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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
Chapter 5
Quantum Mechanics
We get a whole 4 or 5 days to cover material that takes a
graduate course a semester to cover!
Bohr’s model for the atom seems to be on the right track, but
it only works for one-electron atoms…
it doesn’t work for helium…
it doesn’t account for spectral line intensities…
it doesn’t account for splitting of some spectral lines…
it doesn’t account for interactions between atoms…
and we haven’t explained “stationary states.”
Looks like we’ve got some work to do.
You may be on the right track, but… you’ll get run over if you just keep sitting there.
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.
We’ve 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.
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).
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
*

  dV = 1 .
-
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.
 could represent a single particle or an entire system. Let’s
use the “particle” language for a while.
In one dimension, the probability of finding the particle
represented by  between x1 and x2 is

x2
x1
 *  dx .
Let’s do an example. Suppose
(x,t) = Ax, where A is a
constant.
 has no time dependence in it;
it doesn’t change with time, so
we can just write (x) [or (x)].
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 - < x < ?
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.
To normalize, integrate:

1=

 *  dV .

Wait!*  is zero for x < 0 and x > 1, so the integral becomes
1
1 =   Ax   Ax  dx
*
0
1
1 =   Ax  dx
2
0
1 = A2
3 1
x
3
0
3
3


1
0
2
1= A  - 
3 3
*Failure to use appropriate
limits of integration is the
second common mistake.
A2
1=
3
A= 3.
We have just “normalized” :
(x) = 3 x for 0  x  1 .
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?
You’re right. You can do this if you want:
(x, t) = 3 x for 0  x  1 and all t .
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 . That’s quite a powerful statement!
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.
P(x1  x  x 2 ) = 
x2
x1
 *  dx
1/2
1
P(0  x  ) =  ψ*ψ dx
0
2
1/2
1
P(0  x  ) = 
0
2

3x


3 x dx
1/2
1
P(0  x  ) = 3  x 2 dx
0
2
3
1
x
P(0  x  ) = 3
2
3
1/2
0
3
1
1
P(0  x  ) = - 03
2
2
1
1 *Failure to check that the
P(0  x  ) = . result makes sense is the
2
8 third common mistake.
Does this result make sense?* How can we check?
Here’s a plot of the wave function. But remember, we don’t
measure the wave function. What we measure is
proportional to the magnitude of the wave function squared.
Here’s a plot of the probability density (magnitude of wave
function squared).
You “can’t” 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.
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?
Links!
http://phys.educ.ksu.edu/vqm/html/probillustrator.html
http://phys.educ.ksu.edu/vqm/html/qtunneling.html
http://www.phys.ksu.edu/perg/vqmorig/programs/java/qumotion/quantum
_motion.html
http://www.phys.ksu.edu/perg/vqmorig/programs/java/makewave/