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The number is 137. Not 42.

(but then again, check here.)

The reciprocal of a dimensionless constant in atomic physics is approximately 137.036. By dimensionless, it is meant that this is just a number, unlike a distance in inches, which would be a different number of centimeters. A number of people have wondered why this particular constant is this number instead of being larger or smaller; the question of what is special about 137 is considered fascinating by many.

The constant equal to about 1/137.036 is called the fine-structure constant, because one of the things in which its value is reflected are certain details of the spectrum of light emitted by glowing gases.

Just what is the fine structure constant?

The force between two electrically charged objects varies as the product of their electrical charges, and the inverse square of the distance between them.

Thus, the equation giving the electrostatic force between two charged objects, known as Coulomb's Law, is F = ( q(1) * q(2) / (r^2) ) times a constant. This constant, in the MKS system of units, one of the two systems of metric units traditionally used by physicists, and the basis of the current metric system in official everyday use, SI or Système Internationale, is 1 / 4 * pi * epsilon(0), where epsilon(0) stands for the permittivity of free space. The reason the constant in Coulomb's Law isn't just a simple constant is because the underlying value was originally discovered by people working with radio and transformers and capacitors, and so it works out more simply in their equations.

If the two charged objects we are thinking of happen to be an electron and a proton, then they will attract each other. And the electron has a charge of -e, and the proton has a charge of +e. So the force attracting an electron to a proton is given by (e^2) / (4 * pi * epsilon(0)) times 1/(r^2), where r is the distance between them.

In the hydrogen atom, if we consider it in classical rather than quantum mechanical terms, the electron revolves around the proton in such a way that the centrifugal force produced cancels out the attraction of the proton, just as the moon orbits the Earth.

The acceleration felt by something moving around in a circle is (v^2)/r or (omega^2)*r, where r is the radius of the circle, v the speed of the object along the circle, and omega its angular velocity - omega is v divided by r, by definition, and that's why r changes its position in those two formulas. Omega is measured, therefore, in radians per second, not degrees per second; a radian is a distance of about 57 degrees, and represents the angle subtended by an arc of a circle whose length matches its radius.

Force equals mass times acceleration. So at this point we have the equation:

    2                2
   v                e
m --- = --------------------------
   r                            2
         4 * pi * epsilon(0) * r

However, we aren't terribly interested in m, the mass of an electron. Instead, we're interested in another quantity, the angular momentum of that electron. This happens to be m * v * r, and the convention is to use L to stand for it.

So using L to get rid of m, we get:

L * v = --------------------
         4 * pi * epsilon(0)

What makes angular momentum very interesting has to do with quantum mechanics.

In quantum mechanics, the linear momentum of a particle normally can have any value, but it is impossible to determine exactly what that value is, and exactly where the particle is located, both at the same time. The more accurately we know one, the less accurately we know the other. However, if the particle is confined to a box, and all we know about where the particle is located is that it is in the box, then the fuzziness in our knowledge of its momentum changes to something else: now, the particle can have only certain values of linear momentum that are in each direction integer multiples of a function of the dimensions of the box.

Antimatter was discovered in theory before it was actually detected in the laboratory based on this. If someone tried to find out where a particle was, and looked quickly (with help from assistants) in many places at once, it is possible that, looking in two distant locations one might find the particle in both of them, if one happened to look in each one at so close to the same time that both incidents were isolated from each other by the speed of light.

However, to so quickly find a particle's position to such accuracy, one has to pour so much energy into the search that one has contributed enough energy, at the rate of E = m (c^2), to make the extra particle out of nothing, so energy is conserved!

Angular momentum is different from linear momentum. A particle spinning on its own axis, or one rotating around another particle, is always in a box, and a box of a standard size, since there are 360 degrees in every circle.

One of the consequences of this is that angular momentum is better conserved than other things; it can be observed in ways that don't change it. Hence, if you have two particles known to have equal and opposite angular momenta, and you measure both of them at the same time in widely separated locations, they act as if they can talk to each other faster than the speed of light, leading to the unusual phenomenon demonstrated in the Einstein-Podolsky-Rosen experiment.

Because a rotating electron is in a 360 degree box, its angular momentum must be a multiple of a constant value.

For particles spinning on their own axis, the basic value is half of that, and particles whose spin is an odd multiple of that behave in one way, while particles whose spin is an even multiple of that behave in an opposing fashion. Particles of the first kind are called fermions and follow Fermi-Dirac statistics; particles of the second kind are called bosons and follow Bose-Einstein statistics. In the classical world, the behavior of particles follows Maxwell-Boltzmann statistics. The difference between these is: if you throw a pair of dice, the spots that come up follow Maxwell-Boltzmann statistics. If, instead, you shuffle six cards, numbered from 1 to 6, and deal two, the result follows Fermi-Dirac statistics. If, instead, you draw a domino from a set of 21 dominoes, from which all the dominoes containing blanks have been removed, the two numbers on the domino you pick follow Bose-Einstein statistics.

In any case, our electron in a hydrogen atom is constrained to have an angular momentum which is an integer multiple of h divided by 2 times pi, where h is Planck's constant. This value is often written by physicists as "h-bar", a lowercase h with a sloping stroke through the top of it.

Let us think of a hydrogen atom in its ground state. Its angular momentum will just be h/2*pi. So we now have the equation:

   h                     e
-------- * v = --------------------
 2 * pi         4 * pi * epsilon(0)

On the basis of the three physical constants h, e, and epsilon(0), we can calculate the velocity of the electron in orbit in a hydrogen atom.

If we take the ratio of that velocity to c, the speed of light, we get a dimensionless value. Thus:

 v               e
--- = ------------------------
 c     2 * h * c * epsilon(0)

and the value on the right-hand side of this equation is precisely the fine-structure constant. So a simple way to understand what it is is that it is what the ratio of the speed of the electron orbiting the nucleus of a hydrogen atom to the speed of light would normally be. Although electrons don't really move in neat circular orbits in atoms, this number still gives an indication of the magnitude of certain relativistic effects on the behavior of atoms.

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