Thursday, April 05, 2012

Let's fix the value of Planck's constant

Carl Brannen has pointed out the following simple preprint to me:
Quantum Theory without Planck's Constant
by John Ralston of the University of Kansas.

The author shows that if we get rid of the International Prototype Kilogram i.e. if we fix the value of the Planck constant – at least I will generalize this proposal by saying that we will set it to a known constant – we may improve the accuracy of common quantities such as the electron's mass or the electron's charge by two orders of magnitude or so!

Of course, adult theoretical physicists commonly use units with \(\hbar=c=k=1\) but they rarely care about the exact numerical values of less universal constants so many of them are not familiar with the error margins and their origin. It's a very interesting collection of facts.

First, let me review the units in the International System of Units, SI. The elementary units are 1 kilogram, 1 meter, 1 second, 1 ampere, 1 kelvin, 1 candela, and 1 mole. There are seven such basic units and all other units are constructed as products of their powers.

Well, let's eliminate a few of them. I will start from the end.

Mole: chemistry as a vegetable shop

One mole is just a macroscopic amount of material that always contains the same number of atoms or the same number of molecules. We only use the mole because it's convenient for large numbers. However, it's clearly more fundamental to count the actual number of atoms. One mole is equal to\[

N_A = 6.02214179(30)\times 10^{23}~{\rm mol}^{-1}

\] atoms or molecules. The number above is the well-known Avogadro's number. It's not known exactly because the actual definition of one mole says that 1 mole of carbon-12 should have the mass 0.012 kg and one doesn't know the mass of the carbon-12 atom exactly, either. Nevertheless, those who use the unit "1 mole" at all can rarely need the accuracy sketched above. In particle or atomic physics, one would count the actual number of atoms, so the mole and its inaccuracy is therefore irrelevant for physics.

One mole and three motýls, as selected by one Motl.

You might decide that one mole will be redefined so that \(N_A\) is equal to a particular constant number, too. It wouldn't make much difference for those who use the unit, one mole, because high precision isn't the main business of these folks.

Lumo: subjectivity of eyes

One candela is a unit of luminous intensity, meant to roughly agree with one candle (but I should also tell you how large the candle is and what is the material).

It's a bit bizarre concept because luminous intensity is the power emitted by a light source which is however weighted by the standard luminosity function. (Lumo is "light" in Esperanto.) This function describes the sensitivity of the human eye. So be sure that it's not totally accurate or constant. Consequently, 1 candela is a messy unit. You don't expect any amazing high precision here. If you want precision, you need to talk about the actual power in watts and declare exactly how you're treating individual frequencies.

Standard luminosity functions; guess which of them is used to define one candela

Again, no fundamental physicist would talk about 1 candela too often. It's something linked to the human eye, biophysics, medicine, perhaps production of digital cameras.

Kelvin: switching from water to Boltzmann's constant

One kelvin is a unit of temperature difference – or the absolute temperature if measured relatively to the absolute zero. It is still defined so that the triple point of water sits at 273.16 kelvins. The linearity is guaranteed e.g. by demanding that ideal gas at a fixed pressure has a volume that is proportional to the absolute temperature.

Once again, this unit is useful only because one needs a reasonable number of kelvins for macroscopic situations. If you study some detailed microscopic physics, you surely realize that a quadratic degree of freedom at temperature \(T\) carries the average energy \(kT/2\). Equivalently, the entropy is defined as \(k\cdot \log(W)\) where \(W\) is the number of macroscopically indistinguishable microstates (or the volume of the macroscopically indistinguishable region of the phase space in some reasonable, e.g. non-reduced Planck, units). In both cases, \(k\) is Boltzmann'c constant:\[

k=1.3806488(13)\times 10^{-23}~{\rm J/K}

\] If you haven't understood the notation yet, \(2567(45)\) means \(2567\pm 45\) where the error margin is as large as the corrections you get by modifying the same number of the last significant figures.

Again, the number is inaccurate because one kelvin is still being linked to some properties of an arbitrary, messy, and random chemical compound, namely water. A committee proposes to redefine one kelvin so that the value of Boltzmann's constant would be\[

k = 1.3806505\times 10^{-23}~{\rm J/K}

\] exactly and I support the committee. Note that this proposed fixed value of Boltzmann's constant differs by more than 1 sigma from the mean value of the current most accurate measurement but it's not a big deal.

