Today, we're talking about
the wave-particle duality,
and what led to this is what we
talked about in a previous video,
which is that classical theory
broke down when trying to explain
energy and matter and 
how they were so different.
So, instead, what we now have,
is the idea that waves and particles
are not distinct things,
but that there is a continuum 
of waves and particles
so that energy (which was previously 
thought of as a wave)
and light (which is also a wave)
have particle qualities.
And we talked about the fact that we 
now call particles of light "photons."
And if that is true,
there was a guy named deBroglie 
who came along and said,
"Well, if that's true, 
then the opposite must be true,
"in that matter must have 
wave-like qualities.
"If energy behaves like a particle, then
matter must also behave like a wave."
So he came up with this equation,
which is basically a result of combining
E equals mc squared (E = mc2),
which is the classic equation 
relating energy and matter,
and the fact that E equals H new --
which we talked about,
that energy equals 
a constant H times frequency (E = h v),
and the equation that relates
the speed of light to frequency 
and wavelength (c = v * λ)
and he combined them and came up --
and he changed the speed of light
to just a v for velocity,
because he was talking about matter,
and the speed of matter 
will no longer be the speed of light,
but it's just going to be a speed.
So he came up with this equation,
which is known as 
the deBroglie equation,
which says that the wavelength of light
is equal to a constant,
which is Planck's constant (h),
divided by the mass of the object
times the velocity of the object.
So "h" is known as Planck's constant
and is equal to
6.636 x 10 to the minus 34
kg meters squared per second --
(6.636 x 10 -34th) kg m2/s.
So that is Planck's constant, h.
Anyway, it gives us 
the deBroglie equation,
which then allows us to calculate
the wavelength (lambda, λ)
of any object that is moving --
that has a mass (m) and a velocity (v).
So all moving matter has a wavelength.
So why don't we notice this?
Why don't we notice there being 
a wave-like quality -- oops -- to matter?
And the reason is that if you use
this equation for ordinary objects,
you'll find that it's immeasurable.
So let's say you throw a ball.
You'll calculate the wavelength 
of that ball moving through space
as 10 to the minus 34 meters 
(10 -34th m) for a wavelength,
and that's just way too tiny
for us to even see or measure.
So we never notice that ordinary objects
are actually smeared out a little bit
and they have wave-like qualities.
However, it turns out that electrons
are small enough and fast enough
that we do see that they
have wave-like qualities.
Electrons are very small,
and they can move very fast,
and they do show diffraction patterns.
Remember, we said diffraction was
a quality usually assigned to waves.
So electrons display diffraction patterns,
and that is measurable.
So now we realize that electrons
are really smeared out.
They're not in a particular location,
but they have wave-like qualities,
which means they are smeared out.
The fact that electrons are smeared out
and don't have a precise location
directly led to the
Heisenberg uncertainty principle.
This is the equation for 
the Heisenberg uncertainty principle,
and let me tell you about the parts.
First of all, Delta (Δ) in this case
does not mean "change in";
it means "uncertainty in."
So it's a slightly different 
meaning of Delta (Δ).
And "x" is the position of something
(whatever object you are talking about),
and m times v (mv) is momentum.
So there's an inequality here
greater than or equal to (≥).
So the uncertainty in the position (Δx)
times the uncertainty 
of the momentum of an object (Δmv)
is greater than or equal to (≥)
Planck's constant (h) 
divided by 4 pi (4π).
(Δx) (Δmv) ≥ h/4π
So what that means is that
because it's an inequality,
we can't know the position 
and the momentum of an object
at the same time
(at least not precisely).
So if the uncertainty in position 
is great,
then the uncertainty in the momentum 
might be smaller,
but you can't know both 
at the same time.
And this plays out in electrons 
in atoms where you might know
the momentum of the electron 
but you won't know its position exactly.
It's smeared out.
I have a problem for you to try now.
Which of the following would have
the greatest deBroglie wavelength
if they had the same velocity?
So stop the video and think about that.
All right, we have a proton, 
a baseball, and an electron.
What is the equation 
for the deBroglie wavelength?
It is lambda (λ) 
equals Planck's constant (h)
over the mass (m) times the velocity (v).
λ = h/mv
So we look at these three and we see
that the mass is inversely proportional
to wavelength,
so the thing that will have
the greatest wavelength
will be the thing with the smallest mass,
and that would be an electron.
Next, we will deal with the 
quantum mechanical model of the atom.
I've shown an equation here that I 
don't expect you to understand in detail,
but we do want to explain the 
quantum mechanical model at this point,
since we said the classical theory
breaks down for the atom,
so we need to use the
quantum mechanical model.
And this is known as 
the Schrödinger equation here,
and it's a very simple form.
This H with the hat (⌃) on it
is known as the Hamiltonian operator,
and it actually is a series
of very complicated equations.
It's not simple at all.
And E -- this value E -- 
are the energy levels of the atom,
which are discrete.
And then you have psi (Ψ).
Ψ is given down here.
Ψ is known as an orbital.
It is also known as a wave function.
And when we solve 
the Schrödinger equation,
we get the wave functions --
which are essentially 
where the electrons are --
and we get the energy levels (the Es).
And then wave function 
(as I just sort of said)
tells the position of
the electrons in the atom,
but it doesn't have a very good
physical description,
so we usually talk about 
psi squared (Ψ2),
and this is the Greek letter, psi (Ψ),
which is the probability of 
finding an electron in a certain location.
And if we do solve this equation --
the Schrödinger equation --
for one-electron systems, 
like the hydrogen atom,
we do get the Rydberg equation,
which gives us information
about the hydrogen atom.
But for all other systems
with more than one electron,
this equation is 
much, much more complicated.
sro
