>> Welcome back to
Chemistry 131A.
Today we're going to talk
about particles, waves,
the uncertainty principle
and some
of the postulates of
quantum mechanics.
As you'll recall from
the last lecture,
we found out at the turn of the
last century by which I mean
by around 1900, that, in fact,
particles were behaving
strangely
and classical physics
was not accounting
for all the observations and so
a new theory was put forward,
which came forward over a period
of time called quantum mechanics
and like any fundamental
theory it has to be compared
with experiment and so there
were experiments that were done.
We saw some more modern
experiments last time
in which an electron
and an electron microscope
behaved very much like a wave,
the experiment of [inaudible]
at Hitachi, and in fact,
this wave behavior
had been observed
and caused Louis de Broglie,
I believe it's actually
pronounced de Broglie,
but just to keep it clear
we'll say de Broglie anyway.
He proposed in 1924, in fact,
that all particles have
an associated wavelength
that is related to
their momentum
and that this wavelength just
follows the same relationship
as that for a photon.
We saw last time that the
photon momentum was given by H
over lambda and de Broglie
proposed that, in fact,
there was a wavelength in lambda
that was related to H upon P
for a particle not
just for a photon.
In fact in 1927 3 years later,
Davisson and Germer showed
that an electron beam fired
at a nickel crystal
showed a defraction pattern
and that's a wave phenomenon
and furthermore they looked
at what the wavelength
of these electrons
and the beam would have to be
and the wavelength was very,
very close to the exact
prediction that de Broglie made.
So the question is
where is the particle?
In classical mechanics, the
center of mass of a particle has
in principle at least an
exact location at all times.
In quantum mechanics
it's not quite so clear.
In classical mechanics, the
particle follows a trajectory.
What a trajectory is is
it's an exact specification
of the position of the
particle and the momentum
of the particular
or its velocity
if the mass doesn't change at
all times and that's, in fact,
how we do all kinds
of calculations
in classical mechanics whether
I'm going to shoot a shell
and have it land somewhere or
anything along those lines.
That works extremely well for
large objects, but it fails
for small objects because they
show this strange wave behavior.
Now the problem with something
that's demonstrating a wave-like
behavior is that we can't say
for certain where a wave is
because waves tend to
spread out over time.
So we're kind of caught in
a little bit of a difficulty
because if we can't actually
say where the center of mass
of the particle is, if
it appears to be blurry,
we can't specify it
exactly that means
that we can't have a trajectory
and the trajectory following
Newton's laws is exactly how you
calculate where things
are going to end up.
So now you've got
to have a new method
to calculate how things
are going to behave
if they're showing this
wave-like phenomenon and so
that was a big chore
actually and took a lot
of very smart people a
very long time to work out.
In 1927, Werner Heisenberg made
this blurriness more formal
in the famous uncertainty
principle, which states
that no matter how you design
an experiment it is impossible
to measure the momentum
and position of a particle
with arbitrary accuracy.
There's a certain minimum amount
of uncertainty that's left
over given in this
famous expression Delta P,
Delta X is greater than
or equal to H bar over 2.
It's important to emphasize
that this has nothing to do
with your experimental apparatus
having some sort of deficiency.
This is just an idealized
experiment done as best
as you could possibly
do in this universe
and you still cannot
simultaneously specify the two
things at once.
In fact, the quantity H
over 2 pi occurs so often
that we invented a shorthand
notation called H bar,
H with a slash or H cross, was
created for it and you'll see
that often in the formulas that
we use because we get tired
of writing 2 pi so much.
The uncertainty principle says
that we just cannot
simultaneously determine
position and momentum
to arbitrary accuracy no
matter what we do even
in an idealized experiment.
That runs counter to
our everyday experience
where we seem to be able
to watch a rolling marble
and plot its mass and figure
out where it is basically
about as well as we want to
and so there must be something
in this thing that makes it
different for small particles
and the something is
the exact size of H bar.
If H bar were bigger, we
would notice all these things
happening with big objects
but H bar is so small
that we don't notice
it the same way.
