Wave Mechanical Atomic Model

♪ [slow jazz music] ♪

[Prof. Lamb] Well, we have Brian
here with us today. Welcome, Brian.

[Brian] Thank you.

[Prof. Lamb] Brian, do you remember

that when we talked about
the Bohr model of the atom,

we mentioned that there's
a more sophisticated model

that better describes the way
electrons behave in atoms?

[Brian] I think so.

[Prof. Lamb] Well, our objective here today

is to begin learning about this modern
quantum mechanical model of atomic structure.

First, let's recall that Niels Bohr...
[Animated Bohr] Hi.

[Prof. Lamb] ...was able to explain the line
specter of light emitted and absorbed by an atom

with his relatively simple planetary
model of electrons circling the nucleus.

But, do you remember? What was it that
Bohr WASN'T able to explain with his model?

[Brian] Wasn't it why electrons could
adopt certain orbits but not others?

[Prof. Lamb] Yeah, that's exactly right.

Put in other words, he wasn't able to explain

why the energy levels of the electron
are what we call "quantized."

Now, the solution to this mystery
began to reveal itself

when scientists realized that light can be
considered both a particle and a wave.

It was then reasoned that if light can
exhibit both particle and wave behavior,

maybe other things can too.

Things like electrons, for example.

And it was this last logical leap
that led to an understanding

of how electron energy levels
in atoms are quantized.

Would you like to see how?

[Brian] Let's do it.
[Prof. Lamb] Okay.

[Prof. Lamb] To get started, let's see if
we can visualize the electron as a wave.

We're not going to do this in
one single step, but gradually.

Indeed, what you're about to hear

is going to tax your ability to think and
visualize things in three dimensions.

So I'd suggest you get your
cerebral cortex into high gear.

[engine revving]

We'll start out simple and proceed
step-by-step to the full theory.

Let's start with a familiar kind of wave.
Here it is: our old friend, the sine wave.

There are two things about this
wave we need to focus in on.

First is the fact that it can be represented
by a mathematical function, and here it is.

I'm hoping it looks familiar from
one of your basic math classes.

Second is the fact that we can create
a standing wave from this sine wave.

Now, what's a standing wave? Let's see.

Normally, we think of a wave like this
moving horizontally, don't we?

However, if the wave reflects back on itself,

under certain circumstances,
the reflected waves and the incoming waves

may line up to produce a
phenomenon that looks like this.

This is called a standing wave.

You've actually seen
standing waves like this before.

Consider a guitar string.

For a given guitar string length, only certain
wavelengths can produce standing waves.

These are wavelengths that fit neatly
on the string with no string left over.

For example,

a wave whose wavelength is exactly equal
to the length of the string would do the job.

That's the kind of standing wave
we're showing on the string right now.

When we pluck the string,
there are many wavelengths produced,

but the wavelengths that fit along
the string in whole numbers

reflect back and forth off
the two ends of the string

and are reinforced by each other that way.

This produces the standing wave.

Other wavelengths destructively interfere with
each other as they bounce back and forth.

So only the wavelengths which fit in
whole numbers along this string survive,

and that's the note and
its overtones that we hear.

So the string produces one sweet single tone.
[sound of a plucked guitar string]

The same standing wave phenomenon

could be generated if we could somehow bend
the guitar string around on itself like this.

[no audio]

Now notice: The only way
a standing wave will form

is if the path of the wave is the correct length
to accommodate a whole number of wavelengths.

If the circular path is too long or too short,
the wave interferes and cancels itself like this.

When the path isn't just the right length,
the crests and troughs of the wave don't line up.

and we say the wave is destructively
interfering with itself, see?

Now, if we expand the size
of the circle just enough

so that one full wavelength is
added to its circumference,

then once again, a standing wave is formed.

Now, can you see how this approach might help
explain the quantum nature of the Bohr orbits?

I know it's hard to do,

but if we were to think of the electron,
not as a particle, not as a tiny dot,

but as some sort of wave moving
around the circular orbit,

then the electron would
destroy itself in all orbits except

ones where the circumference of the orbit
is a whole number of wavelengths, right?

In other words, the electrons that exist in these
particular orbits but not in orbits in-between.

Isn't that neat? And what you've just heard

is the basic idea of the quantum
mechanical model of the atom.

It doesn't seem to be really
all that hard, does it?

Notice that in our electron standing waves,
there would be nodes;

that is, points exactly between the crests
and troughs where the wave has no amplitude.

Interesting, wouldn't you say?

So what was all the fuss about
getting our brains in high gear?

Well, we're not done yet.

We've got to, now, project this relatively
simple idea into three dimensions.

