows what a day is, and we all get billed in KW-hours. I
doubt that very many people could tell you that a KW-hour/day is a hair
under 42 watts, and even fewer could do it in a blink without a calculator.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google? See http://cfaj.freeshell.org/google/

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
 
J

Jonathan Kirwan

Jan 1, 1970
0
Jonathan Kirwan wrote:
(snip)


I had known that Galileo's first experiments with rolling balls
were an attempt to slow down the fall of gravity, and allow him
to understand the effect. I hadn't known why he decided to do it
that way, without the algebra to show what the result would mean.

Galileo probably began considering motion like this at least as early
as 1586, I think, having written a dialogue on problems of motion that
year. He must have considered inclined plane experiments as early as
1591, since he added them to his De Motu that year. But I seem to
recall that his immersion into building them would have been around
1601-1602.

The details about his thinking, as well as copies of some of his
folios, can be found in Sillman Drake's "Galileo: Pioneer Scientist."
I recommend it.

Jon
 
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