We don't really notice
the momentum of a photon.
Otherwise the lighting that is
on me now would be pushing me
around and I'd have to fight
like a mime to keep my position.
Now, the rationale is
when we think of looking
at something we're looking at
something with the lights on.
We can't look at something
in the dead pitch black
and see anything, but if
you've got a small particle,
then we've learned that light
consists of packets, photons
and these have momentum
and energy.
So when we try to see where a
very small particle is we can't
just look at it.
We have to bounce
something off it.
We have to use something
like light
and the light itself is
going to change the momentum
of the particle and if we
want to get the position
to be very close, then we
have to use a short wavelength
of light but that's a high
frequency and we learned
that the quantum energy of light
is H nu and so if nu is high
that means that we're going to
come in with a ton of energy
like an x-ray and then that's
going to boot the particle
around and so although we could
say it just was there now we
can't say very well
what its momentum is
and these ricochets
happen all the time
and so we have a fundamental
problem if we try to do it.
Now if we turn off all
the lights, then we know
that the particle is moving
with a certain speed, but then,
of course, we can't
tell at all where it is.
So for small objects the
photon kicks the particle
and that creates the
fundamental problem.
So if we don't really know
how big an electron is,
it appears to be a point if
you do experiments with it,
but to tell where it is
we need then a wavelength
of light that's small.
So just the same way you put on
infrared goggles and you look
at night, you can see heat but
everything is much blurrier
because it's not as
sharp as visible light
because it has a
longer wavelength.
So, we need a small wavelength
of light to get a particle
down to what we would consider
to be a reasonable precision
of measurement and that
gives a big kick to something
like the electron
and [inaudible]
and so the electron momentum
in that case becomes
very uncertain.
Conversely you can think of how
you might design an experiment
that would measure the
momentum of a particle.
One way to do it would be
to have a very, very long,
thin tube and have
particles coming in
and if they aren't going
straight along the tube then
they hit the wall and
they're out of there
and you can have some
choppers like fan blades
at certain distances and moving
with certain rotational speed.
If the particle happens to be
going the right speed so it goes
through the whole of this fan
and then the whole of that fan
and so on, then you know for
sure that it has a momentum
within a certain range.
So you've isolated it very well,
but if you really want to get it
to be very, very
small uncertainty,
that means you're going to
have to have a very, very,
very long tube and then the
particle could be anywhere
inside the tube.
So you don't really
know its position.
Of course if you open
it up so it's not dark
and you put light
in, then that fouls
up the momentum as
we said before.
So we can't specify the
position of any kind of object
without light to see
it or something else
which would be even
worse probably.
Now for macroscopic objects the
uncertainty principle does not
limit us.
We're always limited
experimentally
for anything we do
that's a size we can see.
It has nothing to do with that
but for very small objects
like a single electron it
becomes the major factor.
So here's a practice
problem that's meant
to illustrate this difference.
Suppose, it's practice
problem 3,
suppose we take a 1 gram
marble and we know its position
to a tenth of a millimeter,
which is pretty good,
and we know the position of
an electron to 200 picometers,
which is 200 times 10
to the minus 12 meters.
The question is what would
be the minimum uncertainty
according to the uncertainty
principle for the velocity
in each of these cases?
So, first of all the
uncertainty and momentum is
in the velocity and
not in the mass.
The mass stays fixed so
Delta P becomes M Delta X
and then we use the
uncertainty principle.
Delta P, Delta X is greater
than or equal to H over 4 pi.
We have to be a little bit
careful with the units.
We were given the units in
grams but we have to convert
to kilograms and we were given
the uncertainty in millimeters
but we have to convert to meters
because we're using MKS units.
If we take account of
all of those factors,
then we find the Delta V
is H which is 6.62 times 10
to the minus 34 over 4
pi and then we have 10
to the minus 3 kilograms
that's our 1 gram and 10
to the minus 4 meters.
If we work out the
units carefully,
we find that the uncertainty
and velocity is 5.2 times 10
to the minus 28 meters
per second.
