different Planck units

[Martin]

...

 

Alright. If you don't mind then:

hbar is a constant it seems. What exactly is the number' date=' and why was that number chosen? Was it modeled after a physical observation or was it provided through mathematical equations? How does it relate to this:

 

How can they all equal 1?[/quote']

 

most important thing to know is what is Planck's constant h, and then

hbar is easy.

 

hbar is just h/2pi

It has been found more convenient to use in most cases if divided by 2pi.

so hbar appears all thru quantum mechanics.

The official symbol is an h with a bar thru the trunk and in latex

you just write \hbar. Press quote to see the LaTex expression

 

[math]\hbar[/math]

[math]\hbar = \frac{h}{2 \pi}[/math]

 

but before getting confused, get a grasp of simple old h.

 

it is the ratio of a photon's energy to its frequency

by a miracle of nature the two are proportional

(this was Einst. insight in 1905 for which he got Nobel)

the more high frequency the photon buzzes with, the more energy it carries

h is the ratio -----its energy divided by its frequency.

 

THE NUMBER YOU ASSOCIATE WITH h, depends totally on what units you use to measure energy and to measure frequency.

 

so by adjusting your energy unit and your time unit you can make h have any numerical value you want.

 

but as a physical quantity (a product of energy and time) it is constant thru out the universe. As a quantity of energy-multipliedby-time it is a universal constant. but the number we assign to that quantity is to some extent arbitrary.

 

the metric system is beset with ugly numbers, in the metric units of energy (joule) and time (second) the number for h is this:

 

 

[math]h = 6.626068... \times 10^{-34} \text{ joule second}[/math]

 

dividing by 2pi we get

 

[math]\hbar = 1.05457... \times 10^{-34} \text{ joule second}[/math]

 

By taking a small enough time unit instead of the second we can make the number one. But with a small time unit we need to take a small distance unit if we simultaneously want to make the speed of light equal one. And so on. The conventional Planck units are those which have been adjusted or fine tuned to the basic universal proportions so that they simultaneously make hbar and c and G have numerical value one.

[fuhrerkeebs]

How can they all equal 1?

 

They can all equal 1 because you can choose a unit system in which time and length and stuff in SI units is a combination of the constants, and you can create a new set of units by setting the combination of constants in one system of units to 1 in another system of units, and it then follows that the constants equal 1 in the new system of units.

 

Yes indeed. I didn't expect to learn all of this freshman year (though that would be nice). I am currently in AP physics. If you don't know what AP is, they're classes designed by the same people who do the SAT's and they're intended to be like an intro level college course. So, though this class has just begun, it and previous physics work have taught me basic kinematics. Some charge stuff with Coulombs and Volts. Then there was power and watts and joules. We recently started in on deriving graphs of velocity and displacement from not-constant accelerations.

 

Yeah, I'm taking AP physics too, and, at least at my school, they don't spend more than a minute on a new system of units. I personally like it though, because I think it's a waste of time to spend more than a minute on something as easy and trivial as a new system of units. (It doesn't really matter though, because the class is crap anyways...)

[jordan]

Then by setting hbar equal to one, you wont effect your calculations except for the units, right?

 

And when it equals one, what are the units?

[Martin]

Then by setting hbar equal to one' date=' you wont effect your calculations except for the units, right?

 

And when it equals one, what are the units?[/quote']

 

jordan, just setting hbar equal one only uses up a part of your freedom.

 

you still get to decide what number you want to make the speed of light

and what number you want to make the universal G constant have.

 

when defining the basic units of any system you get three wishes and

that determines the system

 

If you were defining SI metric units you could say for instance

 

"A. I want the speed of light to be 299792458 length unit per time unit."

 

"B. I want hbar to be 1.0545 x 10^-34 energy unit per frequency unit."

or another way to say that is

"B. I want hbar to be 1.0545 x 10^-34 energy unit time unit."

and still even that would not completely nail down the sizes of the units, but

then you could say

 

"C. I want G to be 6.673 x 10^-11 cubic length unit per sq. time unit per mass unit."

 

and that would finally force the units to be the familiar sizes of meter and second and kilogram.

 

But we arent concerned with the metric system primarily. You could go thru the same business with different numbers tho. You could say

 

A. I want the speed of light to be 1 length unit per time unit.

B. I want hbar to be 1 energy unit per frequency unit.

C. I want G to be 1 cubic length per sq. time per mass unit.

 

and that would force the units to be what they give at the NIST website for Planck mass, length, time.

 

the point is that it is like solving simultaneous equations in algebra.

once you have specified those three things you can SOLVE for what the time unit has to be and the length unit and mass units have to be, explicitly, in terms of c, hbar, G. there are formulas for what the units of a system have to be in terms of what numerical values it gives to the basic constants.

