Straightening Out Angular Momentum

niels[pictured: Niels Bohr]

Angular Momentum and how it plays into atomic theory has long confused, and thereby, intrigued me.  Reading Manjit Kumar’s Quantum:  Einstein, Bohr, And The Great Debate About About The Nature Of Reality has somewhat illuminated the subject for me.

If you’re like me and enjoy reading about atomic physics (and, really, who doesn’t?), then you’ll often come across something called “angular momentum.”  The idea of Angular Momentum is, in fact, fundamental to understanding the atom—at least as far as anyone can be said to understand the atom.

Plain ol’  “momentum” of an object is simply its:  mass x velocity or m * v.  Momentum is a measure of what I call “pow.”  For example, a block of wood travelling fast, will have more “pow”– feel worse– when it hits your face than a feather travelling slowly.

For an object moving in a circular motion, we often speak of its “angular” momentum. To get the Angular Momentum of an object, you merely multiply the object’s momentum (m * v) times its distance from the circle’s center (also known as its radius, “r”).  You can write that in equation form as:

Ang. Mom. = m * v * r

Angular Momentum is a useful value in physics because it allows you to consider the different momenta of different points on a sphere.  A point on the outside of the sphere will have more momentum– *pow*– than a point of the same mass halfway into the sphere.   Angular Momentum also is convenient because it unites three important descriptors of reality:  mass, velocity, and distance.

One of the earliest instances of Angular Momentum creeping into Quantum Physics was when Bohr was having trouble understanding why, in his model of the atom (which was an improvement over Rutherford’s tiny solar system model) his electrons could only exist in certain orbits.

Bohr’s work had led him to believe that there must be precise gaps between the orbits of electrons. The size of these gaps between “orbits” (or more accurately, between electron “energy levels”) helped produce the colored lines that elements emit when irradiated with energy. These colored lines, aka “spectral lines,” are like the fingerprints of each element; each element producing a different set.

[Aside: One thing I discovered while reading Kumar’s book was that Spectral Lines play a huge, huge role in the development of Quantum Physics. Time and again, discoveries about the Spectral Lines of elements would blow holes in old theories and would force theorists to come up with a new theory that would save their old ones. Maybe I’ll do a post on Spectral Lines soon, but I dunno; like I feared: I’m getting bogged down in Quantum Theory here.]

But back to Angular Momentum and the atom. Nicholson came up with a solution to Bohr’s problem about the orbital gaps.  Nicholson said that electrons could only exist in orbits where (and don’t be intimidated by this; just skim over it for now):

the Angular Momentum of an Electron
[the electron’s Energy divided by its Frequency Of Orbits]
divided by
[two times (its orbit’s Circumference divided by its Diameter)]

The above equation can be drastically simplified (I told you not to be intimidated)…

The “electron’s Energy divided by its Frequency” is a constant called “Planck’s Constant” and is abbreviated simply as “h.” And “Circumference divided by Diameter” is what “pi” is—which we all know from school, and which we all of course remember is– for every circle in the universe!– approximately equal to 3.14.

h , being a “constant” like pi, keeps an eternally constant value; the value of Planck’s Constant (h) is so tiny, so miniscule, that it is hardly worth reporting here, but it is something like a decimal point followed by several trillion zeroes, and then the number 6626 at the end.

So the above (cumbersome) equation for the Angular Momentum of an electron looks much more tame when we write it this way:

Ang. Mom. = h / 2*pi

Eazy-peezy.  Now since, on the right hand side:  h, the number 2, and pi are all unchanging numbers… and since Angular Momentum, as we’ve said, equals:  m * v * r, then what Nicholson was saying was that:

An electron’s mass (m) times its velocity (v) times its radius (r) always equals this certain tiny, tiny number on the right hand side.

There are many energy levels for electrons for which this will be true.  The trick for Bohr was finding levels where it was NOT true.  These would be the gaps between energy levels that Bohr needed to make his atom work.  And these orbital gaps helped explained why, when an element is radiated with energy, a certain pattern of spectral lines will shine from the element.

Nicholson’s “gap rule” will help define what will eventually become the four “quantum numbers” that together define exactly the unique orbital associated with every electron in an atom.  But that’s a story for another day.  And perhaps another blogger (whew!)


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