For reference an atom is about
10 to the minus 10 meters
and so the uncertainty and
velocity is way, way, way, way,
way smaller than anything we
could possibly ever notice.
So it's just as good as
almost infinite precision
as far as we're concerned.
For the electron, however,
if we do the same calculation
and here are the difference
is we put in 9.1 times 10
to the minus 31 kilograms and
then we put in 200 times 10
to the minus 12 meters then what
we find is that the uncertainty
and velocity is about
2.9 times 10
to the plus 5 meters per second.
So that's 200,000
meters per second.
If you've ever run a 10K,
you realize that that's running
pretty fast and so trying
to localize the electron down
to just 200 picometers mean
that its velocity becomes
very, very uncertain.
So the uncertainty in velocity
is very, very large in that case
and the reason why there's a
large uncertainty for the case
of the electron is
that the electron has
such small mass whereas
macroscopic objects
in the 1 gram range
there's no problem.
Okay. Now, de Broglie
said, look,
matter has a wave
associated with it,
which is kind of
a semantic dodge.
It's not as if matter has
suddenly become a wave.
We still can detect particles
in the [inaudible] experiment
when they hit the screen they
give a dot, and we know how
to characterize a wave.
There's a phase, frequency,
there's an amplitude and,
in fact, wave equations
were known
from electromagnetic
radiation, Maxwell's equations,
which seemed to indicate
the light was a wave
and explained many, many of its
properties was a wave equation
and so physicists knew how
to write those things down,
but there's kind
of a question here
as to whether this
associated wave is a real thing
or not a real thing.
Is it a calculational device
or is it a real thing?
Well, Davisson and
Germer seemed to indicate
that it is a real thing; that
this de Broglie wavelength
for small particles this can
be the main thing depending
on what kind of measurement
you're making.
So the question is is the de
Broglie wavelength an actual
measurable thing?
If it is, then what equation
do we write for the wave?
And finally if the wave, if
the behavior is wave like,
then how do we explain
the sharp spots
in the [inaudible] experiment.
When we wanted the electron to
sift through both of the slits,
it was convenient then to
think of it like a wave
because a wave can do that
and a particle cannot,
but when it hits the detector,
it seems like it collapses
like a poorly rigged camping
tent onto just a single point
on the detector and it
certainly doesn't light
up the detector like
a wavy thing.
We see the sharp spots and
only after we measure a lot
of these measurements
do we, in fact,
see the aggregate
behavior is wave
like each individual one
doesn't look wave like.
This took a lot of thought
to sort out for sure.
We don't notice any
wave-like behavior
with macroscopic objects,
we don't notice any,
when we move around,
we don't notice
that there are waves coming off
my hands or that I can't tell
where my fingers are
or anything like that
and the reason why is the
same reason as with the marble
that the small value of Planck's
constant is the explanation.
We're just so much bigger than
H bar that we don't notice that
and for an illustration let's
do another practice problem,
practice problem 4.
Let's figure out the
de Broglie wavelength
of first a 5 milligram
grain of sand moving
at .1 centimeters per second
blowing along the beach and, 2,
an electron moving at
1 kilometer per second.
Well, for the grain
of sand we just plug
into the de Broglie
formula lambda is H upon P
and then the rest of it is
what a lot of chemistry is,
it's units conversion
keeping track of the units,
crossing them all out and
making darn sure at the end
that the units are
what you intend to be.
So in this case,
I explicitly wrote
out a joule is a kilogram meters
squared per second squared cross
out the joules, cross
out the kilograms,
make sure the milligrams
is turned into kilograms
and make sure the centimeter is
turned into meters and I find
that the de Broglie wavelength
for the grain of sand is
about 1.3 times 10 to
the minus 25 meters.
That means the associated
wavelength is much, much, much,
much, much smaller than any
nominal sizes we would think
of of a grain of sand, which
is certainly much bigger
than an atom.
However, for the case
of the electron the de
Broglie wavelength working
out the same math but
using the electron mass
and using the velocity given
now converting kilometers
to meters it works out to 7.3
times 10 to the minus 7 meters.