 

well this post is a bit too complicated for the question. I will get back to talking about sunlight as an example

[Martin]

The temperature of the surface of the sun is T= 20 thousand Fahrenhalf.

 

boltzmann k tells us that a typical photon energy in the light coming off the surface

is kT

 

k = 10^-22 jot per degree

 

and we have this eejay unit of about a quarter of a conventional electron volt, and eejay = 10^-18 jot, so

 

k = 10^-4 eQ per degree

 

So this means the characteristic photon energy at T = 20,000 degrees is

2 eQ

 

Now in any system of units, even the metric system, there is this pure number involving pi and the sum of an infinite series which happens to be 2.7 that relates the typical energy kT with the AVERAGE photon energy.

 

average photon energy = 2.7 kT

 

So for us, and the sun, we multiply 2 times 2.7 and find that the average photon form the sun is carrying 5.4 eQ. We learn this from the temperature.

 

that is in the near infrared.

 

the visible is in the highend tail of the distribution from 7 eQ (red) to 13 eQ(violet).

 

the energy in sunlight is weighted towards the high end because those that have more than average are a bigger piece of the energy, but if you just count photons, the average one in sunlight has 5.4 eQ

and that comes right out of knowing that the temperature of the surface is 20,000 Fahrenhalf.

 

What is the frequency of that average sunlight photon?

 

hbar = 10^-32 jot count = 10^-14 eQ count

 

I just have to divide the energy (5.4 eQ) by hbar:

 

sunlight photon's average (angular) frequency = 5.4 x 10¹⁴ per count.

[Martin]

Again as warmup let's look at the electron's mass in Gbar Planck units.

 

our voltage unit QUARTERVOLT is about 1/4 of conventional volt so the

energy unit eQ is about 1/4 of the conventional eV, so

if you know the mass-energy of the electron in conventional

terms it should be no suprise that the electron mass-energy is

about 2.1 million eQ, more exactly 2,098,000 eQ.

 

the Planck units themselves tend to be somewhat more extreme

(they are the ones making the constants equal one) but we can convert

this to Planck units too.

Planck energy unit is exactly =10¹⁰ jot = 10²⁸ eQ

 

So the electron energy of 2.1E6 eQ must be the same energy as 2.1E-22 Gbar Planck energy units, and then because c is a unit, the electron mass must be the same number of Gbar Planck mass units: electron mass is 2.1 E-22 Gbar mass units.

 

But it will probably be just as convenient if we simply remember 2 million eQ. Or 2.1, or 2.098 depending on how much precision one wants.

-------

 

Now there is this great energy called "Hartree energy" which is twice what it takes to ionize a hydrogen atom from its ground state. And the hartree is ridiculously easy to calculate from this number 2 million eQ.

 

You just have to multiply by (1/137)²

 

so we take 2,098,000 and divide by 137² and we get 112.

That is the Hartree unit: 112 eQ.

Then, dividing by 2, we get 56 eQ. It takes 56 of our voltage units to ionize a hydrogen atom. you will see that's right if you divide by 4 to get it in conventional volts.

[Martin]

As you point out the nice thing about natural units is that you simplify equations. But presumably one cannot simplify all equations at once' date=' so it is best to simplify the ones which are most fundamental. So, for example, you still have c=1, which is good.

...

...

 

The gravitational constant is [math']G = 6.707 \times 10^{-39} \hbar c ({\rm GeV}/c^2)^{-2} = (1.221 \times 10^{19} {\rm GeV})^{-2}[/math]

where I have set

[math]\hbar c=1[/math].

 

This defines the Planck length and Planck energy (so G is the planck energy to the power of -2).

 

...

...

 

In fact, our current energy unit (the eV) is nice because it is fundamental to two of the forces, electromagnetism (obviously) but also QCD, where conincidentally 1GeV is the hadronisation scale.

 

Once we have a theory of everything, we may then have one force with one characteristic distance scale, but until them, isn't any choice we make rather arbitrary?

 

I want to respond to part of Severian's post (for the full post which makes a very interesting suggestion, look back on previous page to post #8 in this thread.)

 

the key thing to notice here is that nowadays Particle Physicists (a very numerous and influential part of the community) are essentially using a system of units where everything is referred to the GeV and one sets hbar = c = 1.

 

HEADSUP: the G in GeV is Giga. the particle physicist's GeV unit is an eevee scaled up by a billion

 

Since hbar = c = 1, the system is only slightly different from Planck units. It differs only in the using a different energy unit. I expect we should get accustomed to the GeV system since it is currently fashionable.

 

In the future one may adopt a different standard energy than GeV.