Ampere: SI is as fixed as CGS

The SI system uses an independent unit for electromagnetic quantities, one ampere. Various other units such as coulomb, henry, weber etc. may be derived from one ampere. There is nothing to adjust about one ampere. One ampere is chosen so that the vacuum permeability is\[

\mu_0 = 4\pi \times 10^{-7}~{\rm N} / {\rm A}^2

\] exactly. The unit of the permeability is one newton per squared ampere (force between wires with some current); I hope that I simplified them correctly. Up to the \(4\pi\) factor that emerged from the "rationalization" (adding the natural \(1/4\pi\) in Coulomb's law) and up to some powers of ten, the electric SI units were chosen to agree with some of the Gauss's CGS units. Already in the 19th century, Gauss was using natural units in which either \(\epsilon_0\) or \(\mu_0\) or some product of their powers was set equal to one. So there is nothing to adjust here and the electric and magnetic units don't add any error margins to our calculations.

Note that a precisely known \(\mu_0\) also means that you know \(\epsilon_0\), the vacuum permittivity, assuming that you know the speed of light, too. It's because \(\epsilon_0\mu_0 c^2=1\).

We're finally approaching the core units of the International System of Units, namely 1 meter, 1 kilogram, and 1 second. They measure the basic objects we can hold in our hands, stretch them in between our hands, or measure with our clocks. Because of the triplet "meter, kilogram, second", the core (or predecessor) of the SI system is also known as the MKS system.

One second

One second has been a very precise units for centuries. It was defined so that 1 average solar day had 86,400 seconds; this number of seconds may be divided to 24 hours per 60 minutes per 60 seconds. The motion of the Earth is rather regular so the accuracy was good enough.

As I mentioned in the recent text about leap seconds, we can already measure time more accurately than what the Solar System can achieve. In particular, atomic clocks measure time much more accurately than that which is why, in 1967, one second was redefined to be
the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
We became independent of the spinning Earth.

An obsolete model of clocks

Note that one rotation of the animated GIF above should take 86,400 seconds. If it's faster, you may buy a slower computer to fix the problem. ;-) By having decoupled ourselves from the spinning planet, we introduced a new problem with the leap second. But we gained many more advantages; we have acquired a decent unit that doesn't depend on Jupiter's mood, Earth's eccentricity, and earthquakes, among tons of other factors.

A modern replacement for the spinning potatoid

We're getting to the last two units: one meter and one kilogram.

One meter

Length is arguably the most elementary dimensionful quantity; Euclidean geometry developed by ancient Greeks (which is all about distances, areas, and volumes) may be viewed as the oldest branch of physics. Some people have measured the distances using inches.

When I learned about one inch, today defined as 0.0254 meters (exactly), which we call one thumb ("palec") in Czech (and most other languages in the world), just to be sure if the details are different in English (where 1 inch is linguistically, via Latin, linked to 1/12 of a foot), I couldn't possibly believe how it was used because the width of my thumb (the finger separated from the other four fingers) was something like half an inch. So how could they have used the actual inch for high-precision measurements, I asked? Of course, the subtlety I neglected was that I was six and my inch was predetermined to grow a little bit more. ;-) Today, with the accuracy of 10% or so, I could use my inch to measure the distance in inches (by placing the greater width of the left inch after the right one, and so on).

Good that you got here. An off-topic video that explains the LHC: "A green politician is on his tour to CERN."

In continental Europe, we would prefer a meter. At some moment, people decided that the circumference of the Earth around a meridian i.e. through the poles should be exactly 40,000,000 meters. That uniquely fixes the value of one meter. Lisa Randall has convinced me that around the same time, some people even considered another – a much less accurate, given our current definition of one meter – definition of one meter: the periodicity of a pendulum of length 1 meter (with all the weight at the bottom) should be exactly two seconds.

So one meter is much closer to the meridian definition. However, it was much less accurate because people were able to measure the size of our planet much less accurately, relatively speaking, than the time between two noons. So they invented a prototype meter at some point. And when the atomic physics measurements got better, they redefined one meter as a multiple of some wavelengths corresponding to a particular transition between excited states of atoms.

A prototype of one meter that was used for a while. Note the X-shaped cross section.

It was a different atom than the caesium isotope used for the definition of one second. Different atoms are not too fundamental. At some point, when the speed of light was known to be 299,792,458 plus minus 1 meter per second (i.e. when the error was 3 parts per billion or so), people wisely decided that such a fundamental and universal constant, the speed of light, should be known exactly. So in the early 1980s, one meter was redefined so that the number is actually accurate and the speed of light in the vacuum is\[

c=299,792,458~{\rm m}/{\rm s}

\] Because it's a known constant now, one meter is also linked to one second and defined using the caesium atom's spectral line. The speed of light doesn't introduce any error to our formulae, either.