An atom is 10 to the minus 10
meters and so now we're talking
about a wavelength of
the electron moving
at this speed that's much,
much bigger than anything we
would think of as the size
of the electron because even
if we don't know
exactly what the size
of the electron is we know
that atoms have electrons
in them and, therefore, the
size of the electron has
to be much smaller than the
size of the atom or they'd be
like beach balls and
they wouldn't fit.
So in this case when
we have this situation,
when the wavelength
that we figure
out from the de Broglie
formula is, in fact,
much bigger than
any nominal size
of whatever it is
we're considering,
at that point we
have to say watch
out because whatever this
thing is it could surely show
quantum behavior.
When the de Broglie
wavelength is very much smaller
than the size of the thing, our
idea of the size of the thing,
then we aren't going to see
any kind of wave behavior
and we might as well
just cut to the chase
and use classical
mechanics to figure
out what's going to happen.
So that's summarized in
this next bullet point.
We don't notice any
wave neighbor
if the particle has a very
small de Broglie wavelength
and if it's larger, then we
surely do notice it at least
in some experiments we will.
Now, what are we going
to use to actually figure
out what something is doing?
For particles we had Newton's
laws, we had ways of figuring
out S equals MA and so forth
what was going to happen.
Now we have this de
Broglie wavelength
but we still don't have a
wave and we don't really want
to call the particle a wave
because that's a
little bit confusing.
It makes you wonder what
a particle ever was.
So what we do is we have kind
of a little bit of a dodge.
We speak of the wave
function of a particle
and the wave function
is associated with it
and it's given the symbol
SI to describe its behavior.
Whenever chemists want
to keep out people,
they switch to Greek
letters because if you switch
to Greek letters then it seems
much, much more intimidating
and you have job security,
but the symbol SI is
universally interpreted
as a probability amplitude, and
I'll get to that in a minute.
The absolute square if we square
it we get a probability density
and there is a reason why
we're using probability
and not certainty and part of
that has to do with uncertainty
and part of it has to do with
the very nature of measurement
as we'll see in the end of
this lecture or possibly
at the beginning of
the next lecture.
We can only know
quantum mechanics
as the probability not the
certainty even as frustrating
as that might seem of finding
the particle somewhere.
Due to its wave nature it
appears to be spread out
and when we measure it we
interfere with it somehow
and that causes the
measurement to collapse
like the tent I mentioned but
the probability is uncertain
until we finally
make the measurement.
It's as if I take a die and I
throw it and I'm not looking
at it and then it lands and
it bounces and it rolls around
and so forth and then I look
at it and it's a 2 and I said,
well, the probability beforehand
of getting a 2 was 1/6 but,
in fact, now that I see it's
2, the probability is 100%
and we're going to
see that phenomenon
when we make a measurement.
Once we know then all the
other possibilities seem
to have just vanished somehow,
which is a little bit mysterious
but that appears
to be what happens
and that's a common
interpretation anyway
of the theory.
Once we measure the position,
we can yield different values
and even though we have exactly
the same electrons coming
through the 2 slits
and everything is
exactly the same each time
and we believe all
electrons are identical,
we still get a distribution
of different values.
We don't necessarily get 1 dot.
In fact, we saw for sure
we didn't get 1 dot;
we got dots all over the
place and when we added them
up it was much like
looking at a billboard.
When you're too close, it's
just dots; when you get back,
you see the big picture and
if you have a lot of dots
and then you see it
looks like a wave.
Keep in mind that this is in a
hypothetical perfect experiment.
It's, of course, not possible
to do a perfect experiment,
but it's possible to get very
close to a perfect experiment
with some kinds of
setups if you're careful
and even then you get a
distribution of results
which is just far, far
greater than any uncertainty
in whatever you setup
and so that looks
like there's something
else at play.
You might think, well,
maybe there's noise,
maybe there's some kind of
thing that we don't perceive
and it's there like road noise.
You only notice it's
absence when you go camping,
but you don't tend to notice it
when it's there even
though it's always there.