In that case it will be very simple to convert everything, because lengths, areas, volumes, masses, densities, energies, will all be expressed in GeV and in powers of GeV, so one will just plug in one conversion factor.

 

We should have a tutorial or an explanation posted as to how one expresses all these traditionally non-energy quantities in terms of GeV.

this will help get me and anyone else listening up to speed with the modern fashion of HEP units.

[Martin]

the point is that natural conversion factors like c^2, and hbar, and hbar c, give standard conversions between energy and other things.

 

So if you have a standard energy quantity GeV

then you get standard quantities from that for everything else.

 

Like c^2 relates energy to mass

So one can measure mass in GeV,

or if one wants to be very correct one can write the unit (GeV/c²)

 

And hbar relates energy to (angular) frequency so one can measure

frequency in GeV, or if finicky in the unit (GeV/hbar)

 

And the reciprocal of frequency is time so one can measure time

in (GeV)^-1, or to be extra correct in terms (GeV/hbar)^-1

 

And c relates time and distance so one can measure distance in units of

(GeV)^-1 too, or else if one wishes in (GeV/hbarc)^-1

 

area

 

[math](\frac{GeV}{\hbar c})^{-2}[/math]

 

volume

 

[math](\frac{GeV}{\hbar c})^{-3}[/math]

 

"per volume" (reciprocal volume)

 

[math](\frac{GeV}{\hbar c})^3[/math]

 

energy per volume (energy density)

 

[math]GeV(\frac{GeV}{\hbar c})^3[/math]

 

and this is getting cluttered so let's just say hbar c=1 and then

energy per volume is just

 

[math](GeV)^4[/math]

 

I was just reading Sean Carroll's blog, he is a prominent world-class cosmologist as well as an entertaining writer. And in one of his pieces about the cosmological constant ("dark energy density" hot topic) he happens to give the CC in terms of an energy density.

 

what units do you think he expresses dark energy density?

I would suppose joules per cubic meter, right?

well that is the correct metric unit, right?

We all use correct metric units of course.

And it is very easy to express that way---dark energy density

is 0.6 joules per cubic kilometer.

But nobody in the major league says it that way!

They express it most often in terms of, you guessed it,

[math](GeV)^4[/math]

well actually they leave out the Giga for this one, because the energy density is so small in this caese, and simply write it in terms of

[math](eV)^4[/math]

 

So this is the contemporary HEP-talk, this the dominant vernacular. We had better get used to it. the nice thing is that it sets hbar = c = 1

so there is only one arbitrary choice of a quantity---the energy GeV.

 

that is roughly the massenergy of a proton, which is nice and convenient for back-of-envelope reckoning

and all you would have to do to get Planck units out of HEP vernacular units would be to plug in Planck energy instead of GeV

 

its cool and Severian also suggests that in future there may be some other standard energy to plug in, we can get to that later

[Severian]

its cool and Severian also suggests that in future there may be some other standard energy to plug in, we can get to that later

 

There has to be one somewhere, otherwise we would be unable to have 'energy'. Any statement about energy which science makes is always with reference to another energy, so at least one fundamental energy scale must be in (or be generated by) the final theory.

[Martin]

echo of Plato in the last post, and Plato might be right

 

since particle physicists (and others influenced by them) like GeV units

let us see what the mass, length, duration etc. actually are

the NIST website has this in a section called

codata recommended values of energy equivalents

http://physics.nist.gov/cuu/Constants/

http://physics.nist.gov/cuu/Constants/energy.html

click on the PDF table, it is easier to see what is happening

I will round off carelessly.

 

GeV/c^2 mass = 1.782E-27 kg

 

hc/GeV length = 1.2398E-15 meter (divide this by 2pi)

 

h/GeV duration = 4.13567E-24 second (divide this by 2pi)

 

the benighted NIST uses Planck's original h, instead of hbar, for their

conversions. so to get the length and time equivalents one has to

divide their numbers sometimes by 2pi.

 

have to go, back later. BTW I like Planck Gbar better than GeV but

similar kettle of fish and HEPs do seem to like Gbar.

[Martin]

Like any rightthinking person, I prefer using hbar (see also Severian's posts) so I will correct the NIST figures accordingly.

 

http://physics.nist.gov/cuu/Constants/energy.html

 

 

mass

[math]\frac{GeV}{c^2} = 1.782 \times 10^{-27} kg[/math]

 

length

[math]\frac{\hbar c}{GeV} = 1.9732 \times 10^{-16} meter [/math]

 

time

[math]\frac{\hbar}{GeV} = 6.582 \times 10^{-25} second [/math]

[Martin]

Now just for comparison we have this alternative system of units running in the background that is Gbar Planck based and has QUARTERVOLT as its unit of voltage, so I guess we can say a billion eequeue is a GeQ (hoping this causes no confusion: there are only so many letters in the laughabet.)

