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Name: Anu
Status: student
Grade: N/A
Location: ID
Country: N/A
Date: 2/21/2005


Question:
Please explain the origin of quantum numbers.


Replies:
The behavior of atoms, molecules, and sub-atomic particles are not correctly described by classical Newtonian mechanics, like our everyday experiences are. Even the motion of planets, stars and galaxies are correctly described by Newtonian mechanics -- but not atoms, molecules, and other sub-atomic particles. Their behavior is correctly described by a different mechanics called quantum mechanics. Instead of the usual equations of classical Newtonian mechanics, another class of equations apply. These "new" types of equations collectively are referred to as "Schroedinger's equation(s)".

The algebraic form of these Schrdinger equations encountered in quantum mechanics are very similar to the classical equations of waves and wave-like motion. As a result they are sometimes referred to as "wave equations". It turns out that not only are the atomic, molecular, and sub-atomic world described by "wave equations", but these particles behave like waves. This is not intuitive, so it is not "fair" to ask WHY, they just do!!

In classical waves (think of strings of a violin or the pipes of an organ) there are certain conditions where standing waves are formed. Specifically for a violin, say, the standing vibrations are multiples of one-half the wave length of the lowest frequency of the "natural" frequency of the string. Picture a string fixed at both ends. Since the ends of the string cannot move, the only "stable" vibrations are those where the ends of the string do not move. In the jargon these conditions are called "boundary conditions" because they result from a restriction on the fixed ends (or boundaries) of the string.

This similarity to waves carries over into quantum mechanics. What one finds when she/he sets out to solve the appropriate Schrdinger equation for atoms, molecules, and sub-atomic particles is that only certain solutions are "stable" in the same sense as with a violin string. That is the solution(s) are restricted by "boundary conditions". These "allowed" solutions are specified by the value of one or more integers that rise out of the "natural" solving of the appropriate Schrdinger equation. Collectively the values of these integers are called "quantum numbers".

The actual process of solving the equations can be involved, but the concept of boundary conditions, i.e. quantum numbers, is pretty simple in principle.

Vince Calder


There is a historical explanation, and a physics explanation, and a chemistry explanation. Of course all these happened at the same time, as people figured this stuff out, in the 1910's to 1920's. Atoms are like cross word puzzles, you have to look at the clues. You cannot see them directly, you have to guess what they are like based on things you really can see. There is no magic here, just a lot of careful thinking, and it took 20 years, and lots of people.

For an account of the history, I suggest a book, The Ascent of Man, by Jacob Bronowski, published in the 1970's. They made a series of public TV shows around his book. But any book on the history of atomic scientists, especially the early years, long before World War II. People like Niels Bohr etc.

The quantum numbers came first then from people looking at how light is absorbed by gases. This goes by the name spectroscopy, and still a powerful tool today, to see what things are made of. People saw that when light is shined through a gas, that certain colors, that is, certain wavelengths, were cut out by the gas. Narrow bands of color would be cut out of the "rainbow". They starting writing down these wavelengths, and started to see patterns. This lead a guy Balmer to write down a mathematical "fit" to the missing lines. The fit was just that, it had no underlying thinking, it just seemed to work. The equation said these lines would be missing, and numbered them 1, 2, 3 etc. These became the quantum numbers. At the same time though, others were just imagining what an atom would be like. They started modeling it with math, writing equations (differential equations) saying that if an electron were flying around a big nucleus, like the earth orbiting the sun, this is what it would do. The math started out simple, with some assumptions which were not justified, but made by intuition. But the two groups began to agree. This told the theory people that they were on the correct path, and they kept going, and what came out is now called quantum mechanics, by about 1932.

It is a long explanation perhaps, but the quantum numbers are these "fit" numbers to the math, or the "fit" numbers to the observations. They were eventually interpreted to be the energy of the electron, and also its angular momentum, and something called "spin". It is easy to get caught up in this, but these are just mental models, pretending that an atom is like a little solar system, little electrons flying around a central nucleus. But this is just a model, convenient for people. The math of quantum mechanics does not always lead to pictures you can so easily put in you head.

Oh, and to the chemist (and spectroscopy person) these quantum numbers get re-named (of course, people can never agree on names!). For one of the quantum numbers (called the principal quantum number) 1 becomes the letter s, then comes p, d, and f. There is another beautiful effort here as this all suddenly explained the periodic table, which you see on the wall of your science class room. It think it is again history, "s" meant that the absorbed line was "sharp", d meant diffuse etc etc -- just observations.

Hopefully you will get the chance to learn some of this, I think my main point here is that this was a puzzle, with lots of clues, and people put it all together over 20 years, and had a mental picture of the atom that was quite powerful, quite useful. It led directly to the atomic bomb, but also to all the electronic, plastics etc all around us. We use it still.

Steve Ross



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