So maybe everywhere when
we're shooting these electrons
and photons there's some kind of
noise or something else around
and if we were smarter
and we figured out what
that source was, what was
jiggering things around,
then we might be
able to get rid of it
and then we might not have this
theory of quantum mechanics,
we would have a theory where we
could measure things the way we
want to and so forth and so on.
In fact, it seems like
that's not the case
that quantum mechanics
really says
that this probability
is the nature of nature
and not the result of an
incomplete theory by leaving
out something hidden that
we just couldn't figure out.
One of the biggest
luminary so far
and physics didn't
like this theory.
Einstein famously said God does
not play dice with the universe.
Keep in mind though that very
bright people can be wrong
about things.
Linus Pauling, who is one
of the brightest guys so far
in chemistry, had a
theory that Vitamin C
in huge doses would
prevent prostate cancer
and that was disproven even
though he believed it ardently.
In this case, apparently
even Einstein was incorrect.
In fact, it seems like there's
nothing but probability.
There is nothing else.
For example, radioactive decay
doesn't depend on pressure,
temperature or anything
else that's been explored
to any appreciable degree;
that's why we can use it
to date things and figure
out how old something is
and so forth and so on.
We believe that all the
nuclei are identical.
They all have the same number
of protons and neutrons
and there they are
in the sample.
Yet all we can know is the
half life, which is the time,
for example, for half the sample
atomic nuclei to decay away
into some other element.
For carbon 14 the half
life is about 5,800 years,
which makes it convenient for
lots of measurements of things
that have been around since
humans have been around
but not too useful for something
that might be extremely old
because then there's no carbon
14 left and so when you try
to count it you just see
nothing and you can say it has
to be older than this but
you can't really narrow it
down too much.
But if we look in our identical
sample and we watch them,
there is no way we can tell
which particular nucleus
out of all the identical
set is going to decay.
We can't beforehand
make any kind
of experiments that's
going to tell us that.
All we can say is that in 5,800
years there are going to be half
as many as there were today.
So this means that really
probability is the whole thing
when it comes to
something like this.
So there are some postulates
of quantum mechanics we need
to understand them
because we need
to have this new
theory down pat.
We need to know what
assumptions it's making
and these postulates
form the basis for all
of our understanding and for
all the detailed calculations
that we might undertake.
The first postulate is
this at any time the state
of a quantum system is
described as fully as possible
by the wave function SI, which
depends on the coordinates
of all the particles
that make up the system.
These could be the electrons
and the nucleus and then atom
or something bigger like
a molecule but in any case
if we know SI we know everything
that it is possible to know
about the system and what
that means comment one
since the wave function contains
all that we can know it follows
that most of the time we do
not know the wave function
because in order to know the
wave function we would have
to be making lots and lots
of very clever measurements
and usually we don't.
So usually we just know
the dog is in the yard
but we don't know
exactly where it is.
So, if we say we know
the wave function
of a system we're making a
very, very strong assertion.
Comment 2 we're often
interested in wave functions
for quantum systems that are
not changing in time like,
for example, the properties
of an isolated atom
or an isolated molecule and in
that case we use a lower case si
to denote the wave function
but in some cases we
might be interested
in a time dependent phenomenon
and then we use an upper case SI
and the problem is
they look very similar.
Usually you put like a
serif on the top to indicate
that you're using
the upper case SI
and usually you set time
apart as a separate variable.
So time is an input
to the wave function.
You have to know the time to
calculate the wave function
and you write it as I've written
on the bottom of Slide 52 here.
The question is how
can we suppose
that the particles whatever they
are have exact positions R1, R2,
R3, and so on when a couple
of slides ago we just decided
that there's an uncertainty
in the position
of any quantum particle, but
isn't this kind of using sort
of bad logic here
to be assuming this?
The answer is no.
It's only with respect to
measurement that we have
to worry about the
uncertainty principle
and even then it's only for
joint measurement of something
like position and momentum
along the same coordinate axis,
X position, X momentum.
The variables here,
the XYZ coordinates
of all the particles, are better
viewed just as simple parameters
on which the wave
function depends.
