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Quantum Mechanics introduction to QM
The task is not to see what no one has seen, but
to think what nobody has thought about that which
everybody sees. E. Schrödinger
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Chapter 5 Quantum Mechanics
We get a whole 4 or 5 days to cover material that
takes a graduate course a semester to cover!
Bohrs model for the atom seems to be on the
right track, but
it only works for one-electron atoms
it doesnt work for helium
it doesnt account for spectral line intensities
it doesnt account for splitting of some spectral
lines
it doesnt account for interactions between atoms
and we havent explained stationary states.
Looks like weve got some work to do.
You may be on the right track, but
youll get run over if you just keep sitting
there.
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5.1 Quantum Mechanics
In Newtonian mechanics, if you know the position
and momentum of a particle, along with all the
forces acting on it, you can predict its behavior
at any time in the future.
Weve already seen that because particles have
wave properties, you can only measure
approximately where a particle is or where it is
going. You can only predict where it probably
will be in the future.
Argh! This could be annoying!
Comments much of chapter 5, especially the
first half, is rather dry and mathematical
(although the math is not difficult). It is also
rather abstract. You may just have to grit your
teeth and bear it.
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Quantum mechanics is a way of expressing the
conservation laws of classical mechanics so that
they encompass the wave -particle duality which
we have been studying.
Quantum mechanics takes the fundamental laws of
classical physics and includes the wave
properties of matter.
I have already mentioned wave functions before.
Let's review wave functions for a minute. Then
we will discuss the wave equation. (Kind of
backwards, huh?)
The symbol we use for the wave function is ?
(si, rhymes with pie), which includes time
dependence, or ?, which depends only on spatial
coordinates.
In other words, ? ?(xyzt) and ? ?(xyz).
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Quantum mechanics is concerned with the wave
function ?, even though ? itself has no direct
physical interpretation.
The absolute magnitude ??? evaluated at a
particular time and place tells us the
probability of finding the system represented by
? in that (xyzt) state.
If the system described by ? exists, then
That is, the system exists in some state at all
times. Such a wave function is normalized.
The wave function must be well-behaved ? and its
derivatives continuous and single-valued
everywhere, and ? must be normalizable. See
Beiser page 163. Good ideas for quiz questions.
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? could represent a single particle or an entire
system. Lets use the particle language for a
while.
In one dimension, the probability of finding the
particle represented by ? between x1 and x2 is
Lets do an example. Suppose ?(x,t) Ax, where
A is a constant.
? has no time dependence in it it doesnt change
with time, so we can just write ?(x) or ?(x).
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Is ? well-behaved? ? and its derivative are
single-valued and continuous, but it is not
normalizable because the integral of ?? blows
up
Here is ?.
Here is ??.
The red area represents the value of the
integral. What do we get if -? lt x lt ??
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However, if we restrict this particle to a box,
then ? is normalizable. So for my example wave
function I will use ?(x) Ax, for 0 x 1, and
?(x) 0 elsewhere, where A is a constant to be
determined.
The first step is always to normalize ? (unless,
of course, ? is already normalized).
If ? has some unknown constant, such as A, in
it, you must normalize!
Failure to normalize is the first common mistake.
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To normalize, integrate
Wait! ? is zero for x lt 0 and x gt 1, so the
integral becomes
Failure to use appropriate limits of integration
is the second common mistake.
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We have just normalized ?
Wait a minute, you told me ? means there was time
in the wave function, and ? means there is no
time in the wave function. Where is time?
Youre right. You can do this if you want
That was a lot of work for a stupid little linear
function. What good is this?
Good question! Answer now that we know ?, in
principle we know (i.e., can calculate)
everything knowable about the particle
represented by ?. Thats quite a powerful
statement!
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OK, so give me an example of something we can
calculate!
Calculate the probability that the a measurement
will find the particle represented by ? between x
0 and x 0.5.
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Failure to check that the result makes sense is
the third common mistake.
Does this result make sense? How can we check?
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Heres a plot of the wave function. But
remember, we dont measure the wave function.
What we measure is proportional to the magnitude
of the wave function squared.
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Heres a plot of the probability density
(magnitude of wave function squared).
You cant talk about the probability that the
particle is at x 0.5 (Heisenberg), but you can
talk about the probability that the particle can
be found within an incremental dx centered at x
0.5.
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The red shaded area represents the probability
that the particle can be found in 0 x 0.5.
The blue shaded area represents the probability
that the particle can be found in 0.5 x 1.0.
I got P(0 x 0.5) 1/8. What would you get
if you calculated P(0.5 x 1.0)?
Does it look like the red shaded area is about
1/8 the total area i.e., the blue shaded area is
about 7 times as big is the red shaded area?
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Links! http//phys.educ.ksu.edu/vqm/html/probillu
strator.html http//phys.educ.ksu.edu/vqm/html/qtu
nneling.html http//www.phys.ksu.edu/perg/vqmorig/
programs/java/qumotion/quantum_motion.html http//
www.phys.ksu.edu/perg/vqmorig/programs/java/makewa
ve/