WHAT is the evidence? WHY did I choose this
evidence? HOW does the evidence show
growth?
I chose
to show my growth in my understanding of inorganic chemistry,
specifically Schrodinger's quantum wave-mechanical model, by comparing
quantum theory notes (with misconceptions) and a sample assessment
question that I
had used in my own classroom before taking Chem506 (my baseline) and
excerpts from notes and a successfully completed problem set from
Chem506 that
shows an improved understanding of the wave-mechanical model (my later
evidence).
I
chose to document my improved understanding of the quantum
wave-mechanical model because this growth shows a more nuanced (and
complicated!) awareness of what the wave function and quantum numbers
do and do not represent and
how they are related to each other. Furthermore, I feel that I
grew in my understanding of scientific theory and my ability to accept
the
wave-mechanical model as a model because of Dr. Berry's continued
stressing that it was
simply a mathematical model that was useful for prediction and
description, but not actually, physically "real" (a rewording of Dr.
Dailey's initial caveat about how mathematical models are made to fit
empirical data and reality, rather than the other way around). I
found the essay by Isaac
Asimov, given
by Dr. Berry, quite helpful in its discussion of the iterative nature
of scientific model-building.
Please see below each piece of evidence for a more detailed discussion
of related content. You may click on
any piece of evidence for a larger image.
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Baseline
Evidence (click on pic for
larger
image)
- Quantum
theory notes (with misconceptions) and sample question used prior to
taking Chem506
- Misconception
of how Schrodinger's wavefunctions and quantum numbers are related
(Sept
2003-June 2008)
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Later Evidence (click on pic for larger
image)
- An excerpt
from my successfully
completed Problem Set #1 (July 3, 2008) from Chem506 (for a larger
excerpt of all the problems on PS#1 related to wavefunctions and
orbitals, click here).
- An excerpt
from a Quiz #1 (July 10, 2008) from Chem506
- Summary of
information learned re: quantum theory, quantum numbers, and
Schrodinger's wavefunction (in the reflection itself)
(Summer
2008)
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Page 1 of notes:
By examining the first page of notes, there is a
conspicuous ABSENCE
of any reference to nodes or any connection of the orbitals to
Schrodinger's wavefunction that had been used to generate these shapes
(most likely because I knew I didn't really understand that aspect of
quantum theory well enough to teach it accurately).
Before taking Chem506, I had an inconsistent understanding of the
quantum wave-mechanical model. While I felt fairly comfortable
teaching students certain aspects of the model, I did not yet really comprehend
- why
wavefunctions
were necessary (stress on the wave)
- how
Schrodinger's wavefunctions actually generated the representations of
s, p, d, and f orbitals,
- (though
I knew the definition of what a node was) why a node was significant
- how
the wavefunctions and four quantum numbers n, l, m, s were associated.
The following second page of notes and a related
assessment question shows my misconception in thinking there was a
simplistic one-to-one correspondence between atomic orbitals and the
magnetic quatum number, m.
Page 2 of notes:
Assessment Question:
*********************
Furthermore, while I do not have hard evidence of
the following error, I
even remember mistakenly thinking that the quantum numbers were the
mathematical solutions to the equation
E(wavefunction
= Hamiltonian (wavefunction)!
In other words, I
though there was some weird four-dimensional wavefunction, f(w,x,y,z)
that when made equal to energy solved out such that (w,x,y,z) = (n, l,
m, s). I even recall telling a student this is how I thought the
quantum numbers and wave functions were related when he asked me about
the connection between them. I really should have just told him
that I had no clue!!! It didn't even make sense to me!
To my credit, this was not a misconception that I willingly taught--in
fact, I tried to avoid discussing Schrodinger's wavefunctions if
possible. Though I only had to air this ugly misconception once
to answer a question asked in class on that single occasion--it sticks
out in my mind because I remember the acute discomfort of being
confronted by my own ignorance.
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In contrast, later
evidence--an excerpt from my successfully completed
Problem Set #1 and an excerpt from Quiz #1 shows that I
am much more comfortable in my understanding of wavefunctions and
quantum theory.
Problem Set #1:
Quiz #1:
In the excerpts from
the problem set and the quiz, it is evident that I know what
Shrodinger's wavefunctions are. I clearly plot out the radial
wavefunction R--the function measuring the intensity of the electron,
when it is considered as a wave rather than a particle. The
positive and the negative have no significance except in regard to
distinguishing when the electron wave goes from having a positive to a
negative amplitude--this is where a node occurs (where the wavefunction
goes from positive to negative). The total number of nodes
for a particular electron is equal to one less than the principal
quantum number (i.e. n-1). The azimuthal (subshell) quantum
number l determines the number of angular nodes (typically a plane).
When
this wavefunction R is squared and multiplied by [4 x pi x radial
distance squared] and plotted over the radial distance (the distance
from the nucleus to the point), the radial electron probability
(sometimes thought of as the electron density) is produced--the source
of our common representations of s, p, d, and f orbitals. In this
squared wavefunction (aka the radial electron probability), the nodes occur in the
same place as where the nodes occur in the radial wavefunction--in this
case, where the radial electron probability equals zero, indicating
zero electron probability or density at that point. It is
important to note that since we are conceiving of electrons as waves,
they do not have a location within the shape provided by the radial
probability, but rather an intensity or probability associated with
each point.
***************
Drawing on my notes from Chem506 (the June 26, 2008
and July 1, 2008 class in particular), the information that I have
learned is that:
- the quantum wave-mechanical model is a based
on a mathematical equation that "works" in describing and predicting
electron's behavior, but it is still just a model used to describe and
predict empirical behavior--it does not fully represent the reality of what
an electron is and its behavior
- the empirical behavior that required
resolution/ the birth of the quantum theory was the observance of
discrete atomic spectral lines--waves
also exhibited this discrete, non-continuous (aka "quanticized") nature
and their amplitude could be described mathematically as a function of
distance from the origin. Wavefunctions also met the requirements
for a function useful for calculating numerical properties, such as
energy, etc (i.e. second derivative of a wavefunction equals a scalar
times the wavefunction).
- like the harmonics on a string, there are
families of wavefunctions. The different values for quantum
numbers impact what subset of wavefunctions you are working with.
The quantum numbers are a set of variable
scalars (which can only have certain allowed values) that are a part of
the wavefunction--they are not the actual variables of the
wavefunction. Distance in 3-D space (typically in polar
coordinates) is the independent variable, electron amplitude (positive
or negative) is the dependent variable.
- also, there is no
one-to-one correspondence between the magnetic quantum number and a
particular orbital, even though the possible number of magnetic
quantum numbers equals the possible orientations or number of orbitals
of a particular type.
- the wavefunction equations for various
atomic orbitals can be combined in various linear combinations to
produce hybrid and molecular orbitals
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Updated
June 28, 2009
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