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s Orbital and Angular Momentum
Name: Nathan T.
Status: student
Age: 18
Location: N/A
Country: N/A
Date: 12/8/2004
Question:
While sitting in my Advanced Placement Chemistry
class, learning about the 3D probability graphs of the orbitals of an
electron in the s orbital. If the angular momentum of the electron is
zero in the s orbital, what is the electron doing? It cannot be moving
because it would then have angular momentum which would mean it was no
longer in the s orbital, but the p orbital (it could move in a straight
line but then it would fly off into space). I would appreciate it if you
could explain the motion of an electron in the s orbital and what it is
doing.
Replies:
Nathan,
This is perhaps a simplified answer to an excellent,
though complex, question, but I will give it a shot.
Imagine a classical particle traveling on a circle.
That particle has angular momentum at each instant,
right? However, if you calculate the average angular
momentum over all angles, that will be zero. So
a classical particle can have instantaneous non-zero
angular momentum, but it can average out to be zero.
Quantum mechanics is a little trickier, and the physics
is harder to describe without mathematics. However, you
may now be prepared to make a conceptual leap
to imagine that a spherical s orbital might behave
in an analogous way...at least, this is the general idea.
Fundamentally, in a quantum state defined by (n,l,ml) we
are guaranteed that if we measure the energy we will get
En, if we measure the angular momentum we will get l, and
if we measure l's projection on the z axis we will get ml.
However, that's not exactly the same as saying that the
angular momentum IS l (or WAS l) when it was measured.
It gets better. Some orbitals have zero probability of finding
the electron at certain places (like the planar node in a
p orbital, or the spherical node in a 2s orbital).
That does not mean that the electron is never there, just
that you will not FIND it there if you look for it. (?)
Quantum mechanics is beautiful and strange, and some of
its results defy a simple classical model. I hope this stimulates
you to learn more.
Best, Dr. Topper
The answer to your inquiry will seem a bit strange, but its "strangeness"
lies in the teaching approach that has a long, but uninspiring, history.
The error in the historical introductory approach to atomic structure is
the "planetary" model for atomic orbitals, ala the Bohr atom. The electron
is NOT like a tiny planet orbiting the nucleus. In an attempt to make
atomic structure "accessible" to students that erroneous analogy has been
stubbornly adhered to because teachers (not just in high school) are
reluctant to tell students the facts of quantum mechanical life,
specifically: THE BEHAVIOR OF ATOMS IS BASED ON QUANTUM MECHANICS AND OUR
INTUITIVE CLASSICAL ANALOGIES SIMPLY DO NOT APPLY. The reason, at least in
part, is the fact that the correct description of even one electron systems
like the hydrogen atom requires some knowledge of calculus, and many high
school students and lower level college undergraduates have not been
exposed to the necessary mathematical "scaffolding" to handle this.
Nonetheless, in my opinion this is no reason to avoid "telling the truth".
The behavior of electrons in atoms is governed by a mathematical functions,
called wave functions, which are the solutions to a certain equation called
the Schroedinger equation. This wave function is expressed most naturally in
spherical polar coordinates and so it has a radial part, and two angular
parts. All waves, classical "standing" waves (regardless of what the
physical system is -- it could be sound waves, light waves, or water waves)
as well as quantum mechanical waves, have stationary (i.e. they do not depend
upon time) solutions for certain values of integers that appear "naturally"
when the mathematics is cranked out, in the jargon the waves are quantized.
This is even visually evident in the vibrations of a string which must have
nodes at (1/2)*n intervals of the fundamental vibration of the string. In
the case of a H-atom there are three such integers --one for the radial
part, and one each for the two angular parts. In addition, the "allowed"
values of these integers, that is those which result in mathematically
acceptable solutions to the wave equation, depend upon one another. In the
case of 's' orbitals the characteristic number (call it 'n') for the radial
part of the allowed solutions require that: n = 1, 2, 3, ..., and the
acceptable solutions of both angular parts have characteristic numbers ( l =
0, and m = 0). These constraints just "fall out of the math, they are not
magic".