Once we've calculated the wave
function then we can use it
to describe all our measurements
that we're going to make
and magically everything
comes out just hunky dory.
Postulate 2 makes an
assertion about probability.
It says now what this
wave function means.
The probability of measuring the
position of a quantum particle
at some position
let's say X not,
within some small
region DX small enough
that the wave function
doesn't change value very much
over the small region,
is given by SI star SI DX
or modulus SI squared DX
for 1 dimensional system.
For a 3 dimensional
system we have to integrate
over all the spatial
variables, DX, DY,
DC or in polar coordinates
DR, D theta,
DFI and in any case now we
have SI R not squared times DV.
Some books use D tau.
I prefer DV because it
reminds me that it's volume.
So our first comment is
what is the asterisk?
What is the SI star?
The answer is that the asterisk
denotes the complex conjugate
because the wave
function is often complex.
Now at first blush that seems
like that might be a problem
because you're saying that
this thing that's associated
with a particle has gotten
an imaginary part to it but,
in fact, not really
because I can write
down a simple algebraic
equation.
Let's say X squared
plus 1 is equal to 0,
and that has a solution but
it has an imaginary part.
So if I just say,
well, I don't want
to have any imaginary
numbers in the wave function,
it might be like algebra.
I won't have a theory
of rooting polynomials
or anything left
over in my theory.
So we accept this,
but of course,
we realize probability is real
and that's exactly why we
take the complex conjugate.
So if you're given a
complex number Z equals X,
X is called the real
part plus IY,
Y is called the imaginary part
and then the complex
conjugate is obtained
by just changing the sign
of the imaginary part.
You can think of a complex
number having an X part
and a Y part and the
complex conjugate,
the X part is the same but
the Y part is reflected
to the other side.
So if it's up here, it goes
down, if it's down here,
it goes up and you just do that
wherever you see I mechanically
and it's very straightforward
to do.
So it's not a big deal
in terms of calculation.
If you then work out what Z
star Z is or absolute Z squared
or the modulus squared,
you'll find that it's X squared
plus Y squared because you have
to recall that I squared
is equal to minus 1.
That's exactly what we want
because then it's a
real positive number
and it corresponds to the
length of something and usually
when you have the
length of something
that means you're going
to be able to add them
up because length adds and
then in our case it's going
to be probabilities that
are going to have to add up
and they're going to
have to add up to 1.
Comment 2, the probability of
finding the particle somewhere
in the universe should be 100%.
That is the particle
shouldn't disappear
and in some cases particles
are annihilated and they turn
into other things, but we aren't
going to consider those cases
in chemistry; those
are for physics.
In terms of us if
we have an electron,
the chance of finding it
somewhere has to be 100%
at least in principle
and that means
that there's another
constraint on the wave function.
The wave function
should be normalized.
That is the integral from minus
infinity to infinity of SI
of X squared DX should be
equal to 1 and that means that,
of course, SI of X whatever
it is has to be a function
that we can integrate
and it'll turn
out that it should also be a
function that we differentiate
as well so we can figure
out where things are going
in time and so forth.
That means that SI of X is
really I am mathematical terms a
very well behaved function.
It's not any exotic
mathematical function
that would cause us problems.
Of course, we put these
limits on the integral plus
and minus infinity and we do not
think the universe is infinite
or rather we think the universe
is not infinite but the math
that we do is much easier when
we make the limits infinity.
This is often true in
all kinds of fields
where if you have an
infinite charge it's very easy
to calculate the electric
field and so forth
and if it's a finite sheet,
it's much harder there are terms
to subtract, and if there's a
funny shaped sheet it's really,
really hard and it doesn't
teach you anything necessarily
different and so just
to get the principle
down you usually take a simple
case and often it's infinity.
So, we're going to assume
that the universe is infinite;
that won't make any
difference for our calculations.
Postulate 3, for every
observable property
that we can measure, energy,
linear momentum, position,
angular momentum, there's
a new player in the game.
It's a linear Hermitian operator
that acts on the wave function.
This is a new object that many
of you may not be
acquainted with.
