But why is it this way? What is going on?
(You have a ..) Let's go back to hydrogen, a proton
and electron going around it.
(What is) What is waving inside the
electron? It is not an electromagnetic wave,
right?
A photon, we said, when we have a wave
packet like this
the thing that's going up and down can
be thought of as the electric field and
magnetic field
at that point in space. But what about it
a
(in a uh) in a hydrogen atom what is the
electron
doing? (What is wave) What is it that that
wave function, Psi, represents?
It turns out, it's actually a wave of
probability. That, um,
it's the probability of finding the
electron at that point in space
at that moment. okay? Or, to be precise,
"probability density", because
the probability of finding it at any particular point goes to zero. But
(uh, so you) how much probability per unit
volume are you talking about, (is what) is
what the wavefunction corresponds to.
That's a kind of a bizarre thing.
What does that mean, probability? Where is it? It's either somewhere or its isnt' somewhere,
right?
And that's how we're used to thinking
about (uh)
real ... real things (in real physics) should,
you know, have an
actual position and a location and a
momentum and a velocity. It's all like
you know billiards, right? Well, in quantum
mechanics,
not so. All we can talk about is that,
well,
for that wave packet its kind of there and
it kind of has that
that momentum and it's traveling at a certain velocity or whatever
but where is the particle? Well, it's sort of
spread out, it's smeared out.
We've smeared out it's position
and smeared out its momentum in a way
that is wave-like, it has a wave nature.
It still has kind of a location and
kind of
a a well-defined wavelength, but it's
not perfectly known.
Well, to get into the mathematics, that
(uh) wave function is actually (described by complex n..) best describe complex
numbers with
with real and imaginary (you know, square root of -1) components.
But that's, you know, well beyond the scope of this course. Now it turns out in quantum
mechanics
there are observables like position, and
velocity, momentum, and so forth.
But if you take pairs of these
observables, they are connected by
the Heisenberg uncertainty principle in
the following way. Um,
too offen, the Heisenberg uncertainty
principle and "quantum"
this and "quantum" that are mischaracterized as meaning "Oh, you can't be
certain about anything" or or
"nothings certain". Not true. Not true. All it is saying,
all the Heisenberg uncertainty principle
says that you cannot
simultaneously measure the position and
momentum
of a particle with joint precision better
than
h-bar (Planck's constant divided by 2 pi)
divided by 2. For deep reasons that
fall out of the mathematics and
Heisenberg was the first one to
enunciate this. (So what is) What does this mean?
Delta x might be the precision with
which you can say
"that particle is at point x" so it's at x
plus or minus a little bit
and it has a momentum p plus or minus a
little bit.
Now it doesn't say that you can't measure
it really really precise - as precisely as
you want. You can make Delta x as small
as you what,
but there's a cost to pay and the cost is
that
Delta p then becomes large. So, in other
words, if you measure, really precisely,
the position
of a particle, you lose information
about its momentum.
Likewise, if you measure its momentum really precisely,
you can't tell where it is in space.
(you) So this product is maintained. Now
(um) like I say, I think Heisenberg's
uncertainty principle gets
a little bit abused into extrapolating
that "Oh, quantum mechanics tells us that
we can say anything (for certain) for
certain anymore." This, it's
really not like that. That's more
philosophy and, you know, sort of
"pop" philosophy rather than real philosophy.
And really it's a statement about
measurement and what measurements you can do.
It is kind of, it's it's a sensible thing
in the following way, that
if you're going to do a measurement (of a)
of the position of a particle you're
going to completely destroy any
information about
its momentum and vice versa.
That kind of makes sense. There's no
measurement that you can do that doesn't
change something. Even something as
simple as, let's say, taking
a temperatures. Where you have a cup of water
(it's at) it's hot and you want to measure its
temperature. What do you do? You take a thermometer and you put it in there, right?
And then the thermometer tells you what
it is, 67 degrees centigrade.
Is that the temperature of the water
before you put the thermometer in? No.
Why? Because a little bit of the heat
from the water went into the thermometer
to measure it. So you've disturbed the
system. That's a,
maybe, a trivial example, but when you
actually get into
measuring the position and momentum
of a
an electron (even in a an atom or or in
space)
you have to disturb it in some way to
measure its position. You have to make it
hit something
(or uh, you know, you you ultimately
destroy it). Now that's more of a deep
thing and that's
(uh, thats that's the
philos) that kind of philosophy is on the right track. Because it's telling you how to
interpret the
Heisenberg uncertainty principle (in a) in
a meaningful way
that connects with how we actually do
measurements in the real physical world.
The thing is
h (this value h) is so small that you almost never run up
against these sorts of effects in
in your everyday experience. Now we we
can demonstrate quantum phenomena
on macroscopic scales. We did so in the
room here. We are able to take (that)
these gas discharge tubes, look at the
spectrum of radiation coming out from
them, and see that there's very discreet (very
well-defined)
uh allowed energy levels within these
atoms
(for whatever the reasons happen to be). So, we can see quantum effects
on macroscopic scale, but if you tried
to really
measure very very precisely those spectral lines, you would find that
they actually have a little bit of a width
to them. That width corresponds to
uh, you know, an uncertainty in the
momentum
of of the electrons in their
orbits around
atoms. So you can't probe it any better
than
than that. Now um
one of the interesting questions that we
didn't (add..) address so far is:
Okay, so you have this hydrogen
(right?), it's got a proton, there's electron
going around, (why
is this, um) why is the electron
(which is attracted to the proton) why
doesn't it just radiate away its energy and
fall completely into the nucleus,
into the proton? Why doesn't that happen? Why do we have
stable hydrogen atoms at all? It actually,
you know, one explanation of that is that the uncertainty principle
does not allow that. If you actually calculate the uncertainty, if you confine an
electron to the size of a proton than
that energy
is is BIGGER than the binding energy of
the electron. So it can't do that.
That's one explanation, is the Heisenberg
uncertainty thing.
You cannot confine an electron to such a
small
place without giving it so much energy
that it leaves the atom altogether.
Well that's an interesting thing, that
the uncertainty in its momentum is
bigger than the binding energy itself.
A particle physics explanation is
actually a better one.
If you take an electron and collide it
with a proton
you can actually make a neutron. A
neutron is slightly
larger mass than the the proton, as it
turns out though,
and decays right back down to to the
proton and electron.
So uh energetically
(it can't be) it would require more
energy to shoot the
electron into the um proton
than than we have (in the) available to us.
The kinetic energy of the electron is not
enough to
combined with the proton to make a
neutron, which is a couple million
electron volts
greater in mass, as it turns out. So
that's a
particle physics explanation. But the, I
think
you know, more appealing explanation is
Heisenberg uncertainty. That's why we
have
matter at all.