Since there is no angular behavior of the electron, it cannot have any
angular momentum. Do not even try to think of the electron as some sort of
planet orbiting the nucleus, that will only cause you grief later on as you
"un-learn" this mistaken picture of the electron behavior in the atom.
The physicists and chemists of the early 20th century had no difficulty
accepting the mathematics (even if some did not believe in atoms yet) because
is closely parallels the mathematics of classical waves.
I believe it serves science best if we collectively tell students, "You
do not yet have the necessary mathematics to understand the results in
quantitative detail so you will just have to accept some things without
proof or derivation." rather than to present false misleading stretched
analogies where no analogy exists. You can find the mathematics in all its
glorious detail in the text: "Introduction to Quantum Mechanics" by Linus
Pauling and E. B. Wilson (Chapter 5). However, if you take a deep breath and
begin reading that chapter -- just the explanatory paragraphs -- and skip
over the messy equations you will gain an appreciation and insight about how
atoms REALLY behave. You will learn, for example, that the location of
electrons are described statistically, i.e. with averages. If you have had a
statistics course it is analogous to averages and probability distributions
you encounter in that context. It really is worth your time to wade through
this "advanced" text, jumping from explanatory island to explanatory island,
leaving the mathematical deep water for a later date.
Vince Calder
Nathan,
I believe you may be going under a misapprehension on what the number
designations are for quantum numbers.
Let's consider the principal quantum number "n" (since it is easier). If we
look at a set of quantum numbers such that n=1, l=0 (ml=0, ms=+1/2) and
compare it to n=2, l=0 (ml=0, ms=+1/2) - we quickly come to the conclusion
that n is somehow related to the distance of the probability from the
nucleus. But does "1" or "2" have any physical significance? That is, does
"1" mean exactly, say, 0.01 picometers and "2" mean 0.02 picometers? While
we can say that n is related to distance, I do not think we can go so far as
to say that it has such a specific physical equivalent value.
On an experimental level, how do we know the energy of an electron in n=1,
l=0 versus that of n=2, l=0? We only get to know this by how much energy it
takes to promote an electron from n=1 to n=2. Thus, we know the *difference*
of the electron orbital levels, but not their exact values.
Similarly, the angular momentum quantum number is not so much designating a
specific physical value, but really is referring to how that particular
value goes into the Schroedinger equation.
Greg (Roberto Gregorius)
A "s" orbital is the de-localized, fuzzy, wave-aspect of an electron.
The wave is trying not to be in exactly one place even though the electron
is trapped in a pointy-bottomed well.
The _very_ bottom of this potential well is in and around the nucleus,
but the electron, being "averse to discipline", or being
"a fixed amount of substance in the variable shape of a dispersed, ghostly
cloud",
just cannot convince itself to fit into that small a space.
So it stays more or less in the smallest space it can convince itself to
stay in, given the energy-incentives at hand.
Nobody knows exactly what situation inside causes this aversion.
So we think of it as a fundamental fact, almost a definition of what an
electron is like.
It is part of the Heisenberg Uncertainty Principle: _all_ particles refuse
to be in exactly one place.
Things smaller than electrons, which would need to be examined to explain
de-localized behavior of particles,
always have even greater uncertainty than the particles themselves, so
they can never be examined,
so we do not ever expect an observational breakthrough that will explain it.
In short, that is what an electron looks like sitting perfectly still in a
pointy-bottomed bucket.
I suppose you are welcome to entertain your own theories why.
Me, I cannot find any that make sense, so I have stopped trying.
Another question to ask yourself is: why does an electron with angular
momentum (p orbital)
look like a dumbbell, instead of a donut around the axis of the angular
momentum?
I do not have the answer. I am not sure it is a correctly informed
question either.
Jim Swenson
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