[to Brian] Now, Brian, let's replace
the familiar form of sine wave

with another that's less familiar.

For those of you who know the terms,

we're going to replace the transverse wave
you just saw with a compression wave.

Here it is.

Now, at first glance, maybe this doesn't
look like a wave to you, but it really is.

The wave isn't in an up-and-down motion

but in the change of color
between yellow and red.

Think of it as a yellow and red wave
moving along a string.

You might think of the yellow
as the wave crests, perhaps,

and the red as the troughs.

This is a truly one-dimensional wave
because everything stays in a line

and the line doesn't bend into a
second dimension to form the wave.

Now, of course, we can bend this string around
on itself, just like the one you saw earlier.

Again, think of the electron
now as being this wave.

If the length of the string--
that is, the circumference of the orbit--

is a whole number of wavelengths,

then the electron will form a standing
wave around the circumference

and won't destroy itself.

Now, just for our purposes here today,

let's think of the electron intensity

being the strongest in this string
in the crests and troughs

(that is, where the color
is pure red or pure yellow).

If that's where the electron's
presence or intensity is strongest,

then Brian, what about the areas
in-between pure yellow and red?

What would those represent, do you suppose?

[Brian] Clearly, the magnitude or
intensity of the redness or yellowness

doesn't instantly drop to
zero outside these points.

It more fades from one color to the other.

[Prof Lamb] That's exactly right.

And so, we don't think of the electron

as having intensity ONLY at those points
on the string (the crests and troughs)

but that the electron intensity gradually
drops to zero at the node and then rises again.

Now remember, we're not thinking of
the electron here as a particle right now.

We're thinking of it as
a wave (a standing wave).

So the electron's standing wave
is smeared out along this string

with greatest intensity at the peaks and
troughs and zero intensity at the nodes.

Now, it's a little hard to see the nodes here.

Brian, where do you suppose the nodes
are in this yellow and red representation?

[Brian] Well probably half-way between

the highest point on the line
and the lowest point.

[Prof. Lamb] Yeah, the points where
the color is the most pure.

Pure yellow, pure red.

Half-way in-between,
there would be a node.

Now let's go to the tricky part, shall we?

We don't live in a one-dimensional world,

so we need to consider

what a three-dimensional
standing wave would look like.

That would represent an electron
in the real world of the atom.

But it's hard to do; it's hard to imagine,
so we need to proceed in steps.

Let's first see if we can imagine a
two-dimensional standing wave, shall we?

This would correspond, not to a wave in a line
(like we saw last) but to a wave in a plane.

Again, we're gonna use a compression wave,

and we can see the crests
and troughs using two colors.

In this two-dimensional system,

we see the waves as rings
of crests and troughs.

The nodes, of course, will also be in the
shape of rings, but they're hard to see.

As you said, they're at the point where
the yellow and red are of equal intensity,

about half-way (in fact, exactly half-way)

between the points where we
see pure yellow and pure red.

[Brian] Okay.

[Prof. Lamb] Now we're ready
to take the last step, this time,

to a three-dimensional standing wave.

Instead of rings in a plane, this system would
tend to look more like shells in a sphere.

You might thing of the waves
like the layers in an onion

or the dolls inside dolls of
a Russian matryoshka doll.

From the outside, this system would
just look like a solid sphere,

but if we cut a cross-section of the sphere,

we would see this beautiful
standing wave-like pattern.

We can then imagine the electron in this form.

The nucleus sits at the center of the sphere
and there are only certain radii

at which the electron wave doesn't
partially or completely destroy itself.

These are the distances form the center

where we see pure yellow
or pure red in our depiction.

You might say that the electron intensity

is highest at the distances
of pure yellow and pure red.

And because of destructive interference,

the electron intensity drops
between these shells until

exactly half-way between pure yellow
and pure red, it drops down to zero.

This latter distance represents a node
where the electron has no intensity at all.

Well, you now should have a
fairly good conceptual picture

of how the modern wave model of the atom works.

As difficult as this is to grasp, however,
it is a much simpler picture than the real thing:

the wave model of the electrons in atoms
developed by a scientist named Schrödinger.

We call his model the Schrödinger model.
(Surprise!)

And it's based on the idea that the
electron can be thought of as a

three-dimensional standing wave.

These waves aren't described in terms
of onions or matryoshka dolls

but in terms of the mathematical equation
that describes the standing wave.

Now, one caveat: Up to this point,

we've just been describing the general
principle of treating an electron as a wave.

We started with a one-dimensional sine wave

then moved it into two dimensions
then three dimensions; but in fact,

the Schrödinger model doesn't actually
use a simple sine wave like this at all.

Instead, it uses waves based on equations
that are much more sophisticated than y=sin x.

