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Question -   Please explain the origin of quantum numbers.
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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 Schrödinger 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 Schrödinger 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 Schrödinger 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
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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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