♪ [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] ♪
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