Yet despite that complexity, just keep in mind

that these so-called wave equations
or wave functions of Schrödinger

are just really variations on this kind of equation.

Here's the simplest Schrödinger wave function.

As is common in equations,
these wave equations contain many variables

and it turns out that some of these variables

adjust the size and position
of the wave in certain ways.

And here's the key:

For the wave to be a standing wave,
these variables can only have certain values

(just as the length of the circular string

could only have certain values to give
us a one-dimensional standing wave).

Now, these variables are called
the quantum numbers,

and they govern the shape and
the size of the standing wave.

It turns out that one of Schrödinger's standing
waves looks a lot like our simple 3D sine wave.

with a spherical shape.

Such standing waves belong to a class
we call the s-type standing wave form.

Oh, and while we're at it, I guess we'd better
give these kinds of standing wave forms a name.

We'll call them orbitals.

So the s-type orbitals are spherical in shape.

Well, it turns out that there are other shapes

for orbitals or standing wave forms
besides the spherical s-type.

Here's another orbital shape here.
I wonder what we ought to call it.

Brian, what does it look like to you?
[Brian] It kind of looks like a dumbbell to me.

[Prof. Lamb, chuckling] Yes, it does.
[Animated man groans]

[Prof. Lamb] Unfortunately,
someone got there before us though

and decided to call these--
not "d-type" for dumbbell

but "p-type" orbitals.

Now, if you're wondering how you might
imagine a standing wave of this type forming,

think of this orbital like a sphere,
say a balloon. I have a balloon here.

If we were to twist this balloon in the
middle (form a node in the middle),

we'd end up with two lobes
that look like this, wouldn't we?

[Brian] Okay.

[Prof. Lamb] That's what a
p-type orbital looks like.

You can see it from different angles.

Alternatively, you might imagine
that this standing wave

is formed by spinning our friend,
the sine wave, around its axis.

If you had a sine wave like this,
if you spun it around its axis,

it would generate in space
these two lobes, wouldn't it?

[Brian] Oh, yeah.

[video is paused during emergency
alert on the bottom of the screen]

[Prof. Lamb] Now, about p-orbitals,

it's interesting that they always
come in sets of three, like blind mice.

[Brian chuckles]

[Prof. Lamb] The three look the same

except that they differ in
their orientation in space.

One is lined up along the x-axis,

one along the y-axis,
and one along the z-axis.

[no audio]

Okay, it turns out that there
are also d-type orbitals.

They are the next type,

and they come, not in sets
of three but in sets of five.

Here they are. Kinda cute, huh?
[Brian chuckles] Sure.

[no audio]

[Prof. Lamb] Okay, now let's take a minute

to talk about these quantum numbers
that show up in the wave functions

(that is, these equations that define
the shape of the standing wave).

The most important quantum number
is called "n," and it's easy to envision

what that quantum number
stands for or corresponds to.

It's called the principal quantum number
and it corresponds to Bohr's orbit numbers:

those numbers that we're already
familiar with from our energy well.

It turns out that not all wave forms or
orbitals are allowed in all Bohr orbits.

Or, in other words, only certain orbitals
are allowed for a given value of n.

Fortunately, the orbitals that ARE allowed

fall into a neat little pattern
hat's easy to remember.

Let's take a look here.

For n=1, you'll see that only the simplest
orbital is allowed. That's the s-type orbital.

[Brian] Oh, okay.

[Prof. Lamb] For n=2, two types
are allowed: the s and the p type.

And since the p-type come in sets of three
we show it that way here on the diagram.

Now, for n=3, we add the d-type,
so we can have 3 s, 3 p, and 3 d.

Brian, I bet you can't guess
what types are allowed for n=4.

What do you think?

[Brian] I don't know. I'd have to say
s, p, d, and-- Isn't there one other one?

[Prof. Lamb] Oh, yeah. Okay.

There's got to be another one,

and it turns out that it's
called the f-type orbitals.

[Brian] Oh, okay.

[Prof. Lamb] They come in sets of seven.

Okay, now we've seen what the orbitals look like

and how many are allowed in each
principal quantum number or Bohr orbit.

One last thought before we go:

For a 1-electron atom
(hydrogen, for example),

the energies of all the orbitals
with a given n value

(that is, with the same n value) are the same.

They have the same energy.

And you see, that's how we've
drawn them here on the diagram,

but here's the important point.

It turns out that they're NOT the
same energy in multi-electron atoms,

and that's very important
to the chemistry of those atoms.

But that's an important topic for another day.

[Brian] Great, I'm excited. [chuckles]
[Prof. Lamb] Good.

♪ [synthesizer jazz music] ♪

END