So, I want to take
a little bit of time
and explain what's going on.
An operator is like the
big brother of a function.
When I think of a function,
I think the function
grabs an input number
and then returns
a function value.
So it grabs a number and
it gives back a number.
An operate so like Y
equals F of X, for example,
an operator takes in a
function, the whole thing,
and then gives a new function
and there are things that take
in operators and
give new operators
and so you can keep
going, but in terms
of this course we don't
need those other objects
so I've written here omega with
a hat takes in the function F
of X and returns
the function G of X.
In this course, operators
are going to be
like gentlemen in the 1950s.
They are not going to
show up without a hat
and usually we just omit
the extra set of parentheses
and we keep the hat on
so we know we're talking
about an operator and not just
a number or something else
and we simply write omega
to the left of the function,
always to the left, because
it's acting on it to the right.
Omega acting on F gives G.
Operators can be as simple
as just multiplying by a
function by X or even a constant
because that gives a new
function or even multiplying
by 1 so we get the same function
back that's still an operation.
So here I've written X
hat, the operator X hat,
operating on F gives
the variable X
without the hat times F and
that's the new function G. So,
if F is X, then X hat on
F is X squared and so on.
Once the operator
is done operating,
the result is a new function
which just has variables
which might be the same.
So one of the things
you're supposed to do
in quantum mechanics when you
see an equation with an operator
in it, is you're supposed to
let the operator do its work
and then get back
to just functions
that you can differentiate
and integrate and so forth.
So that's the goal; don't
leave the operator hanging
around unless you have to.
Comment 4, operators have units.
Multiplying by X is going to add
length units to the new function
and in chemistry we have
to be careful about units.
We can't just be
multiplying by things
and not know what the units are.
We have to make sure we get the
right units whether it's energy
or momentum or position.
This leads to practice problem
5, which is the following.
Does a wave function have units?
If so, what are they?
Well, let's go back to
what the wave function was.
We know the integral of the
wave function squared represents
a probability.
Probability is a ratio
and has no units.
We wrote for a normalized
wave function the integral
of SI star SI is equal to 1,
but the integral is against DX
and DX is like X it has units
of length and, therefore,
SI star SI whatever it is must
have units of inverse length
or length to the minus 1 and,
therefore, since there's 1
of those guys and
changing the sign
of the imaginary part
doesn't change the units.
SI itself must have units of
length to the minus 1/2 power
or 1 over the square
root of length.
For a 3 dimensional wave
function SI has to have units
of length to the
minus 3 halves power.
The expectation value
or average value
of any observable once you
know the wave function is given
by an integral of a
sandwich with SI star,
the operator and SI on DX.
This is for a 1 dimensional
problem.
The expectation value is
usually denoted with brackets
around the thing, which
means an average value
or the value we expect with
a very, very large number
of measurements but no single
measurement need ever return the
expectation value.
For example, if I
flip an unbiased coin
and I count 1 every
time it comes up heads
and I count 0 every
time it comes up tails,
then the expectation value
if I make a very large number
of tosses is 1/2, but
we never get 1/2 in any
of the measurements we do.
We either get 0 or 1.
Here's a challenge if you're
interested, a little bit harder,
what's the expectation
value for throwing a pair
of dice a large number of times?
If you can do that kind of
problem, you may have a history
in playing craps or
other gambling games.
Postulate 4.
The only possible result of
a perfect measurement is one
of the eigenvalues of the
operator corresponding
to the measured observable.
That's quite a mouthful.
What does that mean?
Well, eigen functions
and eigenvalues are
central mathematical objects
in the theory and that's
one reason why you ought
to take math courses up to
and including linear algebra
so that you can learn about
these things without having
to learn them on the fly
while you're also trying
to learn something
else about the subject.
An eigen function is a function
which the operator
returns unchanged except
for multiplication by
a number with units.
First of all here's the form
of the eigenvalue equation.