Finally, we're led to the main topic of this article that you've been patiently waiting for.

One kilogram: consecrating Planck and Dirac and distributing platinum to the needy along the way

One kilogram is defined as something not terribly fundamental. You don't want to have it at home. It looks like this:

For security reasons, it's just a computer-generated portrait of the International Prototype Kilogram. Don't try to steal it. It contains 90% of platinum and 10% of iridium. The price of one kilogram of platinum is $50,000 or so; iridium is about 30% cheaper. It's almost exactly one million Czech crowns; if you're looking for a currency to buy, note that all of our coins and banknotes are made of platinum.

It's such a nice object that one pound – a unit of weight used in the countries that deny the international units such as your country ;-) – is defined as 0.45359237 kg, exactly.

We need to measure the amount of potatoes all the time and the French didn't want to lend the prototype of one kilogram to every shop that was selling potatoes. What a pity. The situation was resolved by producing a hundred of copies of this nice kilogram, national copies and others. They still don't lend the national copies to you if you want to buy a kilogram of potatoes but at least, we may learn something about the accuracy of the prototypes:

The original prototype is defined as 1 kilogram even though it may behave in the same messy way as others. And you see that some of the copies have gained more than 50 micrograms i.e. 50 parts per billion. Of course, it's reasonable to expect that something similar (growth or shrinking) is happening to the original prototype of one kilogram, too (even though we call it "one exact kilogram", by definition). It's really insane to define our unit of mass – and every other quantity that has a kilogram in the expansion into a product of powers of basic SI units – by some crazy piece of metal.

In every second, I mentioned a universal constant that should be naturally set equal to one or at least some totally known and fixed constant, to avoid additional errors. Needless to say, the most fundamental constant to discuss when we add one kilogram to the list of units is the Planck constant\[

h &= 6.62606957(29) \times 10^{-34}~{\rm Js}
\frac{h}{2\pi}\equiv \hbar &= 1.054571726(47)\times 10^{-34}~{\rm Js}

\] The slashed constant is the reduced Planck constant, also known as the Dirac constant, and it's usually considered to be the "more rationalized one" by adult physicists in the same sense as the angular frequency is "more fundamental" than the old-fashioned frequency. Strictly speaking, these values should drift as the international prototype kilogram keeps on evaporating or whatever it does to make sure that some of its copies are gaining 50 micrograms per century. The values above are not here forever.

Note that the standard error margin of the figures above is pretty large; it's about 50 parts per billion! It seems that most of it is linked to the unknown, messy, and variable mass of the international prototype kilogram or the difficulties in using this prototype to measure the Planck constant. Note that the difference between the masses of several prototypes was 50 parts per billion, too. To compare, the speed of light was known at the accuracy 3 parts per billion before its value was fixed – 15+ times more accurately.

Now, my proposal is simple: let's define one kilogram so that\[

\frac{h}{2\pi}\equiv \hbar = 1.0545717\times 10^{-34}~{\rm Js}

\] exactly. Within half a sigma, it agrees with the mean value above. We're free to define a kilogram. Dr John Ralston shows that by eliminating the error margin from the Planck constant, we may improve the accuracy with which we know the numerical values of common constants such as the elementary charge and the electron mass by the factor of 60-120. That sounds good, doesn't it?

You may object: there's a problem with this proposal. If this new definition of one kilogram is adopted, the prototype kilograms will become useless. What will we do with them? However, my proposal is able to solve this difficult problem, too. Each copy of the prototype kilogram will be given to the person of the same citizenship who is most responsible for the transition to the new units. He or she will be obliged to store it or sell it. I admit that I will be responsible for the Czech copy #67. Dr Ralston will probably have to take care of one of the American copies.

It's a big sacrifice but I would accept this burden in the name of a rosier and more accurate future of the mankind.

And that's the memo.

P.S.: You may have proposed to fix the value of Newton's constant. I don't want to argue whether gravity is more fundamental than quantum mechanics. However, even if you thought that gravity were more fundamental than quantum mechanics, and you shouldn't because quantum mechanics is completely universal while \(1/G\) is just the coefficient of one term in the effective action standing next to infinitely many terms, fixing the value of \(G\) would be utterly impractical because its standard error margin is a whopping 1/8,300 of its mean value! No several parts per billion here: the constant governing gravity suffers from a 0.1 part per thousand error!

In January 2014, I will write another article on this topic, focusing on the plans to redefine one ampere.

No comments:

Post a Comment