 

Our Q voltage is about one quarter of a conventional volt and our eQ is about one quarter of a conventional eevee. So we can do something analogous to the previous post. Remember we have a pound-size mass unit, and handbreath length unit, and a time unit which is 222 counts to the minute.

 

 

mass

[math]\frac{GeQ}{c^2} = 10^{-27} pounds[/math]

 

length

[math]\frac{\hbar c}{GeQ} = 10^{-14} hand [/math]

 

time

[math]\frac{\hbar}{GeQ} = 10^{-23} count [/math]

 

So we can imitate HEP-style units and use a billion eequeue instead of a billion eevee, and say everything in terms of GeQ. The main difference is that the equivalences with humanscale units are clean powers of ten. Because all the conversions are powers of ten in this kind of system.

[Martin]

calling out unit voltage "jolt" and abbreviating J does risk confusion with joule.

and our voltage unit is right around a QUARTER of a conventional volt. so I want to tinker around with these units and call the voltage unit a "Q" or a "quartervolt"

 

I guess the electronquartervolt is written eQ and pronounced eequeue.

 

Remember we have pound-size mass and handbreath and a time unit which is 222 count to the minute.

And we have this Fahrenhalf degree which is right around half of a conventional Fahrenheit.

So I am going to express these humanscale units in terms of this eQ energy unit, or a billion eQ.

 

 

mass

[math]\text{1 pound} = 10^{27} \frac{GeQ}{c^2}[/math]

 

length

[math] \text{ 1 hand } = 10^{14} \frac{\hbar c}{GeQ} [/math]

 

time

[math] \text{ 1 count } = 10^{23} \frac{\hbar}{GeQ} [/math]

 

temperature

[math] \text{ 1 degree Fahrenhalf} = 10^{-4} \frac{eQ}{k} = 10^{-13} \frac{GeQ}{k} [/math]

[Martin]

To pick up loose ends of this development, let's review post #17 of this thread, introduce the term "quartervolt" for the voltage unit, and see how things look.

 

I started constructing such a human-scale system in another thread. Am now referring to 8pi G as "Gbar" by analogy with hbar.

 

Still trying out names for some of the units----in this post will use mark for the unit force (which is about half a newton, or a couple of conventional ounces)

 

and dram for the unit charge, which is that of 10¹⁸ electrons

 

Gbar units gradually taking shape. I added electric units (charge current and voltage) and gave the energy unit a name. The names are still placeholders, in case better name ideas show up. the system is essentially a version of Planck units using Gbar = 8pi G instead of the newtonian G constant as Planck originally did.

e is the elementary charge. Our charge unit is exactly 10¹⁸ e, the charge on a billion billion electrons. the unit of energy (called a jot) is about 1/100 of a calorie.

 

The units are defined by assigning these values to the constants

 

Gbar = 10^-7 hand³count^-2pound^-1

c = 10⁹ hand count^-1

hbar = 10^-32 jot count

k = 10^-22 jot degree^-1

e = 10^-18dram (charge unit)

 

[consider the word "jot" as a placeholder, it is this word for the energy unit that I'd like suggestions for, in case anyone can think of a replacement.

that is true for pretty much all these names]

 

In conventional metric,

c = 2.99792458 x 10⁸ meter second^-1

and the other constants are even messier, so there's some attraction

to Gbar units getting simple powers of ten for the constants.

 

with the above powers of ten stipulated then the time unit (count) comes out 222 to the minute.

the temperature degree turns out to be about half a Fahrenheit

the mass unit pound comes out to 434 grams, roughly one pound.

the length unit hand is 8.09 centimeters (around 3 and 1/4 inches)

the force unit mark is 0.4816 newton, a couple of ounces of force.

the energy unit jot works out to around 0.04 joule or 1/100 of a calorie.

the charge unit dram since it is 10¹⁸ electron's worth, has a metric equivalent of 0.1602 Coulomb

the unit current is about 2/3 of a conventional ampere and

the unit voltage quartervolt is about 1/4 of a conventional volt.

the power unit is approx. 1/6 watt.

 

I dont know if the names (like "jot" for the unit energy) would work but I followed Severian's suggestion and tried defining the basic units in terms of a basic voltage and the resulting energy unit: electronquartervolt = eQ.

 

And it looks to me as if the humanscale energy unit, call it a jot, is exactly a billion GeQ.

[Phi for All]

The septenary posts added by TurricaN can be found split off here.

[Martin]

The septenary posts added by TurricaN can be found split off here.

 

 

thanks Phi!

 

turricaN and I both should be grateful---it was not a happy union.

whew!

 

Now I can tell him my reaction to his idea without getting my own thread off-topic.