I've written here omega hat
on F gives little omega,
which is a number with units
on F. The main thing is
that F is the same function
and omega is a constant called
the eigenvalue and what we want
to do in quantum mechanics often
is given an operator we want
to know what it's eigen
functions are because the result
of a measurement is always
one of the eigenvalues
of the operator and if we don't
know it's eigen functions we
can't calculate its
eigenvalues very easily.
So, to go back to the die
suppose we don't know anything
about it.
We might think, well, we could
get 1 and a 1/2 for an answer
because we don't
know how it's shaped,
we don't know what it is, but
in fact, if we look at it,
it's got numbers,
integers, 1, 2, 3, 4, 5, 6,
those are the only
values you can get
by tossing a die onto a table.
You can't get something
else and just knowing
that those are the
only 6 possibilities
that you can have is
knowing a ton compared
to not knowing anything
or thinking there could be
whatever values you might
dream up.
So, given an operator
what we have
to do we often have the task
of finding all the
possible functions F
and all the possible
values omega
that make the eigenvalue
equation true.
In mathematics, the set
of eigenvalues is called
the eigenvalue spectrum.
That's just for you aficionados.
Let's do a practice problem.
Let's consider the
derivative operator D by DX.
What's the set of
eigen functions
and eigenvalues for
this operator?
Well, we set the operator
to the left of the function,
the function is unknown and
the eigenvalue is unknown.
So we're just going to say
the derivative of F is equal
to Z times F. This
eigenvalue equation then
in this case amounts to solving
a first order differential
equation and my advice is that's
another very good math course
to take so that you
know how to do it.
In this case, it's
fairly easy to do.
We separate variables and we
write DF upon F is equal to ZDX
and then we put integrals on
both sides and then we realize
that Z is a constant, the
eigenvalue, that does not depend
on X and, therefore,
we can move Z
on the outside of the integral.
We can look up the
antiderivatives, use Mathematica
if you've had Chem 5
to do the integrals,
or you can actually put
an equation like that
into integrals.com online
and it'll solve it for you
and you'll find that the
natural log of F is equal
to ZX plus some constant because
this is an indefinite integral.
If we exponentiate both
sides, we find the function F
of X is equal to E to the ZX
plus C we can factor that out
as E to the ZX times E to
the C and then we can let E
to the C be some
constant, K, E to the ZX.
As a check, we can always
substitute our solution
into the original equation
and see that it satisfies it.
In fact, when I was doing
differential equations,
one of the most powerful
methods was guessing.
You guessed the solution
and put it back in
and see if it works out.
That can often be quicker than
to try to do it the forward way.
So, here let's do this.
The derivative of F of X is
the derivative of KE to the ZX
and that's K times the
derivative of E to the ZX.
Even I know how to do the
derivative of Z to the EX,
of E to the ZX rather.
That's KZ E to the ZX and that's
just Z times F. So we've shown
that the operator operating
on F gives a number Z
that doesn't depend on X times F
of X. That's exactly
what we have to do.
So the eigen function
of the derivative operator
is the exponential function
and the eigenvalue can
be any complex number Z
as long as it's a constant.
The reason why when you
solve differential equations
that the exponential
function occurs everywhere
in the solutions of
these equations is
because it's an eigen function
of the derivative operator.
One comment the derivative
operator is linear,
the integration operator
is linear
and by linear I mean
this the derivative
of alpha F plus beta G is
alpha times the derivative
of F plus beta times
the derivative of G.
For any functions A, F and G and
any constants alpha and beta.
However, the derivative operator
is not Hermitian and to go back
to how we were going
to characterize observables
the idea was we had
to have a linear
Hermitian operator.
We know what linear
means we have to figure
out what Hermitian
means and it has
to satisfy this following
relationship.
The integral of F star,
omega hat G is equal
to the complex conjugate of the
integral of G star omega hat F.
So we put a star and we swap
the order of the functions.
If it follows that,
it's Hermitian.
Here F and G are any
reasonable functions
that are integrable
in the usual sense.
Okay, that's quite
a bit for today.
So, we're going to take a
break and come back tomorrow
and we're going to pick it up
on what Hermitian operators are
and cover a couple more
postulates in quantum mechanics
and then do a few
interesting problems.
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